Add files using upload-large-folder tool

.gitattributes

@@ -3987,3 +3987,74 @@ q9AzT4oBgHgl3EQfA_op/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -tex
 NNFRT4oBgHgl3EQfGTcp/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
 UtAzT4oBgHgl3EQfX_xg/content/2301.01327v1.pdf filter=lfs diff=lfs merge=lfs -text
 69AzT4oBgHgl3EQfEvpR/content/2301.00998v1.pdf filter=lfs diff=lfs merge=lfs -text
+3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf filter=lfs diff=lfs merge=lfs -text
+GNE0T4oBgHgl3EQfhQF8/content/2301.02429v1.pdf filter=lfs diff=lfs merge=lfs -text
+DdAzT4oBgHgl3EQfwf7y/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+CtE0T4oBgHgl3EQfQQAs/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+UtAzT4oBgHgl3EQfX_xg/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+mNE_T4oBgHgl3EQf6xyS/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+NtE3T4oBgHgl3EQfwwva/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+rdE3T4oBgHgl3EQfMgkz/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+vNAyT4oBgHgl3EQfafc7/content/2301.00242v1.pdf filter=lfs diff=lfs merge=lfs -text
+qdE2T4oBgHgl3EQf0gjz/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf filter=lfs diff=lfs merge=lfs -text
+RdE3T4oBgHgl3EQfygvM/content/2301.04721v1.pdf filter=lfs diff=lfs merge=lfs -text
+DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf filter=lfs diff=lfs merge=lfs -text
+jdE1T4oBgHgl3EQfNQNY/content/2301.02999v1.pdf filter=lfs diff=lfs merge=lfs -text
+CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf filter=lfs diff=lfs merge=lfs -text
+3NFST4oBgHgl3EQfYjhQ/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf filter=lfs diff=lfs merge=lfs -text
+WNE2T4oBgHgl3EQfDgYF/content/2301.03624v1.pdf filter=lfs diff=lfs merge=lfs -text
+69AzT4oBgHgl3EQfEvpR/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+hNE3T4oBgHgl3EQf4AvA/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+vNAyT4oBgHgl3EQfafc7/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+3NAyT4oBgHgl3EQfo_gV/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+wdE3T4oBgHgl3EQflQpC/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+cNE0T4oBgHgl3EQf4wL5/content/2301.02744v1.pdf filter=lfs diff=lfs merge=lfs -text
+3tAyT4oBgHgl3EQfb_f6/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf filter=lfs diff=lfs merge=lfs -text
+wdE3T4oBgHgl3EQflQpC/content/2301.04604v1.pdf filter=lfs diff=lfs merge=lfs -text
+ydE2T4oBgHgl3EQf3wjf/content/2301.04175v1.pdf filter=lfs diff=lfs merge=lfs -text
+stE_T4oBgHgl3EQf8xx6/content/2301.08377v1.pdf filter=lfs diff=lfs merge=lfs -text
+zdAyT4oBgHgl3EQfbPea/content/2301.00259v1.pdf filter=lfs diff=lfs merge=lfs -text
+GNAyT4oBgHgl3EQfSvfA/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+6NE4T4oBgHgl3EQf1g37/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ptAyT4oBgHgl3EQfzflz/content/2301.00702v1.pdf filter=lfs diff=lfs merge=lfs -text
+89E0T4oBgHgl3EQfwgGc/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ntE2T4oBgHgl3EQfJwYe/content/2301.03694v1.pdf filter=lfs diff=lfs merge=lfs -text
+JdAyT4oBgHgl3EQfsPlo/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+a9E4T4oBgHgl3EQfoQ0b/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+bNAyT4oBgHgl3EQfivjL/content/2301.00403v1.pdf filter=lfs diff=lfs merge=lfs -text
+jdA0T4oBgHgl3EQfIv9Y/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+aNE3T4oBgHgl3EQfdAo0/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+zdAyT4oBgHgl3EQfbPea/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ZtAyT4oBgHgl3EQfvvnv/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+cNE0T4oBgHgl3EQf4wL5/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+RdA0T4oBgHgl3EQfDv9U/content/2301.02007v1.pdf filter=lfs diff=lfs merge=lfs -text
+mtFPT4oBgHgl3EQf4zVh/content/2301.13194v1.pdf filter=lfs diff=lfs merge=lfs -text
+a9FLT4oBgHgl3EQfXi8l/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ZtAyT4oBgHgl3EQfvvnv/content/2301.00638v1.pdf filter=lfs diff=lfs merge=lfs -text
+HNA0T4oBgHgl3EQfBv_D/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+b9FPT4oBgHgl3EQfBTTg/content/2301.12985v1.pdf filter=lfs diff=lfs merge=lfs -text
+T9E4T4oBgHgl3EQfmg0h/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+ntE2T4oBgHgl3EQfJwYe/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+HNA0T4oBgHgl3EQfBv_D/content/2301.01981v1.pdf filter=lfs diff=lfs merge=lfs -text
+VtAyT4oBgHgl3EQfV_ek/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf filter=lfs diff=lfs merge=lfs -text
+ldAyT4oBgHgl3EQfk_iC/content/2301.00444v1.pdf filter=lfs diff=lfs merge=lfs -text
+jdA0T4oBgHgl3EQfIv9Y/content/2301.02079v1.pdf filter=lfs diff=lfs merge=lfs -text
+T9E4T4oBgHgl3EQfmg0h/content/2301.05168v1.pdf filter=lfs diff=lfs merge=lfs -text
+vdE4T4oBgHgl3EQfXAw5/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+5NFIT4oBgHgl3EQf7itm/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+c9FIT4oBgHgl3EQfnivD/content/2301.11315v1.pdf filter=lfs diff=lfs merge=lfs -text
+XNAyT4oBgHgl3EQfh_h5/content/2301.00387v1.pdf filter=lfs diff=lfs merge=lfs -text
+3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf filter=lfs diff=lfs merge=lfs -text
+c9FIT4oBgHgl3EQfnivD/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+bdFST4oBgHgl3EQfCjh5/content/2301.13707v1.pdf filter=lfs diff=lfs merge=lfs -text
+xdE0T4oBgHgl3EQf-gIr/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+stFKT4oBgHgl3EQf1y5_/content/2301.11921v1.pdf filter=lfs diff=lfs merge=lfs -text
+XNAyT4oBgHgl3EQfh_h5/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+stFKT4oBgHgl3EQf1y5_/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text
+NtE3T4oBgHgl3EQfwwva/content/2301.04706v1.pdf filter=lfs diff=lfs merge=lfs -text
+a9E4T4oBgHgl3EQfoQ0b/content/2301.05182v1.pdf filter=lfs diff=lfs merge=lfs -text
+9tFJT4oBgHgl3EQfoyxM/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text

0dFQT4oBgHgl3EQf0Daw/content/tmp_files/2301.13415v1.pdf.txt

@@ -0,0 +1,1215 @@
+LOGAI: A LIBRARY FOR LOG ANALYTICS AND INTELLIGENCE
+Qian Cheng, Amrita Saha, Wenzhuo Yang, Chenghao Liu, Doyen Sahoo, Steven Hoi
+Salesforce AI Research
+{qcheng, amrita.saha, wenzhuo.yang, chenghao.liu, dsahoo, shoi}@salesforce.com
+ABSTRACT
+Software and System logs record runtime information about processes executing within a system.
+These logs have become the most critical and ubiquitous forms of observability data that help
+developers understand system behavior, monitor system health and resolve issues. However, the
+volume of logs generated can be humongous (of the order of petabytes per day) especially for complex
+distributed systems, such as cloud, search engine, social media, etc. This has propelled a lot of
+research on developing AI-based log based analytics and intelligence solutions that can process
+huge volume of raw logs and generate insights. In order to enable users to perform multiple types
+of AI-based log analysis tasks in a uniform manner, we introduce LogAI (https://github.com/
+salesforce/logai), a one-stop open source library for log analytics and intelligence. LogAI
+supports tasks such as log summarization, log clustering and log anomaly detection. It adopts the
+OpenTelemetry data model, to enable compatibility with different log management platforms. LogAI
+provides a unified model interface and provides popular time-series, statistical learning and deep
+learning models. Alongside this, LogAI also provides an out-of-the-box GUI for users to conduct
+interactive analysis. With LogAI, we can also easily benchmark popular deep learning algorithms for
+log anomaly detection without putting in redundant effort to process the logs. We have opensourced
+LogAI to cater to a wide range of applications benefiting both academic research and industrial
+prototyping.
+Keywords Log Analysis · Machine Learning · Anomaly Detection · Clustering · Artifical Intelligence · AIOps
+1
+Introduction
+System and Software logs are text messages that are embedded by software and application developers in the source
+code and are designed to carry useful runtime information about the process, which are typically dumped as raw log
+files, once the system starts executing. In modern computer systems, especially for large distributed systems that run
+complex software, such as search engines, social network websites, and cloud platforms, logs are one of the most
+critical observability data. Logs are widely used in a variety of operational tasks, covering use cases such as system
+availability, reliability and security. In scenarios when users have no direct access to the physical servers, logs are often
+the ground truth about the systems and applications. As such, Log Management has become a very important task in
+the industrial landscape. In fact, log management market size grew to $2.29 billion in 2023, at a compound annual
+growth rate (CAGR) of 15.9%, according to the report from The Business [1].
+Ideally, logs should be capturing the runtime information at a very granular level and stored permanently so that
+when any disruptive incident occurs, developers and operators can always look up the correct log file and inspect the
+log messages to debug what caused the incident. In reality though, because of the colossal size of the log dumps,
+storing them permanently in the raw form is often impractical. This challenge can be mitigated with the help of large
+cloud-based logging systems such as AWS Cloudwatch and Microsoft Azure Logs where it is possible to even store
+the entire log data and retain them for a substantial period of time. Moreover, these logging systems also provide
+capabilities to help efficient log querying and visualization, enabling developers and operators to quickly access the log
+dumps or log streams of their software. With these capabilities, the main open question is, how to explore raw logs and
+find the right set of logs associated with an incident? followed by a more advanced one - Is there a way to automatically
+analyze the logs and tell if there are issues with a system, create incidents and provide additional insights?
+arXiv:2301.13415v1  [cs.AI]  31 Jan 2023
+
+Cheng et. al
+Depending on which operational stage logs are involved in, the goal of log analysis in that specific situation could be
+different. Logs can be used for incident detection, where reliability engineers and developers need to continuously
+monitor the log streams in order to detect any unexpected behavior that might be indicative of an incident. For post
+incident detection, log data can play a critical role in root-cause analysis, where operators examine the raw logs to
+identify the loglines that show anomalous patterns and thus localize the anomaly and eventually the root cause of the
+incident to a single service, component or module or a group of them. The situation becomes even more complex in
+large distributed systems, where people (typically reliability engineers) who inspect the logs to resolve incidents may
+not necessarily be the same group of people (i.e. software and application developers) who write the logging statements
+in software code. In these situations, understanding even simple dump logs can take significant amount of time and
+effort, owing to the open-ended nature of the log data.
+Over the past decade there have been various effort targeted at developing both commercial and open-source software
+to cater to automated log analysis. Though, most of the initial work used either domain specific rules or heuristics,
+with the proliferation of AI and ML, more and more data-driven techniques have been adopted and popularized in this
+community. However, most of the AI-driven effort has been applied in an isolated manner, focusing on specific log
+analysis tasks (like how to extract structure out of the raw logs or how to detect anomaly patterns in it). There is still an
+urgent need for bringing together all the AI, ML and NLP techniques to a unified platform that can cater to the entire
+suite of different log analysis tasks. Nevertheless, creating such a one-stop library to serve a diverse set of log-based
+analytics can be quite non-trivial, with some of the potential challenges being, as follows:
+• Lack of unified log data model for log analysis. Different logs are in different formats and as a result
+analysis tools need to be customized for different log formats and schemas. It is not easy to generalize
+analytical algorithms without a unified data model that can handle heterogenous forms of log data.
+• Redundant effort in data preprocessing and information extraction. The current status of log analytics
+in this community is that there is a lack of a consolidated pipeline for data preprocessing and information
+extraction across all log analysis models and tasks - i.e. different log analysis algorithms have been implemented
+independently, with each adopting their own pipelines and workflows. For different tasks, or even different
+algorithms of the same task, developers need to implement multiple redundant preprocessing and information
+extraction process modules.
+• Difficulty in managing log analysis experiments and benchmarking. Empirical benchmarking forms a
+critical part of research and applied science. In the existing literature, there is no unified workflow management
+mechanism to run log analysis benchmarking experiments. For example, while there has been some isolated
+pockets of deep learning research for log anomaly detection, it is quite challenging for other organizations or
+users to adopt them or reproduce their experimental results, due to the lack of a common unified framework
+for log analysis.
+In this inter-disciplinary community of AIOps, users may have different needs while working on log analysis in
+academic and industrial settings when they are in different roles. For example, 1) Machine learning researchers may
+need a hassle-free way to perform benchmarking experiments on public log datasets and reproduce the experimental
+results from peer research groups in order to develop new log analysis algorithms; 2) Industrial data scientists and
+AIOps practitioners may need an intuitive workflow to quickly experiment with existing log analysis algorithms on
+their own log data and select the best performing algorithm, hyperparameters and experimental configurations as their
+log analysis solution, and 3) Data and software engineers need to integrate the selected algorithm into production and
+deploy them in a smooth and efficient way. Unfortunately, we realize there is no existing open source toolkit that can
+satisfy all the above needs.
+We are thus motivated to develop a holistic LogAI solution - a python library aimed for conducting AI-based log
+analytics and intelligence tasks to serve a variety of academic and industrial use-cases. LogAI (https://github.
+com/salesforce/logai) provides a unified way to conduct various of log analysis tasks such as log summarization,
+clustering, anomaly detection. LogAI also provides a unified data model, inheriting from OpenTelemetry log data
+model, to handle logs in different formats. LogAI is also the first open source log analytics library that incorporate
+time-series algorithms, statistical learning algorithms and deep learning algorithms. Moreover, LogAI implemented an
+out-of-the-box GUI portal to conduct log analysis in interactive way, more naturally align with the user experience of
+real-world log analysis.
+Besides, in this technical report we also demonstrate how to use LogAI to easily benchmark deep learning algorithms
+for log anomaly detection without any redundant effort in log preprocessing and cleaning. In this community, there are
+existing libraries like LogLizer and Deep-Loglizer [2, 3] which have consolidated some of the AI/ML effort for the log
+domain. However, they still suffer from a few limitations - for example lacking a unified data processing pipeline that is
+generic across all tasks or algorithms or catering to only anomaly detection as the log analysis task or covering only a
+2
+
+Cheng et. al
+specific types of algorithms. In Section 5, we elaborate on the limitations of these existing libraries and also show how
+LogAI provides a more intuitive framework for designing and managing the experimental settings while performing
+comparable to Deep-Loglizer.
+2
+Related Work
+Recently, researchers and engineers have been working on a variety of problems about automated log analysis in
+academia and industry [4]. Based on the existing solutions, we can summarize a common workflow to conduct
+automated log analysis. The common workflow contains four steps: log collection, log cleaning and preprocessing, log
+information extraction and log analysis and intelligence applications, Figure 1. Log collection is the data loading step
+that collects logs from local log dump files or log management platforms. Log cleaning and preprocessing is the step
+to use predefined rules and domain knowledge to clean noisy log data, remove or replace known log templates. This
+step usually does not involve any ML process. Log information extraction is the step where ML models are involved
+to extract information from log data, and feed the log representation or features to train ML models for analytics and
+intelligence application tasks. Log information extraction usually contains several steps like log parsing, partitioning,
+feature extraction, etc. The final step, log analytics and intelligence, is to train ML models for a specific log downstream
+task. For example, log clustering and summarization are common log analytics tasks, while log based anomaly detection
+and root-cause analysis are common log intelligence tasks.
+Figure 1: Common Log Analytics and Intelligence Workflow. The common workflow contains four steps: 1) log
+collection from local log files or log platforms, 2) log cleaning and preprocessing, 3) log information extraction and 4)
+log analytics tasks (such as clustering and summarization) and log intelligence tasks (such as anomaly detection and
+root-cause analysis).
+Log analysis has a very long history and there are a lot of tools for log analysis. Almost all commercial log management
+software/SaaS have associated log analysis/ log insights offerings. This includes log management products such as
+Splunk, DataDog, NewRelic, etc., as well as cloud providers such as Amazon AWS, Microsoft Azure and Google
+Cloud. In open source community, there are also very popular log management and analysis projects such as GreyLogs,
+Grafana, Prometheus, etc. However, neither these commercial log management platform nor open-source log analysis
+tools are incorporated with comprehensive AI techniques such as deep learning, large language models (LLM), BERT,
+etc.
+Meanwhile, there are a few open-source AI-based log analysis tools that started to support more comprehensive AI
+techniques. For example, LOGPAI (https://github.com/logpai/) is one of the most famous log anaysis community on
+GitHub. LOGPAI provides logparser for automated log parsing. LOGPAI also provides loglizer [5] and deep-loglizer [6]
+for log anomaly detection. Besides LOGPAI, there are other open-source projects, most of which are open source code
+from research outcomes, such as LogClass and Log2Vec from NetManAIOps (https://github.com/orgs/NetManAIOps).
+3
+Design Principles
+In this section we discuss about the design principles of LogAI library. LogAI provides a unified framework for log
+analysis. In order to achieve this, LogAI follows the following design principles: 1) high compatibility with data from
+different log sources, 2) reusable components to avoid reproducing effort, 3) unified setup process for customized
+applications and 4) easy-to-use GUI for out-of-box interactive log analysis.
+3
+
+Log Information
+Log Analytics and
+Log Collection
+Log Cleaning and
+Extraction
+Intelligence
+Preprocessing
+: From local files
+: Cleaning Noisy Data
+• Log Parsing
+: Analytics:
+: From log platforms
+: Log Partitioning
+。 Clustering
+: Remove or Replace
+Custom Log
+. Feature Extraction
+ Summarization
+Templates
+: Intelligence
+: Anomaly Detection
+• Rootcause AnalysisCheng et. al
+3.1
+Compatible with data from different log sources
+One of the attractive qualities of log data is its open-ended form, where developers can design them to capture useful
+runtime and performance information to any arbitrary level of granularity as per the needs of the application. Different
+software can generate very different logs. Even in the same software, there are different levels of logs, such as service
+logs, application logs, systems logs, etc. These logs can be in different formats, either structured, semi-structured or
+unstructured. LogAI takes these factors into consideration and ensures that the data loader can consume and process
+these heterogeneous types of logs in a seamless way, by converting these logs into log record with unified log data
+model.
+3.2
+Reusable components to avoid duplicated effort
+As briefly motivated in Sec 1, a particular challenge of building log analytics in both academic and industrial settings, is
+the lack of an unified framework that allows reusal of data processing and information extraction components across
+different log analysis tasks, even on the same data source or dataset. For instance, engineers and researchers have to
+build separate pipelines to perform log anomaly detection, log clustering or summarization even to deal with the same
+log data source. This burden significantly impacts efficiency in every development stage. from experiments, prototyping
+all the way to productization. Also running multiple pipelines in production increases the system complexity and brings
+additional operational cost. Thus, building a library that unifies the interface of common components across multiple
+downstream tasks is necessary to improve efficiency of all stages of log analysis.
+3.3
+Unified setup process for customized applications
+Even for the same application, the design choice behind the log analysis pipeline might have different variations, based
+on the various needs or limitations of the use-case. For example, log anomaly detection may involve different steps in
+the end-to-end (E2E) workflow. Some may include log parsing, while others might choose to skip this step either due
+to the computational overhead or simply because the downstream analysis models do not need a well-defined parsed
+structure. Also, when converting the raw log text data to machine-readable vectors there can be various choices - either
+to convert log messages into time-series counter vectors or into event sequences by representing each log line as a id
+or as a sequence of natural language tokens. In production setup, adding, removing or replacing a component in the
+E2E workflow could be very time consuming. LogAI is designed to support building customized applications with
+easy plug-in / plug-out components, enabling users to quickly try out various combinations through simple intuitive
+mechanisms like configurable json or yaml files.
+3.4
+Easy-to-use GUI for out-of-box interactive log analysis
+Another learning while we work with different types of log data is about visual examination. Unlike many machine
+learning domains where the model performance evaluation can heavily rely on metrics, such as Precision, Recall,
+F-scores, log analysis tasks usually need more visual examination to validate the performance. Thus, LogAI is developed
+with a graphic user interface (GUI), or a portal, to integrate with interactive analytical features for tasks such as log
+summarization, clustering and anomaly detection. We believe this portal can reduce the cognitive overhead on the
+LogAI users in onboarding to the library and help them execute the log analysis tasks quickly and intuitively.
+4
+Architecture
+LogAI is separated into the GUI module and core library module. The GUI module contains the implementation of a GUI
+portal that talks to backend analysis applications. The portal is supported using Plotly Dash (https://plotly.com/dash/).
+The core library module contains four main layers: data layer, pre-processing layer, information extraction layer and
+analysis layer. Each layer contains the components to process logs in a standard way. LogAI applications, such as log
+summarization, log clustering, unsupervised log anomaly detection, are created on top of the components of the four
+layers.
+4.1
+Core Library Modules
+LogAI is implemented in the architecture described in Figure 2. In this section we describe the technical details of each
+layer. Including the implementation of components and how the components communicate across layers.
+4
+
+Cheng et. al
+Figure 2: LogAI Architecture
+4.1.1
+Data Layer
+Data layer contains two component classes: LogRecordObject class and DataLoader class.
+LogRecordObject class defines the data model of log records. As we mentioned in Introduction, logs are free-form
+text and can be unstructured or semi-structured. Even for structured logs, different software applications may name their
+log data in different ways. LogRecordObject is to adapt log data from different sources to a more unified structure in
+order to provide a data object that can be used in all follow-up processes without modification. In LogAI, data model
+of LogRecordObject is a subset of the log and event record definition by OpenTelemetry (https://opentelemetry.io/),
+containing fields in Table 1.
+Table 1: LogRecordObject Data Model
+Field
+Description
+Timestamp
+Timestamp when event occurred.
+Body
+loglines or the content of log messages.
+Attributes
+a map<key,value> for structured information of log record.
+TraceId
+Request trace id as defined in W3C Trace Context. Can be set for logs that are part of
+request processing and have an assigned trace id. This field is optional.
+SpanId
+Trace flag as defined in W3C Trace Context specification. At the time of writing the
+specification defines one flag - the SAMPLED flag. This field is optional.
+SeverityText
+String represents the severity. This field is optional.
+SeverityNumber
+Numeric values of severity, TRACE(1-4), DEBUG(5-8), INFO(9-12), WARN(13-16),
+ERROR(17-20), FATAL(21-24). This field is optional.
+Resource
+Description of. the source of the log.
+InstrumentationScope
+Multiple occurrences of events coming from the same scope can happen across time and
+they all have the same value of InstrumentationScope.
+DataLoader is a class that implements functions to load data from sources. In current version we implement
+FileDataLoader to load data from local files, e.g. .log,.csv,.tsv,.json. The associated DataLoaderConfig
+class defines the configuration of how data will be loaded. load_data() method will load data from target source and
+return LogRecordObject. In the future versions we will support data loaders with connectors to consume data directly
+from log platforms such as Splunk, Datadog, AWS Cloudwatch, etc.
+4.1.2
+Preprocessing Layer
+Preprocessing. Preprocessor is a class to conduct logline level preprocessing. Users can initialize a preprocessor
+instance with configuration and execute .clean_log() method to obtain cleaned loglines. The supported configuration
+includes custom_delimiters_regex to parse logs with custom delimiters and custom_replace_list to identify
+and replace the substrings that match regex patterns in this list, examples are show in Figure 3.
+5
+
+Data Layer
+preprocessing Layer
+Information Extraction Layer
+Application Layer
+Custom log
+Log
+datafiles
+FileDataLoader
+Preprocess
+Auto-Parsing
+Summarization
+Loglines
+Unstructured)
+Open Log
+OpenDataset
+Log
+Feature
+Log
+Datasets
+Records
+Vectorization
+DataLoader
+Extraction
+Clustering
+Attributes
+(Structured)
+Log streams
+from log
+Connector
+Categorical
+Log Anomaly
+Partitioning
+platforms
+DataLoader
+Encoding
+DetectionCheng et. al
+Figure 3: Example of preprocessor execution
+Partitioning. Partitioner is a class that helps partitioning the logs. As part of the preprocessing, there are needs
+to shuffle, concatenate and sequentialize raw logs into different forms, for example using time-based partitions or
+identifier-based partitions or sliding window partitions of fixed lengths. This class provides optional functions for this
+type of process.
+4.1.3
+Information Extraction Layer
+Information extraction layer contains modules to convert log records into vectors that can be used as input of machine
+learning models for the actual analytical tasks. Current log analysis research and applications indicate three main
+input data types are used in the ML approaches: 1) converting log records into counter vectors to use time-series ML
+techniques, 2) converting log records into feature vectors to use tabular-based ML techniques and 3) converting log
+records into sequences to use sequential ML techniques.
+LogAI implemented four components in the information extraction layer to extract information from the log records
+and convert logs to the target formats. Log parser component implements a series of automatic parsing algorithms
+in order to extract templates from the input loglines. Log vectorizer implements a bag of vectorization algorithms to
+convert free-form log text into numerical representations for each logline. Categorical encoder implements algorithms
+that encoding categorical attributes into numerical representations for each logline. Last but not least, feature extractor
+implements methods to group the logline level representation vectors into log event level representations.
+Automated Log Parsing. LogParser is a class that conducts automated log parsing tasks. Currently LogAI covers
+three automated log parsing algorithms: DRAIN[7], IPLoM[8] and AEL[9]. LogParser takes the unstructured logline
+text as input and generate two sets of results: parsed_logline are the static pattern of all logs in this category,
+parameter_list are the lists of values for each “*” position in the log pattern for the same set of loglines.
+Log Vectorization. LogVectorizer is a class that converts unstructured loglines into semantic vectors. Each semantic
+vector is an array of numeric values that represents this logline text. LogVectorizer supports popular text vectorization
+algorithms such as TF-IDF [10], FastText [11], Word2Vec [12], etc.
+Categorical Encoding. CategoricalEncoder is a class that encodes log attributes, the structured portion of logs.
+The string type attributes will be transformed into categorical representations. CategoricalEncoder supports popular
+categorical encoding algorithms such as label encoding, one-hot encoding, ordinal encoding etc.
+Feature Extraction. FeatureExtractor is a class that conducts final transformation of raw log data into log feature
+set that machine learning models can consume. In LogAI, we primarily cover three types of log features: 1) time-series
+counters, 2) semantic feature sets and 3) sequence vectors. Time-series counters will be used to feed time-series models
+such as ETS, ARIMA. Semantic feature set can be widely used in a variety of machine learning and deep learning
+models. Sequence vectors are a specific type of feature format that are required by sequence-modeling based deep
+learning methods, for example Recurrent Neural Network or Convolutional Neural Networks.
+6
+
+PreprocessorConfig
+Raw logs
+config =PreprocessorConfig(
+20171223-22:15:29:615|Step_LSC|30002312|onExte..
+custom_delimiters_regex=[r"\l"],
+20171223-22:15:29:633|Step_StandReportReceiver...
+3
+20171223-22:15:29:635/StepLSC/30002312|proces..
+custom_replace_list=[
++
+20171223-22:15:29:635|Step_StandStepCounter|30...
+(r'Step_lw+','<Operations>')
+Clean logs
+ParameterList
+20171223-22:15:29:615<Operations>30002312on...
+<Operations>
+20171223-22:15:29:633<Operations>30002312on...
+[Step_LSC]
+3
+20171223-22:15:29:635 <Operations> 30002312 pr...
+2
+[StepStandReportReceiver]
+20171223-22:15:29:635<Operations>30002312 fl...
+3
+[Step_LSC]
+4
+[Step_StandStepCounter]Cheng et. al
+4.1.4
+Analysis Layer
+The analysis layer contains modules that conduct the analysis tasks, including but not limit to semantic anomaly
+detector, time-series anomaly detector, sequence anomaly detector, clustering, etc. Each analysis module provides
+unified interface for multiple underlying algorithms.
+Anomaly Detection. AnomalyDetector is a class to conduct anomaly detection analysis to find abnormal logs from
+semantic perspective. AnomalyDetector takes log features of the given logs as input. The output are the anomaly
+scores. LogAI supports two different types of anomaly detection: 1) anomaly detection based on log counter vectors,
+2) anomaly detection based on log semantic representations. The supported anomaly detection algorithms includes
+univariate and multivariate time-series analysis algorithms from Merlion [13], unsupervised outlier detection models
+like one-class SVM [14] and local outlier filter (LOF) [15] from scikit-learn [16].
+Deep-learning based anomaly detection.
+NNAnomalyDetector class supports deep-learning model based log
+anomaly detection algorithms, most of which are taking log sequence vectors as input. LogAI integrate some of
+the popular deep learning based algorithms like recurrent neural network (RNN) based model LSTM [17], convolutional
+neural network (CNN), Transformers [18] and pretrained Transformer based Language Model BERT [19]. The output
+are anomaly scores for each log sequence.
+Clustering. Clustering is a class to conduct log clustering analysis tasks. The input for log clustering are the
+semantic log features. Clustering is integrated different clustering models, such as k-Means [20], DBSCAN [21] etc.
+The output is a map between each log feature record and a cluster label.
+4.1.5
+E2E Applications
+Depending on the component modules from data layer, preprocessing layer, feature extraction layer and analysis layer,
+LogAI provides the flexibility to build end-to-end log analysis applications. And the applications follows below design
+principles 4. LogAI is launched with several out-of-the-box applications.
+Figure 4: Design Principles of E2E Applications
+Log Summarization. It is very important to understand your logs before using them for downstream tasks. Log
+summarization leverages machine learning to process, aggregate and summarize logs. Please refer to the GUI module
+Section 4.2 for more detail about how to use.
+Log Clustering. Log clustering can be used to categorize logs. Finding meaningful clusters can bring benefits in a
+variety of use cases like anomaly detection, log storage, query, etc. Please refer to the GUI module Section 4.2 for more
+detail about how to use.
+Log Anomaly Detection. Log anomaly detection is an application that detect anomalous loglines. Here in LogAI log
+anomaly detection can detect both time-series anomalies and semantic anomalies. Please refer to the GUI module
+Section 4.2 for more detail about how to use.
+7
+
+Log record
+Data Preparation
+ApplicationWorkflow
+objects
+Workflow Configuration
+Dataloader
+Preprocessing
+IE-component-N
+Analysis-Component
+Configuration
+Configuration
+Configuration
+Configuration
+FileDataloader
+Algorithm-1
+Algorithm-1
+Preprocessing
+OpenSetDataloader
+Algorithm-2
+Algorithm-2
+Partitioning
+OtherDataloader
+Algorithm-3
+Algorithm-3Cheng et. al
+Figure 5: LogAI GUI portal
+4.2
+GUI Module
+The GUI module is implemented to provide a web portal for the out-of-the-box log analysis applications, including
+log summarization, log clustering and log anomaly detection. Figure 5 shows the log summarization of LogAI portal.
+LogAI portal is developed using Plotly Dash framework.
+Control Panel. Control panel is on the left side of the page. In the control panel, users can upload files, configure file
+and algorithm settings. When the user click "Run" button, the analysis execution is triggered. This behavior is uniform
+for all three different applications. After analysis execution completed, the results will be displayed on the right side of
+the page.
+Main Display Panel. On the right side of the page we display the analysis results. Different applications may have
+different layouts. The portal supports interactive visualization. The users can click or hover on parts in the charts to
+drill down and get more detailed information.
+The interaction between frontend and backend of different applications are designed to be unified. The control panel
+collects user input and generate configuration for application and send to backend. Backend consumes the configuration
+to create component instances to execute the workflow. After finishing the job, it will send the result table to frontend.
+The display panel for each application controls how the result table will be rendered for visualization. Users can expand
+the GUI portal to support customized analysis applications by following the same design pattern and reusing the existing
+components.
+4.3
+Summary of Supported ML Algorithms in LogAI
+This section summarizes the machine learning algorithms supported in LogAI. LogAI provides an algorithms compo-
+nent to implement all supported algorithms with algorithm factory. The algorithm contains five algorithmic mod-
+ules, notably: parsing_algo, vectorization_algo, categorical_encoding_algo, clustering_algo, anomaly_detection_algo.
+algorithms component also contains a nn_model module to implement all neural network models. LogAI has defined
+unified algorithm interfaces for each module and we can implement more algorithms and integrated it with LogAI in
+future development. The current LogAI algorithm coverage is shown in Table 4.3.
+The deep-learning models generally being much more parameter-heavy, require more high-end compute devices like
+GPU. In such cases, their LogAI implementations provide options to use different devices (CPU or GPU) or multiple
+GPUs seamlessly through the algorithm parameter configurations.
+8
+
+a LOG Al Powered by Salesforce AI Research
+Al-based Log Analysis
+Summary
+Attributes
+Log.Summarization
+Total Number of Loglines: 20000
+Level
+Log. Clustering
+Total Number of Log Patterns: 15
+INFO
+AnomalyDetection
+Charts
+Trend of Occurrence at Freq(1s)
+File Settings
+Log Type
+HDFS
+50
+1000
+Select Log File
+HDFS_20000.log
+40
+Attributes
+×Level 
+Time Interval
+1s
+10
+Parsing Algortihm
+DRAIN
+20:35:30
+20:36:00
+20:36:30
+20:37:00
+20:37:30
+20:38:00
+20:38:30
+Nov 9, 2008
+RUN
+log pattern
+timestamp
+Log Patterns
+dfs.FSNamesystemBLoCK*NameSystem.addStoredBlockblockMap updated*50010isaddedto* size*
+Dynamic Values
+Position
+Count
+Value
+POSITION_O
+4123
+10.251.215.192,10.251.110.196,10.251.91.159,10.250.18.114,10.251.194.245,10.251.90.239,10.251.203.4,10.251.42.246,10.251.30.6,10.251.
+POSITION_1
+4123
+b1k_-8426566918839220582，blk_8892946833207246710，blk_4685864904040870678，blk_3733773024533525840，blk_1329134914737185064，blk_-542351385442Cheng et. al
+Table 2: Summary of supported machine learning algorithms in LogAI
+Module
+Algorithm
+Task
+Log parser
+DRAIN
+Information Extraction
+IPLoM
+AEL
+Log vectorizer
+Fast-text
+Unstructured Log Representation
+TF-IDF
+Word2Vec
+Semantic
+Sequential
+BertTokenizer
+Categorical Encoder
+Label encoding
+Structured Log Representation
+OneHot Encoding
+Ordinal Encoding
+Clustering
+DBSCAN
+Analysis: Log Clustering
+K-means
+BIRCH
+Anomaly Detection
+One-class SVM
+Analysis: Outlier Detection
+Isolation Forest
+LOF
+Distribution divergence
+ETS
+Analysis: Time-series Anomaly Detection
+Dynamic Baseline
+ARIMA
+NN models
+CNN
+Analysis: Sequential Anomaly Detection
+LSTM
+Transformers
+LogBERT
+Analysis: (Sequential / Non-Sequential) Anomaly Detection
+5
+Experiments: Benchmarking Log Anomaly Detection
+In this section, we elaborate some of the experimental effort at building pipelines for specific log analysis tasks on
+publicly available log datasets. The purpose of this is to benchmark the performance of our LogAI library on these
+standard tasks with the performances reported in existing literature or other well-known log libraries.
+Amongst the different log analysis tasks, log based anomaly detection is perhaps the most objective task, where domain
+experts like reliability and performance engineers can provide some supervision around which log sequences show
+anomalous behavior. The other tasks like log clustering, summarization are much more subjective in nature while log
+based root cause analysis is too specific and tightly coupled with the application or environment it is deployed in. Hence
+for these tasks it is often impossible to collect supervision labels for benchmarking purposes. Consequently most of the
+publicly available log analysis datasets and benchmarks have focused on the anomaly detection task. While a small
+subset of these datasets have also been redesigned to serve log clustering and log summarization in past literature, they
+can at best be considered as pseudo-oracle data for these tasks and are still are not large-scale enough for benchmarking
+purposes. Hence, for this reason, in our LogAI library we focus on benchmarking only the log based anomaly detection
+task.
+Following the advances of Artificial Intelligence (AI), Machine Learning (ML) and Natural Language Processing
+(NLP), for log anomaly detection task also traditional statistical ML based solutions (like SVM, Isolation Forest
+etc.) have gradually given way to more powerful and sophisticated neural models. Some of these newer models can
+leverage self-supervised learning to achieve comparable anomaly detection performance in unsupervised settings in
+comparison to older traditional supervised models. Additionally, the traditional ML models having being around for
+quite a while, have been more extensively studied with fairly well-reproduced benchmarks in existing literature. Hence
+in our benchmarking experiments, we have only focused on the more recent neural models.
+9
+
+Cheng et. al
+5.1
+Limitations of Existing Libraries and Benchmarking Practices
+Over the past decade have been numerous literature [22, 23, 24, 25, 26, 27, 28, 29, 3] reporting the log anomaly
+detection performance on some of the standard open-sourced log datasets, as well as various effort at open-sourcing
+libraries catering the log anomaly detection task. For example, [2, 3] had released libraries (Loglizer and Deep-Loglizer)
+for log based anomaly detection using traditional machine learning and more recent deep learning models, respectively.
+In their library they had consolidated some of the benchmarking effort, bringing together all the popular log anomaly
+detection models for a more fair comparison on a few public log datasets.
+However, despite this, there is still a lack of rigorous standardisation and benchmarking amongst these works, especially
+the ones employing neural models. Below we list some of the specific limitations of Loglizer and Deep-Loglizer library
+which necessitates the need for an unified, generic framework for log analysis tasks:
+• Generic Log Data Processing Pipeline: There is a lack of libraries that provide a generic data processing pipeline
+that is common across different log datasets or different log anomaly algorithms. While Loglizer [5] and Deep-
+Loglizer [3] has achieved this to some degree, they still require some dataset-specific preprocessing and customization
+which are quite open-ended. For users wishing to replicate on their own datasets or other public datasets, there is no
+clear framework guiding the necessary steps and output-structure of the dataset-specific preprocessing to follow. On
+the other hand, LogAI library provides a an unified generic data-processing pipeline across all public datasets and
+log analysis algorithms. It only requires a very minimal dataset-specific customization with a clear template of the
+kind of preprocessing needed for each dataset - for e.g. each dataset has its own way of specifying the fields of the
+LogRecordObject (governed by OpenTelemetry data models) e.g. labels or identifiers of the loglines - which are
+either directly part of the raw log data or have to be derived based on some rules.
+• Catering to multiple Log Analysis Tasks: There is a lack of libraries that can cater to all kinds of log analysis tasks
+(including log clustering, summarization, anomaly detection etc) under a single generic platform. Each of the existing
+log libraries are tailored for a specific kind of log analysis task. For example libraries like loglizer and Deep-Loglizer
+specifically focus on log based anomaly detection, log-parser on parsing log data and log3C cater to clustering and
+correlation specific analysis. On the other hand, logAI enables all of these analysis tasks along with others, like,
+summarization, visualization etc under an unified framework.
+• Coverage of Log Analysis Models: The existing Loglizer library provides the more traditional machine learning
+algorithms for log based anomaly detection, with the Deep-Loglizer being a deep-learning based counterpart of it,
+providing only neural ML models. LogAI on the other hand, provides a generic framework encompassing most of the
+popular AI/ML algorithms - starting from traditional statistical ML models to popular neural models as well as more
+recent pretrained Transformer (BERT) based models. Going ahead, our logAI library can provide a more extended
+platform for integrating with more upcoming and powerful neural models as the mainstream deep learning research
+progresses. For all of these models, logAI provides a single unified data processing platform, that is independent of
+the kind of downstream analysis task or models.
+Thus, with LogAI library, we aim at a more intuitive and easy-to-use log analysis framework for practitioners of
+different areas and levels of expertise to perform log analysis, without being impeded by the technical nuances of the
+task.
+5.2
+Log based Anomaly Detection Workflow
+In order to handle the complex and heterogenous nature of log data, log based anomaly detection typically follows a
+multi-step pipeline. Starting with the raw log data dump or data streams, the log analysis workflow does some initial
+preprocessing and cleaning-up of the raw logs to make them amenable to ML models. This is typically followed by log
+parsing which extracts a loose structure from the semi-structured data and then performs grouping and partitioning of
+the log lines into log sequences in order to model the sequence characteristics of the data. After this, the logs or log
+sequences are vectorized i.e. represented as a machine-readable vector, by first tokenizing each instance and converting
+each token to a d-dimensional embedding. On this vectorized version of the log data, various anomaly detection models
+can be applied.
+The choices of each of these steps (for e.g. whether to apply parsing or not, or whether to partition based on sessions or
+sliding windows, or whether to apply clustering or not) can be guided by various factors - nature of the application
+generating the log data or the model requirements or other efficiency or performance related constraints.
+i) Log Preprocessing: In LogAI, this step involves handling the formatting of timestamps, logline-identifiers and any
+associated labels (e.g. anomaly labels) in the raw log data to make it compatible to openTelemetry data. Additionally it
+also provides customised filtering of specific regular expression patterns (like IP addresses or memory locations or file
+paths) that are deemed irrelevant for the actual log analysis.
+10
+
+Cheng et. al
+Figure 6: Example of Log Parsing
+ii)Log Parsing: To enable downstream processing, un-
+structured log messages first need to be parsed into a
+structured event template (i.e. constant part that was ac-
+tually designed by the developers) and parameters (i.e.
+variable part which contain the dynamic runtime informa-
+tion). Figure 6 provides one such example of parsing a
+logline. In LogAI library we provide three popular log
+parsers which use heuristic-based techniques - Drain [30],
+IPLoM [31] and AEL [32].
+iii) Log Partitioning: After parsing the next step is to
+partition the log data into groups, based on some seman-
+tics where each group represents a finite chunk of log lines or log sequences. The main purpose behind this is to
+decompose the original log dump, which typically consists of millions of log lines into logical chunks, so as to enable
+explicit modeling on these chunks and allow the models to capture anomaly patterns over sequences of log templates or
+log parameter values or both. In literature, various Log partitioning techniques have been applied [27, 33]. In LogAI we
+provide different schemes like - Fixed or Sliding window based partitions, where the length of window is determined by
+length of log sequence or a period of time, and Identifier based partitions where logs are partitioned based on some
+identifier (e.g. the session or process they originate from). Figure 7 illustrates these different choices of log grouping
+and partitioning. A log event is eventually deemed to be anomalous or not, either at the level of a log line or a log
+partition.
+Figure 7: Different types of log partition-
+ing
+iv) Log Vectorization: After log partitioning, the next step is to represent
+each partition in a machine-readable way (e.g. a vector or a matrix) by
+extracting features from them. This can be done in various ways [34, 33].
+In LogAI we provide the following vectorization techniques -
+• i) sequential representation which converts each partition to an ordered
+sequence of log event ids
+• ii) quantitative representation which uses count vectors, weighted by the
+term and inverse document frequency information of the log events
+• iii) semantic representation captures the linguistic meaning from the se-
+quence of language tokens in the log events and learns a high-dimensional
+embedding vector for each token in the dataset.
+The nature of log representation chosen has direct consequence in terms of
+which patterns of anomalies they can support - for example, for capturing
+keyword based anomalies, semantic representation might be key, while for anomalies related to template count and
+variable distribution, quantitative representations are possibly more appropriate. The semantic embedding vectors
+themselves can be either obtained using pretrained neural language models like GloVe, FastText, pretrained Transformer
+like BERT, RoBERTa etc. Or they can also be learnt from scratch on the available training data, by building custom
+vocabulary and using these neural language models.
+v) Log Anomaly Detection Models for benchmarking: The task of log based anomaly detection is to analyze a dump
+of log data, consisting of a series of timestamped log lines and identify the anomalous log lines that are potentially
+incident-indicating. Based on the kind of application, log anomaly signals can either be used to detect or localize an
+already occurred incident or disruption or used to forecast future potential faults or failures. In literature, log based
+anomaly detection models have been broadly categorized into two types - Supervised and Unsupervised, based on the
+kind of training framework they follow. Since our objective is to benchmark only neural models, we limit our discussion
+in this section to this class of models alone.
+Supervised Anomaly Detection models require the anomaly label to be available at the level of each log line or a log
+group or partition. Furthermore, they typically assume that each of the training, development and test data will contain
+a mix of both anomalous and non-anomalous log data. These models use the supervised losses like cross entropy loss or
+squared error loss. But they can suffer due to the under-representativeness of the anomalous class of logs, especially if
+they occur very rarely in the training and development data. Due to their direct dependency on modeling the anomalous
+class explicitly these models also lack robustness when the anomaly distribution changes.
+11
+
+LogLine: 081109 204655 556 INFO dfs.DataNode$PacketResp0nder:
+Received block blk_3587508140051953248 of size 67108864 from
+/10.251.42.84
+Timestamp: 081109 204655 556
+Level: INFO
+Component: dfs.DataNodesPacketResponder
+Template: Received block <*> of size <*> from <*>
+Parameter:["blk 3587508140051953248",“67108864",“10.251.42.84"]Fixed Partitions
+Sliding Partitions
+Identifier Partitions
+1
+2
+1
+1
+2
+2
+3
+3
+3
+1
+2
+1
+1
+2
+2
+3
+3
+3
+2
+2
+2
+3
+3
+3Cheng et. al
+Unsupervised Anomaly Detection models do not require any anomaly label for the log data. But the existing
+unsupervised models in the literature typically assume that the entire training data is comprised of only normal or
+non-anomalous logs and generally show a sharp decline in performance when the training data is adulterated with even
+a small fraction of anomalous logs. Amongst the most popular unsupervised anomaly detection models, mainly two
+paradigms have been followed:
+• Forecasting based models: These models learn the representations of the log lines through a forecasting based
+self-supervision i.e. by learning to predict the label of next log line given an input context of log sequence. For
+all of these models, following Deep-Loglizer paper, the label has been taken as the event id of the next log line.
+This category of models includes various sequence encoding networks that have been popular in deep-learning -
+like recurrent neural network or convolutional neural network based models or the more recent, more powerful
+self-attention based Transformer models. These models are typically trained a cross-entropy loss between the true
+and predicted distributions, which aims to maximise the likelihood of the true label, conditional to the given input
+sequence.
+• Reconstruction based models: This includes Auto-encoder based models which try to reconstruct a given sequence
+of loglines through a learnable hidden layer that learn an n-dimensional representation of each log-line. The other
+more recent models in this category are Transformer based models which are trained using masked-language modeling
+principles. During training a certain fraction of the input tokens would be masked and the model would learn to
+predict these tokens using the remaining input context; and in the process learning the contextual representation of
+each token in a log-line or a log-sequence. This is the fundamental principle behind BERT Language model with the
+masked language modeling providing the learning objective when training on the log data in a self-supervised way.
+Forecasting based Anomaly Detection: For our benchmarking with forecasting based models, we select three core
+deep learning models which have been the basis of the some of the most popular recent neural log anomaly detection
+methods
+• LSTM: This model corresponds to a long-short term memory (LSTM) network to encode a given log sequence. It
+also provides various options - i) whether to utilize uni-directional or bi-directional encoding of tokens in a given
+input sequence ii) whether to have a learnable attention network over the input sequence, which linearly combines the
+hidden representations with the attention weights.
+• CNN: This model corresponds to a convolutional neural network (CNN) to encode a given log sequence. Different
+convolutional layers with different shape settings are applied on the input followed by a 1-d max-pooling operation.
+The outputs from each of these are then concatenated and fed into a fully-connected layer.
+• Transformer: This model corresponds to a Transformer based encoder network with a multi-headed self-attention
+mechanism to encode a given log sequence. Since the Transformer outputs a d-dimensional representation for
+each token in the input log-sequence, a mean-pooling operation is applied over those representations, to get a fixed
+representation for the entire sequence.
+Since the LSTM, CNN and Transformer models need a d-dimensional representation of each log, first an embedding
+layer is applied to the raw log input features. In case of sequential feature representation, each log event id is embedded
+as a d-dimensional vector, while for semantic feature representation, the embedding layer is initialized with the
+pretrained embeddings (e.g. Word2Vec or FastText etc) and embeds each log token id to a d-dimensional vector.
+The output of the LSTM, CNN or Transformer a fixed d-dimensional representation of the input sequence which is
+then downward projected to 1-d space, followed by a softmax layer. For supervised versions of these models, since the
+explicit label (anomalous or not) exists for each log-line or log-sequence, the output of the softmax layer is aimed to
+directly predict this label. For forecasting based unsupervised versions, the output of the softmax layer is aimed to
+predict the id of the next log-line, that is succeeding the given input log sequence. During inference, for forecasting
+based unsupervised models make a prediction for a given input log sequence, which is then compared against the actual
+log event following the input sequence. We follow the similar inference strategy as [3] and predict a test instance as
+anomalous if the ground truth is not one of the k (=10) most probable log events predicted by the model. A smaller k
+imposes more demanding requirements on the model’s performance.
+In literature, LSTM based models have been used by DeepLog [35], LogAnomaly [34] and LogRobust [36]. While
+DeepLog uses sequential representations, where each log message is represented by the index of its log event,
+LogAnomaly uses semantic representations. While both of these use unidirectional LSTM in an unsupervised setting,
+LogRobust uses supervised version of an bi-directional LSTM with the attention network. CNN has been used by [37]
+but only in a supervised setting. Transformer based model has been applied in LogSy [38], but they additionally use
+auxiliary log datasets as pseudo-anomalous data. This helps them to learn a better representation of normal log data
+from the target system of interest while regularizing against overfitting. In order to ensure better reproducibility, in our
+benchmarking we do not use any additional log datasets and hence in some of the supervised settings, our Transformer
+based models suffer from overfitting issues and yield somewhat poorer results and are not directly comparable to the
+12
+
+Cheng et. al
+results obtained by [37]. Following [3] for all of these models, in both the supervised and unsupervised settings, we
+report the F1-Scores.
+Reconstruction based Anomaly Detection: For our benchmarking with reconstruction based models, we select the
+LogBERT model from the work LanoBERT [39]. Following that literature, the preprocessing configurations are set
+before the BERT model can be applied - i) Since LogBERT is a parser-free technique, no log parsing is applied. ii)
+For obtaining the vectorized log representation, the preprocessed log sequences are tokenized using the WordPiece
+(Wu et al. 2016) model used in BERT. iii) The tokenizer is trained from scratch on each log dataset to ensure that the
+dataset-specific custom vocabulary can be learned. During training the usual masked language modeling principles
+of BERT is followed. During inference, multiple masked versions of each test instance is generated, by passing a
+fixed-size masking window over the token sequence, ignoring masking of special characters. Thus a test instance of
+sequence length N will result in an average of N
+n masked instances, each have a masked n-gram of length upto n. After
+running the inference on the masked test instance, the anomaly score is obtained as the average of the top-prediction
+probabilities (or log-probabilities) over the k-most confident masked tokens. Following LanoBERT, we report AUROC
+(Area under ROC) metric over this anomaly score.
+All unsupervised models, (forecasting or reconstruction based) are trained only on the normal log data. Following
+Deep-Loglizer, for the forecasting based models, around 20% of the data and for LogBERT, following LanoBERT,
+around 30% of the data is sequestered for test. These percentages include the entire set of anomalous logs in the dataset.
+In LogAI, we take out 10% of the training data as development set for validation and model selection purposes.
+5.3
+Datasets:
+Following Deep-Loglizer and LanoBERT, we perform our benchmarking experiments on two of the most popular
+public log anomaly detection datasets - HDFS and BGL. Additionally for LogBERT we also benchmark on the public
+dataset, Thunderbird. Further, similar to Deep-Loglizer, for BGL dataset we also perform a fixed-window based log
+partitioning by grouping log-lines over every 6-hourly window. However for LogBERT model, following LanoBERT,
+we treat each individual log-line as a train or test instance, without doing any log partitioning. On the other hand, for
+HDFS dataset, since anomaly labels are available only at the level of each session-id (which is also known as BLOCK
+in the raw dataset), we use identifier based log partitioning, by constructing log-sequences for each session-id. These
+resulting log partitions are treated as the training or test instances for all algorithms.
+5.4
+Experimental Settings and Results:
+For our benchmarking we conduct experiments on the above choice of anomaly detection algorithms under various
+settings and compare our experimental results with those published in Deep-Loglizer [3] and LanoBERT [39] papers In
+Table 3 we list the performance of the different supervised and unsupervised forecasting-based models (LSTM, CNN
+and Transformer), while 4 shows the results using the unsupervised reconstruction-based LogBERT model.
+Evaluation Metrics: In order to compare the performances, for all supervised and unsupervised forecasting-based
+models we use F1-Score as the metric, following Deep-Loglizer paper. Whereas, for LogBERT, following LanoBERT
+paper we report the AUROC metric. LanoBERT paper also provides F1-Score, but the F1-Score calculation needs
+fixing a threshold, which is challenging to do over the training data that only has normal logs. According to the paper,
+their reported scores are the best F1 value that was calculated using the threshold that yields the best performance for
+the test dataset. This is not a fair metric, as it involves label-knowledge of the blind test set and hence we only compare
+using AUROC metric.
+Configuration Settings for Evaluation: For each of LSTM and Transformer models, we benchmark on 8 different
+configuration settings for each dataset - based on the kind of supervision (supervised or unsupervised), whether log
+parsing is applied or not, whether the log representation is sequential or semantics based. For CNN models, we found
+the semantics based log representation results in very slow convergence rate, hence we have benchmarked our results
+using only the sequential feature representations of the logs. On the other hand, Deep-Loglizer showcases only specific
+settings for these models - for e.g. forecasting based unsupervised anomaly detection is done using Unidirectional
+LSTM with no-attention and Transformer network while supervised models are Bidirectional LSTM with attention and
+CNN network, whereas all of these methods can be applied on both supervised and unsupervised settings. Each of their
+models use the Log Parsing step and have two variants that use sequential and semantic feature representations for the
+logs. However Deep-Loglizer paper [3] provides only 8 configurations for each dataset whereas LogAI is benchmarked
+on a more exhaustive set of 20 configurations per dataset.
+Performance Comparison: In most of these configurations the performance achieved by LogAI is comparable to that
+of Deep-Loglizer. The 2-3% difference in performance between the models is not quite statistically significant and can
+13
+
+Cheng et. al
+Model
+Details
+Supervision
+Log Pars-
+ing
+Log
+Represen-
+tation
+HDFS
+BGL
+LogAI
+Deep-
+Loglizer LogAI
+Deep-
+Loglizer
+LSTM
+Unidirectional, No
+Attention
+Unsupervised
+
+sequential
+0.981
+0.944
+0.938
+0.961
+semantic
+0.981
+0.945
+0.924
+0.967
+
+sequential
+0.979
+-
+0.925
+-
+semantic
+0.981
+-
+0.924
+-
+Bidirectional, With
+Attention
+Supervised
+
+sequential
+0.984
+0.96
+0.983
+0.983
+semantic
+0.964
+0.964
+0.95
+0.983
+
+sequential
+0.989
+-
+0.931
+-
+semantic
+0.971
+-
+0.983
+-
+CNN
+2-D Convolution with
+1-D Max pooling
+Unsupervised
+
+sequential
+0.981
+-
+0.929
+-
+
+sequential
+0.981
+-
+0.922
+-
+Supervised
+
+sequential
+0.943
+0.97
+0.983
+0.972
+
+sequential
+0.946
+-
+0.990
+-
+Transformer
+Multihead
+single-
+layer self-attention
+model,
+trained
+from scratch
+Unsupervised
+
+sequential
+0.971
+0.905
+0.933
+0.956
+semantic
+0.978
+0.925
+0.921
+0.957
+
+sequential
+0.98
+-
+0.92
+-
+semantic
+0.975
+-
+0.917
+-
+Supervised
+
+sequential
+0.934
+-
+0.986
+-
+semantic
+0.784
+-
+0.963
+-
+
+sequential
+0.945
+-
+0.994
+-
+semantic
+0.915
+-
+0.977
+-
+Table 3: Comparison between different supervised and unsupervised Forecasting-based neural anomaly detection
+models in LogAI and Deep-Loglizer library [3], using F1-Score as the performance metric. The dashed (-) cells indicate
+that there are no reported numbers in the Deep-Loglizer paper corresponding to those configurations.
+mostly be attributed to the following factors: Following the implementation open-sourced by authors of Deep-Loglizer
+in https://github.com/logpai/deep-loglizer, it is evident that the library does not utilize any development (or
+validation) set and directly performs model selection based on the test performance. LogAI on the other hand, selects
+the model checkpoint on the validation performance and reports the results on the blind test set. Secondly, because of
+the same reason the resulting the training and test splits used by LogAI and Deep-Loglizer are not identical. Especially
+for BGL data, perhaps the performance difference is somewhat more observeable, since both libraries apply fixed
+time-partitions of 6 hours and reports the evaluation at the level of the partitions, instead of the logline level evaluation.
+This also adds to the possibility of more significant differences in the training/test data setup between the two models.
+For Transformer based models, especially in the supervised setting, we observe a reduced performance. Similar effect
+had been studied in the original work [38] that used Transformer model as Log Anomaly Detector in the supervised
+setting. Their model suffered from overfitting on the target system’s log data due to the presence of only rare and sparse
+anomaly patterns in the train data. To overcome the overfitting issue, they additionally involve other external system’s
+logs as auxiliary data - treating them as pesudo “anomalous” logs. But in order to keep our benchmarking reproducible,
+we do not use any additional auxiliary data abd subsequently report a poorer performance. The Deep-Loglizer paper
+also benchmarks with only the unsupervised setting of the Transformer model, which is much less prone to overfitting.
+For LogBERT model, we benchmark the test results taking various inferencing strategies. Given a test instance, which
+has been converted to multiple masked versions (each having a continuous n-gram masked), either we average the
+inference score either over all the masked tokens or over the top-6 most confident ones, based on the the model
+prediction likelihood. For the latter we consider different inference scores - mean predictive loss or maximum predictive
+probability or log probability or the entropy of the prediction distribution. All of these metrics are quite correlated and
+our objective is to simply show that our LogBERT implementation yields reasonably stable results across these different
+inferencing strategies. While LanoBERT also uses Predictive Loss and Probability based scores, they provide AUROC
+evaluation metric metric only for the latter and they also evaluate only HDFS and BGL dataset. In the predictive
+probability based inference strategy, results obtained by LogAI and LanoBERT are quite comparable, with small
+differences owing to the variability of the train, test splits used in the two implementations (The authors of LanoBERT
+have used their own train test split due to the general lack of standardized data splits for these datasets).
+14
+
+Cheng et. al
+Inference Strategy
+Datasets
+HDFS
+BGL
+Thunderbird
+LogAI
+LanoBERT
+LogAI
+LanoBERT
+LogAI
+LanoBERT
+Averaged over all masked tokens
+Mean Predictive Loss
+0.983
+-
+0.998
+-
+0.953
+-
+Averaged over top-6 most-confident masked tokens
+Mean Predictive Loss
+0.98
+-
+0.964
+-
+0.937
+-
+Max Predictive Prob.
+0.976
+0.99
+0.972
+0.972
+0.953
+-
+Max Predictive LogProb.
+0.976
+-
+0.969
+-
+0.917
+-
+Mean Predictive Entropy
+0.976
+-
+0.973
+-
+0.967
+-
+Table 4: Comparison of LogBERT model performance achieved by our LogAI library and by LanoBERT [39], using the
+AUROC metric. Both versions of the model are in unsupervised setting (trained on normal logs only) and do not need
+any log parsing. The dashed (-) cells indicate that there are no reported numbers in the LanoBERT paper corresponding
+to those configurations.
+Overall our experiments on the suite deep learning based log anomaly detection models suggests that their implemen-
+tations in the LogAI library is able to reproduce the established performance benchmarks on standard open-source
+datasets with reasonable accuracy. Additionally, owing to a more generic data processing pipeline we are seamlessly
+able to extend to a more exhaustive set of experimental settings, than what has been explored or implemented before in
+existing literature and libraries.
+6
+Conclusion
+In this technical report we introduced LogAI, an open source library for AI-based log analytics and intelligence. LogAI
+library uses the same unified log data model as OpenTelemetry to ensure the analytical processes to be agnostic to any
+log platforms that supports OpenTelemetry. LogAI also abstracts common processes in different downstream tasks and
+provides reusable components to execute these processes. LogAI also provides a large varieties of AI capabilities, from
+time-series anlaysis, traditional statistical learning to deep learning and pre-trained transformer models. We showed
+how LogAI can be used to conduct a variety of common log analysis tasks such as log summarization, clustering and
+anomaly detection and also provide extensive benchmarking results on Log Anomaly Detection. LogAI version v0.1.0
+is released as open-source code under BSD-3-Clause license. Our team will provide continuous support and further
+improvements in the future versions.
+Acknowledgments
+We would like to thank a number of leaders and colleagues from Salesforce.com Inc. who have provided strong support,
+advice, and contributions to this open-source project.
+References
+[1] The Business Research Company. Log Management Global Market Report. 2023.
+[2] Shilin He, Jieming Zhu, Pinjia He, and Michael R. Lyu. Experience report: System log analysis for anomaly
+detection. In 2016 IEEE 27th International Symposium on Software Reliability Engineering (ISSRE), pages
+207–218, 2016.
+[3] Zhuangbin Chen, Jinyang Liu, Wenwei Gu, Yuxin Su, and Michael R. Lyu. Experience report: Deep learning-based
+system log analysis for anomaly detection. CoRR, abs/2107.05908, 2021.
+[4] Jiang Zhaoxue, Li Tong, Zhang Zhenguo, Ge Jingguo, You Junling, and Li Liangxiong. A survey on log research
+of aiops: Methods and trends. Mobile Networks and Applications, pages 1–12, 2022.
+[5] Shilin He, Jieming Zhu, Pinjia He, and Michael R. Lyu. Experience report: System log analysis for anomaly
+detection. In 27th IEEE International Symposium on Software Reliability Engineering, ISSRE 2016, Ottawa, ON,
+Canada, October 23-27, 2016, pages 207–218. IEEE Computer Society, 2016.
+[6] Zhuangbin Chen, Jinyang Liu, Wenwei Gu, Yuxin Su, and Michael R. Lyu. Experience report: Deep learning-based
+system log analysis for anomaly detection, 2021.
+15
+
+Cheng et. al
+[7] Pinjia He, Jieming Zhu, Zibin Zheng, and Michael R Lyu. Drain: An online log parsing approach with fixed depth
+tree. In 2017 IEEE international conference on web services (ICWS), pages 33–40. IEEE, 2017.
+[8] Adetokunbo AO Makanju, A Nur Zincir-Heywood, and Evangelos E Milios. Clustering event logs using iterative
+partitioning. In Proceedings of the 15th ACM SIGKDD international conference on Knowledge discovery and
+data mining, pages 1255–1264, 2009.
+[9] Zhen Ming Jiang, Ahmed E Hassan, Gilbert Hamann, and Parminder Flora. An automated approach for abstracting
+execution logs to execution events. Journal of Software Maintenance and Evolution: Research and Practice,
+20(4):249–267, 2008.
+[10] Juan Ramos et al. Using tf-idf to determine word relevance in document queries. In Proceedings of the first
+instructional conference on machine learning, volume 242, pages 29–48. Citeseer, 2003.
+[11] Piotr Bojanowski, Edouard Grave, Armand Joulin, and Tomas Mikolov. Enriching word vectors with subword
+information. Transactions of the association for computational linguistics, 5:135–146, 2017.
+[12] Tomas Mikolov, Kai Chen, Greg Corrado, and Jeffrey Dean. Efficient estimation of word representations in vector
+space. arXiv preprint arXiv:1301.3781, 2013.
+[13] Aadyot Bhatnagar, Paul Kassianik, Chenghao Liu, Tian Lan, Wenzhuo Yang, Rowan Cassius, Doyen Sahoo,
+Devansh Arpit, Sri Subramanian, Gerald Woo, Amrita Saha, Arun Kumar Jagota, Gokulakrishnan Gopalakrishnan,
+Manpreet Singh, K C Krithika, Sukumar Maddineni, Daeki Cho, Bo Zong, Yingbo Zhou, Caiming Xiong, Silvio
+Savarese, Steven Hoi, and Huan Wang. Merlion: A machine learning library for time series. 2021.
+[14] Bernhard Schölkopf, John C. Platt, John C. Shawe-Taylor, Alex J. Smola, and Robert C. Williamson. Estimating
+the support of a high-dimensional distribution. Neural Comput., 13(7):1443–1471, jul 2001.
+[15] Markus M. Breunig, Hans-Peter Kriegel, Raymond T. Ng, and Jörg Sander. Lof: Identifying density-based local
+outliers. In Proceedings of the 2000 ACM SIGMOD International Conference on Management of Data, SIGMOD
+’00, page 93–104, New York, NY, USA, 2000. Association for Computing Machinery.
+[16] F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss,
+V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn:
+Machine learning in Python. Journal of Machine Learning Research, 12:2825–2830, 2011.
+[17] Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural Computation, 9(8):1735–1780, 1997.
+[18] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Łukasz Kaiser,
+and Illia Polosukhin. Attention is all you need. In Proceedings of the 31st International Conference on Neural
+Information Processing Systems, NIPS’17, page 6000–6010, Red Hook, NY, USA, 2017. Curran Associates Inc.
+[19] Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. BERT: pre-training of deep bidirectional
+transformers for language understanding. In Jill Burstein, Christy Doran, and Thamar Solorio, editors, Proceedings
+of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human
+Language Technologies, NAACL-HLT 2019, Minneapolis, MN, USA, June 2-7, 2019, Volume 1 (Long and Short
+Papers), pages 4171–4186. Association for Computational Linguistics, 2019.
+[20] D. Sculley. Web-scale k-means clustering. In Proceedings of the 19th International Conference on World Wide
+Web, WWW ’10, page 1177–1178, New York, NY, USA, 2010. Association for Computing Machinery.
+[21] Erich Schubert, Jörg Sander, Martin Ester, Hans Peter Kriegel, and Xiaowei Xu. Dbscan revisited, revisited: Why
+and how you should (still) use dbscan. ACM Trans. Database Syst., 42(3), jul 2017.
+[22] Jiang Zhaoxue, Li Tong, Zhang Zhenguo, Ge Jingguo, You Junling, and Li Liangxiong. A survey on log research
+of aiops: Methods and trends. Mob. Netw. Appl., 26(6):2353–2364, dec 2021.
+[23] Shilin He, Pinjia He, Zhuangbin Chen, Tianyi Yang, Yuxin Su, and Michael R. Lyu. A survey on automated log
+analysis for reliability engineering. ACM Comput. Surv., 54(6), jul 2021.
+[24] Paolo Notaro, Jorge Cardoso, and Michael Gerndt. A survey of aiops methods for failure management. ACM
+Trans. Intell. Syst. Technol., 12(6), nov 2021.
+[25] Xiao Han and Shuhan Yuan. Unsupervised cross-system log anomaly detection via domain adaptation. In
+Proceedings of the 30th ACM International Conference on Information & Knowledge Management, CIKM ’21,
+page 3068–3072, New York, NY, USA, 2021. Association for Computing Machinery.
+[26] Van-Hoang Le and Hongyu Zhang. Log-based anomaly detection with deep learning: How far are we? In
+Proceedings of the 44th International Conference on Software Engineering, ICSE ’22, page 1356–1367, New
+York, NY, USA, 2022. Association for Computing Machinery.
+16
+
+Cheng et. al
+[27] Nengwen Zhao, Honglin Wang, Zeyan Li, Xiao Peng, Gang Wang, Zhu Pan, Yong Wu, Zhen Feng, Xidao Wen,
+Wenchi Zhang, Kaixin Sui, and Dan Pei. An empirical investigation of practical log anomaly detection for online
+service systems. In Proceedings of the 29th ACM Joint Meeting on European Software Engineering Conference
+and Symposium on the Foundations of Software Engineering, ESEC/FSE 2021, page 1404–1415, New York, NY,
+USA, 2021. Association for Computing Machinery.
+[28] Yichen Zhu, Weibin Meng, Ying Liu, Shenglin Zhang, Tao Han, Shimin Tao, and Dan Pei. Unilog: Deploy one
+model and specialize it for all log analysis tasks. CoRR, abs/2112.03159, 2021.
+[29] Jacopo Soldani and Antonio Brogi. Anomaly detection and failure root cause analysis in (micro) service-based
+cloud applications: A survey. ACM Comput. Surv., 55(3), feb 2022.
+[30] Pinjia He, Jieming Zhu, Zibin Zheng, and Michael R. Lyu. Drain: An online log parsing approach with fixed
+depth tree. In 2017 IEEE International Conference on Web Services (ICWS), pages 33–40, 2017.
+[31] Adetokunbo A.O. Makanju, A. Nur Zincir-Heywood, and Evangelos E. Milios. Clustering event logs using
+iterative partitioning. In Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery
+and Data Mining, KDD ’09, page 1255–1264, New York, NY, USA, 2009. Association for Computing Machinery.
+[32] Zhen Ming Jiang, Ahmed E. Hassan, Parminder Flora, and Gilbert Hamann. Abstracting execution logs to
+execution events for enterprise applications (short paper). In 2008 The Eighth International Conference on Quality
+Software, pages 181–186, 2008.
+[33] Mostafa Farshchi, Jean-Guy Schneider, Ingo Weber, and John Grundy. Experience report: Anomaly detection of
+cloud application operations using log and cloud metric correlation analysis. In 2015 IEEE 26th International
+Symposium on Software Reliability Engineering (ISSRE), pages 24–34, 2015.
+[34] Weibin Meng, Ying Liu, Yichen Zhu, Shenglin Zhang, Dan Pei, Yuqing Liu, Yihao Chen, Ruizhi Zhang, Shimin
+Tao, Pei Sun, and Rong Zhou. Loganomaly: Unsupervised detection of sequential and quantitative anomalies in
+unstructured logs. In Proceedings of the 28th International Joint Conference on Artificial Intelligence, IJCAI’19,
+page 4739–4745. AAAI Press, 2019.
+[35] Min Du, Feifei Li, Guineng Zheng, and Vivek Srikumar. Deeplog: Anomaly detection and diagnosis from
+system logs through deep learning. In Proceedings of the 2017 ACM SIGSAC Conference on Computer and
+Communications Security, CCS ’17, page 1285–1298, New York, NY, USA, 2017. Association for Computing
+Machinery.
+[36] Xu Zhang, Yong Xu, Qingwei Lin, Bo Qiao, Hongyu Zhang, Yingnong Dang, Chunyu Xie, Xinsheng Yang, Qian
+Cheng, Ze Li, Junjie Chen, Xiaoting He, Randolph Yao, Jian-Guang Lou, Murali Chintalapati, Furao Shen, and
+Dongmei Zhang. Robust log-based anomaly detection on unstable log data. In Proceedings of the 2019 27th ACM
+Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software
+Engineering, ESEC/FSE 2019, page 807–817, New York, NY, USA, 2019. Association for Computing Machinery.
+[37] Siyang Lu, Xiang Wei, Yandong Li, and Liqiang Wang. Detecting anomaly in big data system logs using
+convolutional neural network. In 2018 IEEE 16th Intl Conf on Dependable, Autonomic and Secure Computing,
+16th Intl Conf on Pervasive Intelligence and Computing, 4th Intl Conf on Big Data Intelligence and Computing and
+Cyber Science and Technology Congress, DASC/PiCom/DataCom/CyberSciTech 2018, Athens, Greece, August
+12-15, 2018, pages 151–158. IEEE Computer Society, 2018.
+[38] Sasho Nedelkoski, Jasmin Bogatinovski, Alexander Acker, Jorge Cardoso, and Odej Kao.
+Self-attentive
+classification-based anomaly detection in unstructured logs. In 2020 IEEE International Conference on Data
+Mining (ICDM), pages 1196–1201, 2020.
+[39] Yukyung Lee, Jina Kim, and Pilsung Kang. Lanobert : System log anomaly detection based on BERT masked
+language model. CoRR, abs/2111.09564, 2021.
+17
+

1dAyT4oBgHgl3EQf1fkT/content/tmp_files/2301.00734v1.pdf.txt

@@ -0,0 +1,1547 @@
+Nonlinear Non-Hermitian Landau-Zener-St¨uckelberg-Majorana interferometry
+Xin Wang,1 H. D. Liu,1, ∗ and L. B. Fu2, †
+1Center for Quantum Sciences and School of Physics, Northeast Normal University, Changchun 130024, China
+2Graduate School of China Academy of Engineering Physics,
+No. 10 Xibeiwang East Road, Haidian District, Beijing, 100193, China
+(Dated: January 3, 2023)
+In this work, we have studied the non-Hermitian nonlinear LZSM interferometry in a non-Hermitian N-body
+interacting boson system in which the non-Hermicity is from the nonreciprocal tunnelings between the bosons.
+By using the mean-field approximation and projective Hilbert space, the effect of nonreciprocity and nonlin-
+earity on the energy spectrum, the dynamics, and the formation of the interference fringes have been studied.
+The different symmetries and the impact of the two different types of reciprocity, i.e. the in-phase tunneling and
+anti-phase tunneling, on the energy spectrum and the phase transition between the Josephson oscillation and the
+self-trapping have been investigated. For the LZSM interferometry, the strength of the nonreciprocity is found
+to take an essential role in the population of the projective state and the strengths of the interference patterns in
+the projective space. While the conditions of destructive and constructive interference under the weak-coupling
+approximation still only depend on the strength of nonlinearity. Our result provides an application of the non-
+linear non-Hermitian LZSM interferometry in studying the parameters of a non-Hermitian nonlinear two-level
+system which related to the nonlinearity and the non-Hermicity.
+I.
+INTRODUCTION
+The quantum two-level system (TLS) is the most basic part
+of physical systems. Among them, the Landau-Zener (LZ)
+transition between two levels at an avoided crossing [1–3]
+has received widespread attention. When these two-level sys-
+tems are under a strong periodic driving field, a series of
+LZ transitions occur and the transitions probability exhibit a
+periodic dependence on the phase (St¨uckelberg phase) accu-
+mulated between transitions [1, 4]. The periodic change is
+called Landau-Zener-St¨uckelberg-Majorana(LZSM) interfer-
+ometry [5, 6]. With the development of research, LZSM inter-
+ferometry has become an important phenomenon in quantum
+science and technology. On the one hand, LZSM interfer-
+ometry is used for ultra-fast universal quantum control of a
+quantum-dot charge qubit [7] and characterized qubit dephas-
+ing [8], etc. On the other hand, it has involved many fields
+so far, such as molecular nanomagnets [9, 10], quasi-one-
+dimensional layered materials [11, 12], ultracold molecules
+[13], quantum noise [14], Bose-Einstein condensates [15–19],
+Rydberg atoms [20], etc. Interestingly, if a two-level system
+takes account of the nonlinear interaction, it may produce un-
+expected interference features [21–26]. For the non-linear LZ
+model, the self-trapping phase transition may occur in LZSM
+interferometry [27–31], and there may be exceptional ring
+structures in the energy spectra [32, 33].
+In recent years, the non-Hermitian quantum systems with
+real energy spectra received widespread attention in the-
+ory and experiment [34–41]. There are two kinds of non-
+Hermicity, asymmetric coupling strengths in nonreciprocal
+systems and the gain-loss in reciprocal system.
+There are
+two kinds of non-Hermitian Hamiltonians, describing nonre-
+ciprocal systems with asymmetric coupling strengths [42–46]
+∗ liuhd100@nenu.edu.cn
+† lbfu@gscaep.ac.cn
+and gain-loss systems [37–41]. Bender and Boettcher dis-
+covered a series of parity-time (PT) -symmetric Hamiltonians
+[47], which could result in real energy spectra. Mostafazadeh
+generalized this type of Hamiltonian to a η-pseudo-Hermitian
+quantum theory which explains the conditions for the non-
+Hermitian system to have the real energy spectra (η is a pos-
+itive Hermitian operator) [48–50]. The theory has been ap-
+plied in many fields for more than ten years of development,
+such as quantum field theory [51–55], super-symmetric quan-
+tum mechanics [56, 57], non-commutative field theory [58],
+quantum information [59], etc. Especially, there always ex-
+ists some exceptional points (EPs) in the real energy spec-
+trum of the non-Hermitian system [60, 61], at which two or
+more eigenstates of the system coalesce. These EPs of the en-
+ergy spectrum in the parameter space are closely related to the
+symmetry, topological properties, and phase transitions of the
+system [34–36]. Consequently, efforts have been put forward
+to extend the study of LZ problem to non-Hermitian system
+[6, 62–65]. Therefore, for non-Hermitian systems and nonlin-
+ear LZSM interference, it is natural to ask how will the en-
+ergy spectrum of the nonlinear LZ system changes if the non-
+Hermiticity emerges? Will non-linearity affect EPs? Since
+the populations of the bare states on the adiabatic eigenstates
+normally can not be normalized by a time-independent coeffi-
+cient [66]. Can the interesting self-trapping effect in the case
+of nonlinear non-Hermitian still be observed? We shed lights
+on these questions in this paper. By setting up the projec-
+tive Hilbert space, we show that the populations of the projec-
+tive quantum states can still achieve LZSM interferometry and
+analyzed the influence of non-Hermicity and nonlinearity on
+the energy spectra and the interference. Then, we discussed
+the influence of non-Hermitian on the self-trapping effect. Fi-
+nally, under the weak-coupling approximation of the projec-
+tive quantum states, we further demonstrated the validity and
+accuracy of the proposed method.
+The structure of the paper is as follows.
+In Sec.II, we
+introduce a non-Hermitian N-body interacting boson system
+which is equivalent to a nonlinear nonreciprocal two-level
+arXiv:2301.00734v1  [quant-ph]  2 Jan 2023
+
+2
+system with periodic driving in the mean-field approxima-
+tion, and discussed the energy spectrum of this two-level sys-
+tem, In Sec.III, the influence of nonlinear strength and non-
+Hermiticity on LZSM interferometry and the self-trapping ef-
+fects has been studied. Under the weak-coupling limit, the
+non-Hermicity does not affect the conditions of destructive
+interference and constructive interference. Finally, the con-
+clusions are summarized in Sec.IV.
+II.
+NONLINEAR NONHERMITIAN TWO-LEVEL MODEL
+The second quantized Hamiltonian of a nonreciprocal
+interacting-boson system is
+ˆH0 = γ
+2(ˆa†ˆa − ˆb†ˆb) + ∆2
+2 ˆa†ˆb + ∆1
+2 ˆaˆb† − c
+4N (ˆa†ˆa − ˆb†ˆb)2, (1)
+where annihilation operators ˆa, ˆb and generation operators
+ˆa†, ˆb† are for the different quantum states that are the left and
+right well in the double-well BEC system. γ = A sin(ωt) + ϵ0
+is the monochromatic driving field with amplitude A, fre-
+quency ω, and offset ϵ0. c is the interaction strength between
+bosons, ∆i (i = 1, 2) is the tunneling amplitude. When the
+total number of bosons N → ∞, all particles are assumed to
+be in the same spin coherent state in the mean-field approx-
+imation [67, 68]. Considering that the quantum states of the
+non-Hermitian system are in a dual Hilbert space to keep the
+normalize condition [50], the selected coherent states need to
+be defined by both left and right states as
+|Ψr
+sc⟩ =
+1
+√
+N!
+(α1ˆa† + β1ˆb†)N|∅⟩,
+|Ψl
+sc⟩ =
+1
+√
+N!
+(α2ˆa† + β2ˆb†)N|∅⟩,
+(2)
+Based on this, we derive the semi-classical Hamiltonian (see
+Appendix. A)
+ˆHM = ⟨Ψl
+sc| ˆH0|Ψr
+sc⟩
+N
+= γ
+2(α1α∗
+2 − β1β∗
+2) + ∆2
+2 α∗
+2β1 + ∆1
+2 α1β∗
+2 − c
+4(β1β∗
+2 − α1α∗
+2)2,
+(3)
+by the dynamical evolution of the semiclassical Hamiltonian
+[67]
+i˙α1 = ∂ ˆHm
+∂α∗
+2
+,
+i˙β1 = ∂ ˆHm
+∂β∗
+2
+,
+(4)
+we can construct the following dimensionless Schr¨odinger
+equation
+i ∂
+∂t
+�
+α1
+β1
+�
+= ˆHmF
+�
+α1
+β1
+�
+,
+(5)
+with the MF Hamiltonian
+ˆHmF =
+� γ
+2 + c
+2(β1β∗
+2 − α1α∗
+2)
+∆1
+2
+∆2
+2
+− γ
+2 − c
+2(β1β∗
+2 − α1α∗
+2)
+�
+,
+(6)
+t3
+t1
+t2
+t3
+t2
+t1
+ωt/π
+(b) ϵ0 = 5
+(a) ϵ0 = 0
+En(t)
+0
+1
+2
+3
+4
+-6
+-4
+-2
+0
+2
+4
+0
+1
+2
+3
+4
+-5
+0
+5
+En(t)
+c/Δ=0
+c/Δ=3
+FIG. 1. Time evolution of the energy levels for different offsets: (a)
+ϵ0 = 0 and (b) ϵ0 = 5, where A = 10, ω = 1 and ∆1∆2 > 0. The
+time-dependent adiabatic energy levels (i.e., ∆ = 1) are shown by the
+red (c = 0) and black (c = 3) dashed lines, while the diabatic energy
+levels (i.e., ∆ = 0 ) are shown by the blue (c = 0) and green (c = 3)
+solid lines.
+and state |ψr⟩ = (α1, β1)T. Therefore, the model Hamiltonian
+under periodic driving can be described by a nonlinear nonre-
+ciprocal two-level Hamiltonian
+ˆH = ∆1 + ∆2
+4
+ˆσx+ ∆1 − ∆2
+4
+i ˆσy+ γ(t) + c(β1β∗
+2 − α1α∗
+2)
+2
+ˆσz (7)
+where ˆσx,y,z are the Pauli matrices, α1, α2, β1, β2 are the prob-
+ability amplitudes. The dynamic equations of the system are
+[50]
+i ∂
+∂t|ψr⟩ = ˆH|ψr⟩,
+i ∂
+∂t|ψl⟩ = ˆH†|ψl⟩,
+(8)
+where ⟨ψl|ψr⟩ = 1 and the quantum states
+|ψr⟩ = α1 |↑⟩ + β1 |↓⟩ ,
+|ψl⟩ = α2 |↑⟩ + β2| |↓⟩
+(9)
+are represented under the diabatic basis {|↑⟩ , |↓⟩} with spin
+eigenstates |↑⟩ and |↓⟩.
+For the adiabatic basis, the left and right instantaneous
+eigenstates of the time-dependent Hamiltonian ˆH are derived
+by[50]
+ˆH|φr
+n⟩ = En|φr
+n⟩,
+ˆH†|φl
+n⟩ = E∗
+n|φl
+n⟩,
+(10)
+where ⟨φl
+m|φr
+n⟩ = δnm (n = 1, 2), the eigenenergies En(t) are
+determined by the quartic equation (see Appendix. B)
+E4+cE3+ 1
+4(c2−γ2−∆1∆2)E2− c∆1∆2
+4
+E− ∆1∆2c2
+16
+= 0. (11)
+By solving equation (11), we draw the energy spectrum of the
+system (7) (see Fig.1 and Fig.2). The two parameters
+∆ ≡
+�
+|∆1∆2|,
+k ≡
+�
+|∆1/∆2|
+(12)
+
+3
+Ep
+t3
+t2
+t1
+t1
+Ep
+(b) ϵ0 = 5
+c/Δ=0
+c/Δ=3
+(a) ϵ0 = 0
+0
+1
+2
+3
+4
+-6
+-4
+-2
+0
+2
+4
+1
+2
+3
+4
+-5
+0
+5
+En(t)
+En(t)
+ωt/π
+t3
+FIG. 2. Time evolution of the energy levels for different offsets: (a)
+ϵ0 = 0 and (b) ϵ0 = 5, where A = 10, ω = 1 and ∆1∆2 < 0. The
+time-dependent adiabatic energy levels (i.e., ∆ = √|∆1∆2| = 1) are
+shown by the red (c = 0) and black (c = 3) dashed lines, while the
+diabatic energy levels (i.e., ∆ = 0 ) are shown by the blue (c = 0) and
+green (c = 3) solid lines.
+are introduced to describe the mean tunneling amplitude and
+the nonreciprocity.
+In the in-phase tunneling case ∆1∆2 > 0 as shown in Fig.1,
+the energy spectrum of the system (7) is the same as the Her-
+mitian Hamiltonian ˆHh = ∆
+2 ˆσx + γ(t)+c(|β|2−|α|2)
+2
+ˆσz. Therefore,
+the Hamiltonian ˆH and quantum states |ψr⟩ of the two non-
+reciprocal systems can be related to the Hermitian system by
+following relation
+ˆHh = ˆS ˆH ˆS −1,
+|ψ⟩ = ˆS |ψr⟩ =
+�
+α1
+kβ1
+�
+.
+(13)
+where ˆS =
+�
+1 0
+0 k
+�
+. Compared with ˆHh, the nonreciproc-
+ity, which only affects the eigenstates of the system, neither
+changes the eigenvalue nor destroys the symmetry of the sys-
+tem. In the anti-phase tunneling case ∆1∆2 < 0 as shown in
+Fig.2 , the non-adiabatic energy levels have a series of de-
+generate points (EPs) when c = 0 (see the crossing points of
+red dash lines in Fig.2, and the imaginary parts of En are not
+shown). Interestingly, when the nonlinearity is added (c � 0),
+the EPs disappear and the near-degenerate regions are formed
+(see the black dashed lines in Fig.2). When considering the
+offset (ϵ0 � 0), the near-degenerate regions disappear near the
+times t
+′
+n =
+t1+t3
+2
++ 2nπ
+ω (with n being an integer), the period
+changes from nπ
+ω to 2nπ
+ω , and the ring energy levels will tend to
+degenerate at times t1 + 2mπ
+ω (with m being an integer) as ϵ0 in-
+creases as shown in Fig.2. Obviously, the nonlinearity affects
+the EPs. By equation (11), En = 0 is the root of the equation
+iff c∆1∆2 = 0. Therefore, the existence of c does not allow the
+existence of EPs in the anti-phase tunneling case ∆1∆2 < 0.
+Next, we analyzed the cases of the existence of real roots of
+0
+1
+2
+3
+4
+5
+c/
+-4
+-2
+-1012
+4
+(t)/
+FIG. 3. Different regions for parameter space of
+c
+∆ and
+γ
+∆ in the
+anti-phase tunneling case. Region I for f( c
+∆, γ
+∆) < 0, Region II for
+γ2
+∆2 > 1 when f( c
+∆, γ
+∆) > 0, Region III for γ2
+∆2 < 1. Naturally, when
+f( c
+∆, γ
+∆) < 0, the inequality γ2
+∆2 > 1 is guaranteed.
+the energy spectrum.
+For the special cases c = 0, the eigenenergies of the system
+are ±
+�
+γ2(t) + ∆1∆2. It is easy to find that the EPs emerge
+at γ2(t) = −∆1∆2 in the anti-phase tunneling case ∆1∆2 < 0.
+For c � 0, the nature (real or not) of the roots of the energy
+equation (11) depend on the sign of
+δ = −c2γ2∆1∆2ξ,
+(14)
+with ξ = ((c2 − γ2 − ∆1∆2)3 − 27c2γ2∆1∆2).
+When δ > 0, there are two real roots and a pair of conjugate
+complex roots. The system will always have real eigenener-
+gies. When δ < 0, the equation has four unequal real roots if
+c2 + 2(∆1∆2 + γ2) and (∆1∆2 + γ2)(2c2 + ∆1∆2 + γ2) are both
+positive. Otherwise, the equation has two pairs of unequal
+conjugate complex roots. Obviously, for the in-phase tunnel-
+ing case ∆1∆2 > 0, there always exists real eigenenergies of
+the system.
+For the anti-phase tunneling case with δ < 0, the conditions
+that the energy equation has real roots can be simply described
+as γ2
+∆2 > 1 in f( c
+∆, γ
+∆) = [( c
+∆)2−( γ
+∆)2+1]3+27( c
+∆)2( γ
+∆)2 < 0. In-
+terestingly, γ
+∆ = ±1 are exactly the tangent lines of f( c
+∆, γ
+∆) =
+0. Therefore, the condition is naturally satisfied (as shown in
+Fig.3), so we get the same conclusion as ∆1∆2 > 0.
+Finally, we consider another two special case: γ = 0 and
+ξ = 0. The energy spectrum are all complex only when δ = 0,
+c(∆1∆2 − γ2) = 0, (∆1∆2 + γ2)(2c2 + ∆1∆2 + γ2) = 0 and
+c2 + 2(∆1∆2 + γ2) < 0. For, c � 0 and ∆1∆2 � 0, these
+conditions cannot be satisfied at the same time.
+In a word, the system will always have real eigen energies.
+These results on the nature of the eigenenergies can be ex-
+plained by the symmetry related to the different types of non-
+reciprocal. For the in-phase tunneling case ∆1∆2 > 0, the
+symmetry of the system is unbroken since the system can be
+transformed into a Hermitian one with ˆS . Therefore, the real
+eigen energies are guaranteed. While it is not a necessary re-
+sult for the anti-phase case ∆1∆2 < 0 . Although the non-
+linearity c makes EPs disappear in the evolution of En, the
+eigenvalues of one energy state are still complex. For these
+two cases, it is inevitable to have different effects on the evo-
+lution of states. So next we will analyze the dynamic evolution
+
+4
+FIG. 4.
+The interference patterns of the population probability
+|α1|2 at time t = 50/∆ as a function of ϵ0/∆ and ω/∆ in the state
+(α1(0), β1(0)) = (0, 1), (α2(0), β2(0)) = (0, 1) with (a) c/∆ = 0,
+∆1∆2 > 0, (b) c/∆ = 1.05, ∆1∆2 > 0, (c) c/∆ = 0, ∆1∆2 < 0,
+and (d) c/∆ = 1.05, ∆1∆2 < 0. The other parameters are chosen
+as k = 2, A/∆ = 2.5. The white area is singular, and |α1|2 tends to
+infinity.
+of the two cases based on the method of the projective Hilbert
+space.
+III.
+NONLINEAR NON-HERMITIAN LZSM
+INTERFEROMETRY
+In the nonlinear Hermitian LZ system, The LZSM inter-
+ference patterns can be destructive or constructive, which are
+determined by the St¨uckelberg phases and the nonlinearity can
+strongly change the features of the LZSM interferometry. As
+shown in Fig. 4, the interference pattern of |α1|2 is axisymmet-
+ric for the linear in-phase tunneling case (c = 0, ∆1∆2 > 0). In
+the nonlinear case (c � 0), the symmetry of the interference
+pattern is destroyed (as shown in Fig. 4b). When c = 0 and
+∆1∆2 < 0, the Eps make the interference patterns divergent
+and form a singular region (white area in Fig. 4c). It is hard
+to study the influence of each parameter on the features of the
+LZSM interferometry. Next, we propose the concept of pro-
+jective Hilbert space (see AppendixC for detail) and find the
+effect of the nonreciprocity k.
+Through equations (8), without losing generality, the quan-
+tum state |ψr⟩ can be defined as
+|ψr⟩ = eµ(t)+iν(t)| ˜ψ⟩ = eµ(t)+iν(t)
+� ˜a
+˜b
+�
+,
+(15)
+with the normalization relation ⟨ ˜ψ| ˜ψ⟩ = 1 (µ and ν are two real
+parameters), where | ˜ψ⟩ =
+� ˜a
+˜b
+�
+is the quantum state in the pro-
+jective Hilbert space. Then, we draw the normalized interfer-
+ence patterns |˜a|2 = |α1|2/(|α1|2+|β1|2) (see Fig.5). Comparing
+with |α1|2, the regulation of the parameters on the |˜a|2 interfer-
+ence pattern are emerge when c = 0. This is because the
+LZSM interference is determined by the St¨uckelberg phases.
+The phases accumulated in the evolution process are retained
+in the quantum states | ˜ψ⟩ in the projective Hilbert space by
+FIG. 5. The interference patterns of the projective state population
+probability |˜a|2 at time t = 50/∆ as a function of ϵ0/∆ and ω/∆ in the
+state (α1(t0), β1(t0)) = (0, 1), (α2(t0), β2(t0)) = (0, 1) in the anti-phase
+tunneling case ∆1∆2 < 0 with (a) c/∆ = 0, k = 2, (b) c/∆ = 1.05, k =
+2, (c) c/∆ = 0, k = 1/2, and (d) c/∆ = 1.05, k = 1/2.
+removing the divergence caused by the non-Hermitian term
+em(t). In Fig.5, when c = 0, the populations of the correspond-
+ing the projective quantum states in the singular region of the
+quantum states are limited to the values affected by the nonre-
+ciprocity k. To further reveal the influence of parameter k, we
+next start from the simplest case with c = 0 and then analyze
+the case with c � 0. Then, we demonstrated the validity and
+accuracy of the proposed method and numerical results in the
+weak-coupling limit.
+A.
+The effect of noncrciprocity and the projective quantum
+states in the linear non-Hermitian system
+Assuming c = 0, the Hamiltonian of the system (7) be-
+comes
+ˆHmF =
+�
+γ
+2
+∆1
+2
+∆2
+2
+− γ
+2
+�
+,
+(16)
+where ∆1∆2 < 0. Consider the quantum state |ψr⟩ = eµ+iν| ˜ψ⟩ =
+eµ+iν
+� ˜a
+˜b
+�
+, and Eq. (8), one can get
+˙µ = − i
+2⟨ ˜ψ| ˆH − ˆH†| ˜ψ⟩,
+˙ν = −1
+2⟨ ˜ψ| ˆH + ˆH†| ˜ψ⟩ + i⟨ ˜ψ| ˙˜ψ⟩,
+(17)
+Substituting Eq.
+(17) and the definition | ˜ψ⟩ =
+� ˜a
+˜b
+�
+≡
+�
+sin θ
+2eiϕ
+cos θ
+2
+�
+into equation (8), we have (see Appendix C for
+
+43
+3
+(a)c/△=0
+(b)c/ △=1
+0
+9
+6
+3
+3
+(c)c/ △=0
+(d)c/ △=1
+0
+-9
+-6
+-3
+0
+3
+6
+9
+-9
+-6
+-3
+0
+3
+6
+Eo/△
+Eo/△2
+.05
+0
+30
+20
+10
+.05
+0
+99
+60.8
+0.63
+3
+(a) c/ △=0
+(b) c
+0
+9
+6
+3
+3
+(c) c/ △=0
+(d) c
+0
+-9
+-6
+-3
+0
+3
+6
+9
+-9
+-6
+-3
+0
+3
+/△
+E0/△
+Eo0.4
+0.2
+/△=1.05
+0
+0.2
+0.1
+△=1.05
+0
+6
+99
+65
+FIG. 6. The dynamical evolution trajectory of the projective right
+quantum state of the system (16) on the Bloch sphere with the dif-
+ferent non-Hermitian: (a) k = 2 and (b) k = 1/2. The numerical
+simulation parameters:
+A
+∆ = 2.5, ϵ0 = 0 and the initial condition is
+(˜a, ˜b) = (0, 1). The z-axis coordinates of the points of the red dashed
+circle on the Bloch sphere are z0 = cos θ0 = 1−k2
+1+k2 .
+details)
+˙θ = −∆1 sin ϕ cos2 θ
+2 − ∆2 sin ϕ sin2 θ
+2,
+˙ϕ = −γ − ∆1
+2 cot θ
+2 cos ϕ + ∆2
+2 tan θ
+2 cos ϕ,
+˙µ = ∆2 − ∆1
+4
+sin θ sin ϕ,
+˙ν = γ
+2 − ∆2
+2 tan θ
+2 cos ϕ.
+(18)
+For ϵ0 = 0, when the time is long enough, the projective state
+will always be on a certain circle (˙θ = 0) of the Bloch sphere
+(see Fig.6). By Eq. (18), we can get the equation of the circle
+where the projective quantum state finally lies. surprisingly,
+we find the correlation between k and θ0 = limt→∞ θ as
+k2 = tan2 θ0
+2 .
+(19)
+Therefore, in combination with Fig.5, we can explain why
+|˜a|2 is limited to a certain value in the singular region.
+B.
+The influence of interaction and non-Hermitian on
+population in the projective Hilbert space
+In the nonlinear Hermitian system[33], i.e ∆ = ∆1 = ∆2,
+when ϵ0 = 0 and A ≪ ω, the population of the system will
+have the self-trapping phase transition and the Josephson os-
+cillation under the different nonlinearities, and the boundary
+line is c/∆ = 2[67, 69]. Based on this, we next study the non-
+linear non-Hermitian LZSM interference patterns for ϵ0 = 0
+with different nonlinearities c, non-Hermitian parameters k
+and mean amplitudes ∆ [see Fig.7 and Fig.9].
+Firstly, we consider the in-phase tunneling case ∆1∆2 > 0,
+where the symmetry of the system is unbroken. For the Her-
+mitian Hamiltonian ˆHh, near the boundary of two different os-
+cillations, the maximum population of the self-trapping region
+is 0.5, and then the amplitude gradually decreases with the in-
+crease of c/∆. The populations of the state for non-Hermitian
+FIG. 7. The nonlinear non-Hermitian LZSM interference patterns
+with different nonlinearities (a) k = 2, and (b) k = 1/2 for weak
+driving at ϵ0 = 0 and the in-phase tunneling case ∆1∆2 > 0: the
+projective population |˜a|2 as a function of ∆/ω and c/ω for A/ω =
+0.05 from the initial time t0 = 0 to t = 2π/ω , The red dashed-dotted
+line (with slope 1/2) is plotted to denote the boundary between the
+different oscillations.
+Hamiltonian ˆH with ∆1 � ∆2 is only different from those for
+the Hermitian Hamiltonian ˆHh in a weight of k as shown in
+Eq. (13). Therefore, we can get |˜a|2 = k2|˜b|2 at the boundary
+similar with the Hermitian case. Therefore, the boundary line
+c/∆ = 2 (red dashed line in Fig.7) between the two regions
+(self-trapping and Josephson oscillation) is the same as that in
+the Hermitian system. The amplitude of the population of the
+projective quantum state is determined by the nonreciprocal k
+as show in Fig.7(a) and (b). Then, we consider the dynamical
+evolution of the projective quantum state near the boundary,
+by Eq. (8) and (15), one can obtain
+˙θr =ImA sin θr − ∆1 sin ϕr cos2 θr
+2 − ∆2 sin ϕr sin2 θr
+2 ,
+˙ϕr = − γ − ReA − ∆1
+2 cot θr
+2 cos ϕr + ∆2
+2 tan θr
+2 cos ϕr,
+˙µr = − ImA
+2
+cos θr + ∆2 − ∆1
+4
+sin θr sin ϕr,
+˙νr =γ
+2 + ReA
+2
+− ∆2
+2 tan θr
+2 cos ϕr.
+(20)
+with the right quantum state |ψr⟩ =
+�
+α1
+β1
+�
+= eµr+iνr � ˜a
+˜b
+�
+=
+eµr+iνr �
+sin θr
+2 eiϕr
+cos θr
+2
+�
+, and
+˙θl = − ImA sin θl − ∆2 sin ϕl cos2 θl
+2 − ∆1 sin ϕl sin2 θl
+2 ,
+˙ϕl = − γ − ReA − ∆2
+2 cot θl
+2 cos ϕl + ∆1
+2 tan θl
+2 cos ϕl,
+˙µl =ImA
+2
+cos θl + ∆1 − ∆2
+4
+sin θl sin ϕl,
+˙νl =γ
+2 + ReA
+2
+− ∆1
+2 tan θl
+2 cos ϕl.
+(21)
+with the left quantum state |ψl⟩ =
+�
+α2
+β2
+�
+= eµl+iνl � ˜al
+˜bl
+�
+=
+eµl+iνl �
+sin θl
+2 eiϕl
+cos θl
+2
+�
+, where A ≡ c(α1α∗
+2 − β1β∗
+2). By numerical
+simulation, we give the dynamical evolution of the projective
+right state on the Bloch sphere near the boundary c/∆ = 2 in
+Fig.8.
+
+1/2XZ
+(a) k=2
+(b) k=13
+5
+(a) k=2
+0
+0
+50.5
+0
+01013
+5
+(b) k=1/2
+0
+0
+50.5
+0
+0106
+FIG. 8. The dynamics of the projective states represented by the
+trajectories spherical coordinates (θ, φ) on the Bloch sphere in the
+in-phase tunneling case ∆1∆2 > 0 with different strengths of nonlin-
+earity and nonreciprocity: (a) c/∆ = 1.9, k = 2, (b) c/∆ = 2, k = 2,
+(c) c/∆ = 2.1, k = 2, (d) c/∆ = 1.9, k = 1/2, (e) c/∆ = 2, k = 1/2,
+and (f) c/∆ = 2.1, k = 1/2. The other parameters are chosen as
+A
+ω = 0.05, ϵ0 = 3, and the initial state is (˜a, ˜b) = (0, 1). The z-
+axis axis coordinates of the red dashed circle on the Bloch sphere
+are z0 = cos θ0 = 1−k2
+1+k2 , and the z-axis axis coordinates of the green
+dashed circle on the Bloch sphere are z
+′
+0 = 0.
+When c/∆ > 2, the projective states can only evolve on
+the surface of the Bloch sphere above the red dashed circle as
+shown in Fig. 8 (b), (c), (e) and (f). The red circle represent
+the projective states of which the relative population differ-
+ence |˜b|2 − |˜a|2 is 1−k2
+k2+1 = cos θ0. By |˜a|2 = k2|˜b|2 and the nor-
+malization condition, cos θ0 = |˜b|2 − |˜a|2 labels the boundary
+between the self-trapping region and the Josephson oscilla-
+tion region. As we discussed before, the nonreciprocal k does
+not affect the constructive interference and destructive inter-
+ference, but affects the the relative population difference of
+the state. When k is larger, the relative population difference
+at the boundary between the two regions are smaller [see the
+red circle in Fig. 8(a-c) and (d-f)] and the projective popula-
+tion probability |˜a|2 are smaller [see Fig. 7 (a) and (b)].
+For
+the anti-phase tunneling case ∆1∆2 < 0, because of the exis-
+tence of EPs in the linear case c = 0, the projective quantum
+states reaches self-trapping region no matter how weak the
+nonlinearity is. The trajectories of the projective states on the
+Bloch sphere will always above the red dashed circles which
+label the boundaries between the self-trapping region and the
+Josephson oscillation region as shown in Fig.9. the maximum
+population of the projective quantum state is still affected by
+the nonreciprocity k as shown in Eq. (19) and Fig.10(a-d).
+FIG. 9. The nonlinear non-Hermitian LZSM interference patterns
+with different nonlinearities (a) k = 2, and (b) k = 1/2 for weak
+driving at ϵ0 = 0 and the anti-phase tunneling case ∆1∆2 < 0: the
+projective population |˜a|2 as a function of ∆/ω and c/ω for A/ω =
+0.05 from the initial time t0 = 0 to t = 2π/ω.
+FIG. 10. The dynamics of the projective states represented by the tra-
+jectories spherical coordinates (θ, φ) on the Bloch sphere in the anti-
+phase tunneling case ∆1∆2 < 0 with different strengths of nonlinear-
+ity and nonreciprocity: (a) c/∆ = 0.1, k = 2, (b) c/∆ = 1, k = 2, (c)
+c/∆ = 0.1, k = 1/2, and (d) c/∆ = 1, k = 1/2. The other parameters
+are chosen as A
+ω = 0.05, ϵ0 = 3, and the initial state is (˜a, ˜b) = (0, 1).
+The z-axis coordinates of the red dashed circle on the Bloch sphere
+are z0 = cos θ0 = 1−k2
+1+k2 , and the z-axis coordinates of the green dashed
+circle on the Bloch sphere are z
+′
+0 = 0.
+Compare Fig Fig.10(b) and (d) with Fig.10(a) and (c), it is
+easy to find that the stronger the nonlinearity, the stronger the
+degree of self-trapping effect.
+C.
+Weak-coupling limit of the projective quantum states:
+∆ ≪ ω
+When the weak-coupling limit is considered, the adia-
+batic energy levels will be difficult to transition in the near-
+degenerate region. However, in this approximation, we only
+make |˜ag(t)|2 ∼ |˜ag(t0)|2 and |˜bg(t)|2 ∼ |˜bg(t0)|2 where g = r, l.
+Assuming that the initial condition is (˜ag(t0), ˜bg(t0)) = (0, 1),
+the quantum state can always be written in the following form:
+|ψg(t)⟩ = eµg(t)+iνg(t)
+�
+0
+1
+�
+,
+(22)
+
+0.83
+5
+(a) k=2
+0
+0
+50.6
+0.4
+0.2
+0
+10100.23
+5
+A
+(b) k=1/2
+0
+0
+50.1
+0
+10101(c) c/△=0.1
+(d) c/ △=1(a) c/△=0.1
+(b) c/ △=
+Z↑ Z
+X(c) c/△=2.1
+(d) c/ △=1.
+(e) c/ △=2
+(f) c/ △=2.9(a) c/△=1.9
+(b) c/ △=27
+0
+0.02
+0.04
+0
+2
+4
+10-3
+0
+0.02
+0.04
+0
+1
+2
+10-3
+0
+5
+10
+0
+1
+2
+10-4
+0
+5
+10
+0
+1
+2
+10-3
+exact
+approximate
+(a) c/ =0
+(b) c/ =0.5
+(d) c/ =0
+(e) c/ =0.5
+(c) c/ =1
+(f) c/ =1
+FIG. 11. Time evolution of the projective population probability |˜a|2
+for weak coupling in the in-phase tunneling case ∆1∆2 > 0, with
+different nonlinearities: (a) c/ω = 0, k = 2, (b) c/ω = 0.5, k = 2 and
+(c) c/ω = 1, k = 2. (d) c/ω = 0, k = 1/2, (e) c/ω = 0.5, k = 1/2
+and (f) c/ω = 1, k = 1/2. The other parameters are A/ω = 10.5,
+∆/ω = 0.05, and ϵ0/ω = 3.
+where g = r, l. By Eqs. (8),(17) and (22), we get ˙µr(t)+i˙νr(t)+
+˙µl(t) − i˙νl(t) = 0. This means
+β1(t)β∗
+2(t) − α1(t)α∗
+2(t) ∼ β1(t0)β∗
+2(t0) − α1(t0)α∗
+2(t0),
+(23)
+Based on this approximation, we can transform the dynamic
+of the system from Schr¨odinger picture to Dirac picture by in-
+troducing the gauge transformation φr(t) = U(t)ϕr(t) [U(t) =
+ϵ0
+2t − A cos(ωt)
+2ω
++ c
+2(β1β∗
+2 − α1α∗
+2) with ϕr(t) = [˜α1, ˜β1]T ] [33].
+Under the new basis, the nonlinear dynamic Eqs. (8) become
+(Assuming ∆1 > 0):
+i ∂
+∂t
+� ˜α1
+˜β1
+�
+=
+�
+0
+kΩ
+(−1)j
+k Ω∗
+0
+� � ˜α1
+˜β1
+�
+,
+(24)
+and
+i ∂
+∂t
+� ˜α2
+˜β2
+�
+=
+�
+0
+(−1)j
+k Ω∗
+kΩ
+0
+� � ˜α2
+˜β2
+�
+(25)
+with
+Ω = ∆
+2 eiΦ(t),
+Φ(t) = ϵ0t − A cos(ωt)
+ω
++ ct,
+(26)
+and j = 1, 2 corresponding to the anti-phase case ∆2 < 0
+and in-phase case ∆2 > 0, respectively. Ω denotes the field-
+induced Rabi frequency where Φ(t) is the relative phase of
+two diabatic energy levels. The nonreciprocity k in front of
+0
+0.02
+0.04
+0
+2
+4
+10-3
+0
+5
+10
+0
+1
+2
+10-3
+0
+5
+10
+0
+1
+2
+10-4
+(d) c/ =0.5
+(b) c/ =0.5
+(c) c/ =0
+(a) c/ =0
+exact
+approximate
+FIG. 12. Time evolution of the Projective quantum state population
+probability |˜a|2 for weak coupling in the anti-phase tunneling case
+∆1∆2 < 0, with different nonlinearities: (a) c/ω = 0, k = 2 and (b)
+c/ω = 0.5, k = 2. (c) c/ω = 0, k = 1/2 and (d) c/ω = 0.5, k = 1/2.
+The other parameters are A/ω = 10.5, ∆/ω = 0.05, and ϵ0/ω = 3.
+Ω correspond to the weight of the populations of the projec-
+tive quantum state. Thus, we can understand the fact that the
+maximums value of the populations under the self-trapping
+regions change with k2 in the in-phase case ∆1∆2 > 0. In a
+full cycle, Φ(t) can be approximately written as
+Φ(t) ⋍
+� t3
+t1
+(ϵ0 + c − nω)dt = 2π
+ω (ϵ0 + c − nω)
+(27)
+with n = 0, ±1, ±2, .... When Φm = 2mπ, i.e. c + ϵ0 ≃ (n +
+m)ω = dω (m, d = 0, ±1, ±2, ...), the patterns are constructive.
+While, the patterns will be destructive when Φm = (2m+ 1
+2)π,.
+By calculating the nonlinear equation (8), the linear equa-
+tion(24), we can get the exact solution and approximate solu-
+tion respectively. In Fig.11, we show multi-period LZSM in-
+terference fringes with different characteristics in the in-phase
+tunneling case ∆2 > 0. when c = 0, 1, i.e., Φm = 2mπ,
+the patterns are constructive, and when c = 0.5, 1.5, i.e.,
+Φm = (2m + 1
+2)π, the patterns are destructive. In all non-
+linear cases, the two are consistent. In Fig.12, we show the
+anti-phase tunneling case ∆2 < 0. Like the in-phase tunneling
+case, the constructive interference and destructive interference
+only depend on m, and the nonreciprocity k only affect the
+maximal value of the projective population probability |˜a|2.
+IV.
+CONCLUSION
+In this work, we have studied the non-Hermitian nonlin-
+ear LZSM interferometry in which the non-Hermicity is from
+the nonreciprocal tunnelings between the bosons. By using
+the mean-field approximation and projective Hilbert space,
+the effect of nonreciprocity and nonlinearity on the energy
+
+8
+spectrum, the dynamics, and the formation of the interfer-
+ence fringes have been studied. The results show that dif-
+ferent types of reciprocity correspond to different types of
+symmetries of the system. For the in-phase tunneling case
+∆1∆2 > 0, the system can be transformed into a Hermitian one
+with a nonunitary transformation. It has the same energy spec-
+trum and boundary between the Josephson region and the self-
+trapping region as the Hermitian one. While it is not a neces-
+sary result for the anti-phase case ∆1∆2 < 0. The EPs can only
+exist in its linear case c = 0 and the eigenvalues of one en-
+ergy state will be complex in its nonlinear case. There is only
+a self-trapping region in this case since the evolution of the
+projective states will always be above the boundary when the
+nonlinearity exists. For the LZSM interferometry, the strength
+of the nonreciprocity k is found to take an essential role in the
+population of the projective state and determine the maximal
+values and strengths of the interference patterns in the projec-
+tive space. Finally, under the weak-coupling approximation,
+we found that the types and strengths of the nonreciprocity do
+not affect the conditions of destructive and constructive inter-
+ference. It only depends on the strength of nonlinearity. Our
+result provides a possible way to study the parameters of a
+non-Hermitian nonlinear two-level system and its related ex-
+ternal fields by the LZSM interferometry.
+ACKNOWLEDGMENTS
+We thank S. C. Li and F. Q. Dou for their helpful discus-
+sions. This work is supported by the National Natural Sci-
+ence Foundation of China (NSFC) (Grants Nos. 11875103,
+12147206, 11725417, 12088101, 12047548, and U1930403),
+and Science Challenge Project (Grant No. TZ2018005)).
+Appendix A: Semi-classical Hamiltonian
+In the non-Hermitian system, let ˆH be a non-Hermitian Hamiltonian with a complete biorthonormal eigenbasis {|ψr
+n⟩, |ψl
+n⟩},
+the orthogonal normalization of the quantum states are
+⟨ψr
+n|ψl
+m⟩ = δnm.
+(A1)
+Similarly, for system (1), in the mean-field approximation, the coherent state should be written as
+|Ψr
+sc⟩ =
+1
+√
+N!
+(α1ˆa† + β1ˆb†)N|∅⟩,
+(A2)
+|Ψl
+sc⟩ =
+1
+√
+N!
+(α2ˆa† + β2ˆb†)N|∅⟩,
+(A3)
+According to the normalization condition ⟨Ψl
+sc|Ψr
+sc⟩ = 1:
+α1α∗
+2 + β1β∗
+2 = 1.
+(A4)
+Then, applying the Hamiltonian of system (1) to the right quantum state |Ψr
+sc⟩ , one can obtain
+ˆH|ψr
+SC⟩ =
+�γ
+2 ˆa†ˆa − ˆb†ˆb + ∆2
+2 ˆa†ˆb + ∆1
+2 ˆaˆb† − c
+4N (ˆa†ˆa − ˆb†ˆb)2)
+�
+1
+√
+N!
+N
+�
+r=0
+Cr
+N(α1ˆa†)N−r(β1ˆb†)r|∅⟩,
+(A5)
+When calculating the expectation value of an observable, the quantum states of the systems are normalized. So in the system
+(1), the expectation value of ˆH0 should be written as
+⟨Ψl
+sc| ˆH0|Ψr
+sc⟩ =Nγ
+2
+N
+�
+r=0
+(N − 1)!
+(N − r − 1)!r!(α1α∗
+2)N−r−1(β1β∗
+2)rα1α∗
+2 − Nγ
+2
+N
+�
+r=0
+(N − 1)!
+(N − r)!(r − 1)!(α1α∗
+2)N−r(β1β∗
+2)r−1β1β∗
+2
++N(∆2
+2
+N
+�
+r=0
+Cr
+N−1(N − r)(α1α∗
+2)N−r−1(β1β∗
+2)rα∗
+2β1 + ∆1
+2
+N
+�
+r=0
+Cr−1
+N−1r(α1α∗
+2)N−r(β1β∗
+2)r−1α1β∗
+2)
++
+N
+�
+r=0
+Cr−1
+N−1r(α1α∗
+2)N−r(β1β∗
+2)r−1α1β∗
+2) − cN
+4 (β1β∗
+2 − α1α∗
+2)2
+=Nγ
+2 (α1α∗
+2 − β1β∗
+2) + N∆2
+2 (α∗
+2β1) + N∆1
+2 (α1β∗
+2) − cN
+4 (β1β∗
+2 − α1α∗
+2)2,
+(A6)
+The expectation value of each particle is
+ˆHM = ⟨Ψl
+sc| ˆH0|Ψr
+sc⟩
+N
+= −c
+4(β1β∗
+2 − α1α∗
+2)2 + ∆2
+2 (α∗
+2β1) + ∆2
+2 (α1β∗
+2) + γ
+2(α1α∗
+2 − β1β∗
+2).
+(A7)
+
+9
+Appendix B: Derivation of the Energy level equation
+In the non-Hermitian system, the Hamiltonian ˆH has a complete biorthonormal eigenbasis {|ψr
+n⟩, |ψl
+n⟩} of satisfying
+ˆH|φr
+n⟩ = En|φr
+n⟩,
+(B1)
+ˆH†|φl
+n⟩ = E∗
+n|φl
+n⟩,
+(B2)
+⟨φl
+m|φr
+n⟩ = δmn,
+(n = 1, 2, ...)
+(B3)
+By equations (B1), we can naturally conclude that the adiabatic basis of the system (7) satisfies
+Fα1 + i∆
+2 β1 = Eα1,
+i∆
+2 α1 − Fβ1 = Eβ1,
+(B4)
+F∗α2 − i∆
+2 β2 = E∗α1,
+− i∆
+2 α2 − F∗β2 = E∗β2,
+(B5)
+α1α∗
+2 + β1β∗
+2 = 1.
+(B6)
+where F ≡ γ
+2 + c
+2(β1β∗
+2 − α1α∗
+2). To derive non-trivial solutions of Eqs. (B1) and (B2), we must ensure that | ˆH − E ˆI| = 0 and
+| ˆH† − E∗ ˆI| = 0 (ˆI is an identity matrix). Namely,
+E2 − F2 + ∆2
+4 = 0,
+(B7)
+E∗2 − F∗2 + ∆2
+4 = 0,
+(B8)
+By (B4) and the complex conjugate of Eq. (B5), we have
+α1α∗
+2
+β1β∗
+2
+= −4(E + F)2
+∆2
+,
+(B9)
+By the normalization (B6) and Eq. (B7), it becomes
+β1β∗
+2 = E − F
+2E
+,
+(B10)
+Therefore,
+F ≡ γ
+2 + c
+2(β1β∗
+2 − α1α∗
+2) = γ
+2 − cF
+2E .
+(B11)
+Substitute Eq. (B11) into Eq. (B7), we finally have
+E4 + cE3 + 1
+4(c2 − γ2 + ∆2)E2 + c∆2
+4 E + ∆2c2
+16
+= 0.
+(B12)
+Appendix C: The projective space for non-Hermitian quantum system
+Consider the following Schr¨odinger equation
+i d
+dt|ψ(t)⟩ = ˆH|ψ(t)⟩,
+(C1)
+
+10
+where ˆH is generally a non-Hermitian Hamiltonian. Let us define |ψ(t)⟩ = eµ+iν| ˜ψ(t)⟩ with the normalization relation ⟨ ˜ψ(t)| ˜ψ(t)⟩ =
+1 (µ and ν are two real parameters). From Eq. (C1) and its Hermitian conjugation, one can get
+˙µ = − i
+2⟨ ˜ψ| ˆH − ˆH†| ˜ψ⟩,
+(C2)
+and
+˙ν = −1
+2⟨ ˜ψ| ˆH + ˆH†| ˜ψ⟩ + i⟨ ˜ψ| ˙˜ψ⟩.
+(C3)
+One has to keep mind that the above deduction is some different from what had been done by using adjoint equation of (C1).
+In quantum theory with Hermitian Hamiltonian systems, |ψ(t)⟩ and | ˜ψ(t)⟩ are equivalence, since the time evolution is unitary
+(probability preserving) and they are only different in a global phase. Under this equivalence, | ˜ψ(t)⟩ can be employed as a vector
+on so-called projective Hilbert space of the system. However, for a system with a non-Hermitian Hamiltonian, the time evolution
+is not unitary. Hence, though the state vectors only differ in norms, they may describe different system states. Nevertheless, we
+can still formally set up the projective Hilbert space for a non-Hermitian system by using | ˜ψ(t)⟩ as a state on it.
+Based on the above definition, from Eqs. (C2) and (C3), we can see that one can obtain the norm increment and the global
+phase of the state acquiring in its time evolution only from the trace in the projective space, the latter is as the same as for
+Hermitian systems. The global phase and its relation with the projective Hilbert space plays significant role in geometric
+(topology) properties of Hermitian quantum systems. Therefore, it may be interesting to study the geometric properties of a
+non-Hermitian system in such a point of view.
+In order to show such discussions clearly, we employ a two-level system, describing physics of two coupled sites with gain
+and loss, of which the counterpart Hermitian system also plays a role in illustrating the geometric properties of quantum systems.
+The time evolution of such a two-level system is described by a 2 × 2 matrix Hamiltonian system by the following equation,
+i d
+dt
+�
+a
+b
+�
+=
+�
+H11 H12
+H21 H22
+� �
+a
+b
+�
+,
+(C4)
+Then following the definition |ψ(t)⟩ = eµ+iν| ˜ψ(t)⟩, one can get
+d
+dt(iµ − ν)˜a + i d
+dt ˜a = H11˜a + H12˜b,
+(C5)
+d
+dt(iµ − ν)˜b + i d
+dt
+˜b = H21˜a + H22˜b,
+(C6)
+Combining with their complex conjugations, and considering |˜a|2 + |˜b|2 = 1, we can easily verify the equations (C2) and (C3).
+For convenience and without losing generality, we then construct the vector in the projective space for a state |ψ(t)⟩ =
+�
+a
+b
+�
+with | ˜ψ(t)⟩ =
+� ˜aeiϕ
+˜b
+�
+, ˜a =
+a
+√
+|a|2+|b|2 , ˜b =
+b
+√
+|a|2+|b|2 , and ϕ = arg(a) − arg(b). By denoting z = |b|2 − |a|2 which is just the relative
+population difference of the two levels, it then can be mapped to a sphere, the so-called Bloch sphere, with the coordinates (ϕ, z).
+From Eq. (C3), we can obtain the evolution of the total phase
+d
+dtβ = −1/2⟨ ˜ψ| ˆH + ˆH†| ˜ψ⟩ + 1/2(1 − z)dϕ
+dt .
+(C7)
+This equation is the same as what had been obtained for Hermitian systems by Aharonov and Anandan excepting that in the
+dynamic part Hermitian Hamiltonian ˆH is replaced by ( ˆH + ˆH†)/2. The second part in the right hand of the above equation
+is known as the geometric part. One can easily prove that, if the trace of the evolution is closed in the projective space, the
+geometric phase just equals to the half of solid angle of the close path on the Bloch sphere, which is just the so-called AA phase,
+the geometric phase of cyclic state.
+[1] L. D. Landau, Phys. Z. Sowjetunion 2 , 46 (1932).
+[2] C. Zener and R. H. Fowler, Proc. R. Soc. Lond. A 137, 696
+
+11
+(1932).
+[3] E. C. G. Stueckelberg, Helv. Phys Acta 5, 369 (1932).
+[4] L. D. Landau, Phys. Z. Sowjetunion 1 , 88 (1932).
+[5] S. Shevchenko, S. Ashhab, and F. Nori, Physics Reports 492, 1
+(2010).
+[6] B. T. Torosov and N. V. Vitanov, Phys. Rev. A 96, 013845
+(2017).
+[7] G. Cao, H. O. Li, T. Tu, L. Wang, C. Zhou, M. Xiao, G. C. Guo,
+H. W. Jiang, and G. P. Guo, Nat. Commun. 4, 1401 (2013).
+[8] F. Forster, G. Petersen, S. Manus, P. H¨anggi, D. Schuh,
+W. Wegscheider, S. Kohler, and S. Ludwig, Phys. Rev. Lett.
+112, 116803 (2014).
+[9] P. F¨oldi, M. G. Benedict, J. M. Pereira, and F. M. Peeters, Phys.
+Rev. B 75, 104430 (2007).
+[10] C. Calero, E. M. Chudnovsky, and D. A. Garanin, Phys. Rev.
+B 72, 024409 (2005).
+[11] B. K. Cooper and V. M. Yakovenko, Phys. Rev. Lett. 96, 037001
+(2006).
+[12] A. Banerjee and V. M. Yakovenko, Phys. Rev. B 78, 125404
+(2008).
+[13] M. Mark, T. Kraemer, P. Waldburger, J. Herbig, C. Chin, H.-C.
+N¨agerl, and R. Grimm, Phys. Rev. Lett. 99, 113201 (2007).
+[14] L. Du, M. Wang, and Y. Yu, Phys. Rev. B 82, 045128 (2010).
+[15] Q. Niu, X.-G. Zhao, G. A. Georgakis, and M. G. Raizen, Phys.
+Rev. Lett. 76, 4504 (1996).
+[16] O. Morsch, J. H. M¨uller, M. Cristiani, D. Ciampini, and E. Ari-
+mondo, Phys. Rev. Lett. 87, 140402 (2001).
+[17] Y. A. Chen, S. D. Huber, S. Trotzky, I. Bloch, and E. Altman,
+Nat. Phys. 7, 61 (2011).
+[18] M. Cristiani, O. Morsch, J. H. M¨uller, D. Ciampini, and E. Ari-
+mondo, Phys. Rev. A 65, 063612 (2002).
+[19] Q. Zhang, P. H¨anggi, and J. Gong, Phys. Rev. A 77, 053607
+(2008).
+[20] C. S. E. van Ditzhuijzen, A. Tauschinsky, and H. B. van Linden
+van den Heuvell, Phys. Rev. A 80, 063407 (2009).
+[21] J. Liu, L. Fu, B.-Y. Y. Ou, S.-G. G. Chen, D.-i. I. Choi, B. Wu,
+and Q. Niu, Phys. Rev. A 66, 1 (2002), 0105140.
+[22] S.-C. Li, L.-B. Fu, W.-S. Duan, and J. Liu, Phys. Rev. A 78,
+063621 (2008).
+[23] L.-B. Fu, D.-F. Ye, C. Lee, W. Zhang, and J. Liu, Phys. Rev. A
+80, 013619 (2009).
+[24] D.-F. Ye, L.-B. Fu, and J. Liu, Phys. Rev. A 77, 013402 (2008).
+[25] S.-C. Li, Journal of Physics B: Atomic, Molecular and Optical
+Physics 43, 205303 (2010).
+[26] S.-C. Li and L.-B. Fu, Phys. Rev. A 102, 033323 (2020); Phys.
+Rev. A 101, 023618 (2020); Phys. Rev. A 102, 033313 (2020).
+[27] J. Liu, L. Fu, B.-Y. Ou, S.-G. Chen, D.-I. Choi, B. Wu, and
+Q. Niu, Phys. Rev. A 66, 023404 (2002).
+[28] G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, Phys.
+Rev. A 55, 4318 (1997).
+[29] A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, Phys.
+Rev. Lett. 79, 4950 (1997).
+[30] S. Kohler and F. Sols, Phys. Rev. Lett. 89, 060403 (2002).
+[31] O. V. Ivakhnenko, S. N. Shevchenko, and F. Nori, Physics Re-
+ports 995, 1 (2023).
+[32] B. Wu and Q. Niu, Phys. Rev. A 61, 023402 (2000).
+[33] S.-C. Li, L.-B. Fu, and J. Liu, Phys. Rev. A 98, 013601 (2018).
+[34] R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Mussli-
+mani, S. Rotter, and D. N. Christodoulides, Nat. Phys. 14, 11
+(2018).
+[35] Y. Ashida, Z. Gong, and M. Ueda, Advances in Physics 69, 249
+(2020).
+[36] M. A. Miri and A. Al`u, Science 363, eaar7709 (2019).
+[37] W. Zhu, X. Fang, D. Li, Y. Sun, Y. Li, Y. Jing, and H. Chen,
+Phys. Rev. Lett. 121, 124501 (2018).
+[38] Y. Wu, W. Liu, J. Geng, X. Song, X. Ye, C.-K. Duan, X. Rong,
+and J. Du, Science 364, 878 (2019).
+[39] J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and
+L. Luo, Nat. Commun. 10, 855 (2019), arXiv:1608.05061.
+[40] W. Xiong, Z. Li, Y. Song, J. Chen, G.-Q. Zhang, and M. Wang,
+Phys. Rev. A 104, 063508 (2021).
+[41] W. Xiong, Z. Li, G.-Q. Zhang, M. Wang, H.-C. Li, X.-Q. Luo,
+and J. Chen, Phys. Rev. A 106, 033518 (2022).
+[42] S. Yao and Z. Wang, Phys. Rev. Lett. 121, 086803 (2018).
+[43] C. Yin, H. Jiang, L. Li, R. L¨u, and S. Chen, Phys. Rev. A 97,
+052115 (2018).
+[44] C. H. Lee and R. Thomale, Phys. Rev. B 99, 201103 (2019).
+[45] L. Li, C. H. Lee, and J. Gong, Phys. Rev. Lett. 124, 250402
+(2020).
+[46] X. Huang, C. Lu, C. Liang, H. Tao, and Y. C. Liu, Light Sci.
+Appl. 10 (2021), 10.1038/s41377-021-00464-2.
+[47] C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243
+(1998).
+[48] J. Wong, J. Math. Phys. 8, 2039 (1967).
+[49] F. H. M. Faisal and J. V. Moloney, J. Phys. B 16, 3109 (1983).
+[50] A. Mostafazadeh, J. Math. Phys. 43, 205 (2002); J. Math. Phys.
+43, 2814 (2002); J. Math. Phys. 43, 3944 (2002); J. Math. Phys.
+43, 6343 (2002); J. Math. Phys. 44, 974 (2003); J. Math. Phys.
+45, 932 (2004); Nuclear Physics B 640, 419 (2002).
+[51] C. M. Bender, K. A. Milton, and V. M. Savage, Phys. Rev. D
+62, 085001 (2000).
+[52] C. M. Bender, S. Boettcher, H. Jones, P. N. Meisinger, and
+M. Simsek, Phys. Lett. A 291, 197 (2001).
+[53] C. M. Bender, D. C. Brody, and H. F. Jones, Phys. Rev. Lett.
+93, 251601 (2004); Phys. Rev. D 70, 025001 (2004).
+[54] A. Mostafazadeh, Int. J. Mod. Phys. A 21, 2553 (2006).
+[55] C. M. Bender, V. Branchina, and E. Messina, Phys. Rev. D 85,
+085001 (2012).
+[56] C. M. Bender and K. A. Milton, Phys. Rev. D 57, 3595 (1998).
+[57] P. Dorey, C. Dunning, and R. Tateo, J. Phys. A 34, L391 (2001).
+[58] Y.-G. Miao, H. J. M¨uller-Kirsten, and D. K. Park, Journal of
+High Energy Physics 2003, 038 (2003).
+[59] L. Jin and Z. Song, Phys. Rev. A 80, 052107 (2009); Phys. Rev.
+A 85, 012111 (2012); J. Phys. A 44, 375304 (2011).
+[60] F. Minganti, A. Miranowicz, R. W. Chhajlany,
+and F. Nori,
+Phys. Rev. A 100, 062131 (2019).
+[61] B. Longstaff and E.-M. Graefe, Phys. Rev. A 100, 052119
+(2019).
+[62] E. M. Graefe, H. J. Korsch, and A. E. Niederle, Phys. Rev. Lett.
+101, 150408 (2008).
+[63] B. Longstaff and E.-M. Graefe, Phys. Rev. A 100, 052119
+(2019).
+[64] X. Shen, F. Wang, Z. Li, and Z. Wu, Phys. Rev. A 100, 062514
+(2019).
+[65] W.-Y. Wang, B. Sun, and J. Liu, Phys. Rev. A 106, 063708
+(2022).
+[66] S. Ib´a˜nez and J. G. Muga, Phys. Rev. A 89, 033403 (2014).
+[67] H.-D. Liu, J. Fang, and T.-Y. Zheng, Commun. Theor. Phys.
+68, 439 (2017).
+[68] J. I. Cirac, M. Lewenstein, K. Mølmer, and P. Zoller, Phys. Rev.
+A 57, 1208 (1998).
+[69] W. Wang, L. B. Fu, and X. X. Yi, Phys. Rev. A 75, 045601
+(2007).
+

1tE4T4oBgHgl3EQfaQwc/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:5ac013d166c2dc030d8f7bd9bf82d690a5bf6753853593a747b5b1f1374c952e
+size 177623

29E1T4oBgHgl3EQfAQI2/content/tmp_files/2301.02836v1.pdf.txt

@@ -0,0 +1,1606 @@
+Dynamic Local Feature Aggregation for Learning on Point Clouds
+Zihao Lia, Pan Gaoa, Hui Yuanb, Ran Weic
+aNanjing University of Aeronautics and Astronautics, Nanjing ,China
+bShandong University, Jinan, China
+cScience and Technology on Electro-optic Control Laboratory, Luoyang, China
+Abstract
+Existing point cloud learning methods aggregate features from neighbouring points relying on constructing graph in the
+spatial domain, which results in feature update for each point based on spatially-fixed neighbours throughout layers.
+In this paper, we propose a dynamic feature aggregation (DFA) method that can transfer information by constructing
+local graphs in the feature domain without spatial constraints. By finding k-nearest neighbors in the feature domain,
+we perform relative position encoding and semantic feature encoding to explore latent position and feature similarity
+information, respectively, so that rich local features can be learned. At the same time, we also learn low-dimensional global
+features from the original point cloud for enhancing feature representation. Between DFA layers, we dynamically update
+the constructed local graph structure, so that we can learn richer information, which greatly improves adaptability
+and efficiency.
+We demonstrate the superiority of our method by conducting extensive experiments on point cloud
+classification and segmentation tasks. Implementation code is available: https://github.com/jiamang/DFA.
+Keywords:
+dynamic feature aggregation, point cloud, relative position encoding, semantic feature encoding,
+classification, segmentation
+1. Introduction
+The collection of points that express the spatial distri-
+bution and surface features of the target is called point
+cloud data, which represents the 3D target in an unstruc-
+tured form. The point cloud obtained by combining the
+laser principle and the photography principle mainly con-
+tains three-dimensional position coordinates (X, Y, Z),
+laser reflection intensity and color information (R, G, B).
+Common point cloud data formats include RGB-D dual-
+modality format and Point Cloud space format.
+RGB-
+D dual-modality data records the color information and
+depth information of the surface of the target object. The
+Email addresses: pride_19@163.com (Zihao Li),
+Pan.Gao@nuaa.edu.cn (Pan Gao), huiyuan@sdu.edu.cn (Hui Yuan),
+115946873@qq.com (Ran Wei)
+Point Cloud space format records three-dimensional coor-
+dinates of the sampling points on the surface of the object,
+reflecting the spatial contour information.
+Learning features from point clouds often requires a lot
+of advanced processing.
+Traditional methods proposed
+to solve these problems include capturing the geometric
+characteristics of point clouds by using the hand-crafted
+features [1]. With the breakthrough of convolution neu-
+ral network and deep learning, significantly better perfor-
+mance is achieved in various tasks of point cloud process-
+ing. However, standard deep neural network needs nor-
+mative input data, but the point cloud data does not need
+to be irregular, and operations such as translation and
+rotation will not change its own nature. Some methods
+consider converting to a normative 3D grid and then send
+the grid into the network for training, but it will cause ad-
+Preprint submitted to Journal of LATEX Templates
+January 10, 2023
+arXiv:2301.02836v1  [cs.CV]  7 Jan 2023
+
+ditional memory occupation and information loss. Point-
+net proposed by [2] creates a precedent for learning and
+processing directly on the original point cloud, where the
+multi-layer perceptron is applied to each point.
+However, since Pointnet [2] cannot capture the contex-
+tual information, many recent studies have introduced dif-
+ferent modules to learn more abundant local structures,
+which can be divided into the following categories:
+1)
+Feature update based on constructing graph structure
+[3][4][5][6][7]; 2) Feature pooling based on neighboring
+points [8][9][10][11][12]; 3) Convolution based on a series
+of kernels [13][14][15][16][17][15][18][19]; 4) Learning based
+on attention mechanism [20][21][22][23]. These methods
+have achieved good results in classification and segmen-
+tation, but the construction of local feature learners and
+calculation of attention weight have very expensive com-
+puting cost and memory occupation. In addition, the fea-
+ture extractors proposed by some methods are not efficient
+enough, and there are many parts worth improving.
+The goal of this paper is to design an efficient local
+feature extractor without adding much complexity, and
+then use the learned efficient features to represent objects,
+which will improve the point cloud classification and seg-
+mentation tasks.
+So we propose a dynamic feature ag-
+gregation (DFA) module, which extracts and learns latent
+features by finding k-nearest neighbors in the feature do-
+main, encoding location information and semantic feature
+information simultaneously, and concatenating these two
+parts.
+In the classification and segmentation task, this
+module is stacked to extract rich local features.
+Using
+the network structure like Pointnet [2], we extract low-
+dimensional global features from the initial point cloud,
+and then concatenate them with local features extracted
+by multiple DFAs. Finally, high-dimensional global fea-
+tures are obtained for classification and segmentation. For
+segmentation, we concatenate the high-dimensional global
+features again with local features, and perform the MLP
+operation to predict the category of each point.
+In general, we design an efficient local feature extrac-
+tor that utilizes multi-level and multi-source features to
+effectively characterize objects.
+Multi-level features are
+reflected in that by stacking several layers of DFA, we can
+gradually obtain deeper contextual features. Multi-source
+features are reflected in that we combine multiple types of
+features of location information, feature differences, fea-
+tures themselves, and low-dimensional global features to
+perform deeper and higher-dimensional feature learning.
+In order to test its efficiency, we have done relevant tests
+on the ModelNet40 [24], shapeNet [25] and S3DIS [26]
+datasets. Furthermore, we also do many visualization re-
+sults and ablation experiments. Our main contributions
+are summarized as follows:
+• We propose a new operation DFA, which finds k-
+nearest neighbors in the feature domain to construct
+a local graph structure for feature aggregation at each
+time. The graph between DFA layers is dynamically
+updated, which is more adaptable.
+• In each DFA layer, we can learn rich latent position
+and feature difference information through proposed
+relative position encoding and semantic feature en-
+coding, respectively. To the best of our knowledge,
+simultaneously aggregating the relative position and
+feature information in the feature domain has not
+been studied before.
+• We make full use of the learned local features and low-
+dimensional global features for point cloud classifica-
+tion and segmentation tasks, and test on benchmark
+datasets with outstanding quantitative and qualita-
+tive results.
+2. Related work
+2.1. Voxel-based Network.
+Converting point cloud data into regular voxel structure
+can preserve and express spatial distribution. In 2016, Qi
+2
+
+et al. [27] improved voxel CNN and proposed two differ-
+ent voxel CNN network structures. Afterwards, Tchapmi
+et al. [28] jointly proposed segcloud based on voxel-based
+3D full convolution neural network and point based con-
+ditional random field. Wang et al. [29] proposed O-CNN.
+Its core idea is to use octree to represent 3D shapes, and
+only the sparse octree occupied by the shape boundary
+is subject to CNN operation. In order to effectively en-
+code the distribution of voxel midpoint, Meng et al. [30]
+proposed the voxel variational self encoder network VV-
+net, and the point distribution in each voxel is captured
+by the self encoder. In 2020, Shao et al. [31] proposed
+the data structure of opportunity space hash, designed
+hash2col and col2hash, so that CNN operations such as
+convolution and pooling can be parallelized.
+2.2. View-based Network.
+Usually, the point cloud is projected into the 2D image
+first, and then the 2D CNN is used to extract the image
+features. Due to the limitations of the existing deep learn-
+ing network, this kind of method can only recognize the
+point cloud model from a specific angle. In 2017, Lawin et
+al. [32] generated images with different pitch angles and
+translation distances by controlling the equidistant angle.
+Snapnet-r proposed by Gueery et al. [33] can use 2D im-
+ages and 3D as spatial structure information at the same
+time. The mvpnet proposed by Jaritz et al. [34] in 2019
+can aggregate 2D image features into 3D. The relationship
+network proposed by Yang et al. [35] comprehensively con-
+siders the relationship between different views and regions,
+and also uses the attention mechanism to generate scores
+to reflect the relative discrimination ability of views.
+2.3. Point-based Network.
+Direct processing of point clouds contains complete orig-
+inal information. Qi et al. [2] proposed Pointnet network,
+which is the first deep neural network to directly process
+disordered point clouds. Since it does not consider local
+features, they [36] further proposed Pointnet++ to extract
+local features at multiple levels. Later Atzmon et al. [37]
+proposed point convolution neural network, which uses ex-
+pansion operator and constraint operator to generate con-
+volution. In response to the problem of inflexibility of fixed
+grids, Thomas et al. [19] proposed KPconv, which is lo-
+cated in Euclidean space and is very effective in classifying
+point clouds with different densities. In addition, Point-
+Conv [15] and PointCNN [38] use 3D convolution kernels to
+extract features instead of sharing MLP. The PointConv
+[15] can be extended to deconvolution to achieve better
+segmentation results. And PointCNN [38] introduced the
+x-transform to rearrange the points into a potentially regu-
+lar order, and then use convolution to extract local features
+from the point cloud.
+Graph-based Methods. By constructing a local or global
+graph structure to update delivery messages and learn fea-
+tures. In general, the graph structure of the spatial domain
+relies on finding k-nearest neighbors for message passing,
+and the graph structure of the spectral domain needs to
+be realized by methods such as Laplace matrix spectral
+decomposition and Chebyshev polynomial approximation.
+KCNet [4] defines a point set kernel as a set of learnable
+3D points. It aggregates repetitive features at 3D locations
+on the nearest neighbor graph based on geometric rela-
+tionships and local high-dimensional features measured by
+kernel correlations. Wang et al. [5] proposed DGCNN to
+learn the embedding of edges by constructing local graphs.
+Unlike DGCNN [5], 3DGCN [39] defines learnable ker-
+nels using graph max pooling mechanism, and introduces
+shift invariance and scale invariance into deep learning net-
+works. DeepGCNs [40] uses residual connections and di-
+lated convolutions to train deeper graph structures, and
+experiments confirm the positive effect of depth.
+Transformer-based Methods. Since the great success of
+transformers in the NLP field, a lot of work has also in-
+troduced attention mechanisms to related tasks in point
+clouds recently. PCT [41] adopts a similar architecture to
+3
+
+Concat
+𝑓�′
+Pool
+𝑥��, 𝑓��
+𝑥�, 𝑓�
+𝑥��, 𝑓��
+𝑥��, 𝑓��
+𝑥��, 𝑓��
+𝑥��, 𝑓��
+ℎ���
+ℎ���
+ℎ���
+ℎ���
+ℎ���
+Feature Potential Encoding(FeaPE)
+Concat
+MLP
+shared
+ℎ���
+ℎ���
+ℎ���
+ℎ���
+ℎ���
+ℎ��
+shared
+𝐸��
+ 𝑥�-  𝑥��
+ 𝑥�
+ 𝑥��
+MLP
+𝐸���
+ 𝑥�-  𝑥��
+ 𝑥�
+ 𝑥��
+𝐸��
+ 𝑥�-  𝑥��
+ 𝑥�
+ 𝑥��
+ℎ��
+ℎ��
+Relative position encoding
+ℎ��
+Concat
+(𝑓�- 𝑓��)
+𝑓�
+(𝑓�- 𝑓��)
+𝑓�
+(𝑓�- 𝑓��)
+𝑓�
+ℎ��
+ℎ��
+Semantic feature encoding
+ℎ��
+ℎ��
+ℎ��
+ℎ��
+ℎ��
+ℎ��
+ℎ��
+ℎ��
+ℎ��
+ℎ��
+Figure 1: Illustration of feature extraction by DFA layer. The color closeness represents the adjacent points in the feature domain rather than
+the spatial neighbors. Rich information is obtained through relative position encoding and semantic feature encoding. The edge features of
+each adjacent point are obtained by sharing MLP, and finally the features of the central point are updated by maximum pooling operation.
+The subscript j1 · · · j5 index the feature-domain neighbors for center xi.
+pointnet [2], using neighbor information embedding, and
+improved offset transformer for feature learning, so that it
+has achieved good results in classification and segmenta-
+tion tasks. Similarly, there are also some research works
+based on the pointnet++ [36] network, such as PT [42]
+and BL-Net [43] . The PT [42] proposed by Zhao et al.
+is to add a layer of transformer to extract features after
+each downsampling or upsampling. The transformer has
+been modified to measure the difference between the cor-
+responding channels between two eigenvectors (Q and K).
+BL-Net [43] newly designed position feedback module to
+perform feature-guided point shifting. In addition, Yan et
+al. [44] also used the attention mechanism and proposed
+PointASNL that can effectively process point clouds with
+noise.
+3. Methodology
+Extracting and utilizing effective features is crucial in
+point cloud tasks. We construct a local graph structure
+through dynamic updating, and the information can dif-
+fuse nonlocally in the whole point cloud. Based on the
+graph structure, we explore both the latent location and
+semantic features of different layers. Further, we make full
+use of global features and local features containing detailed
+information. We describe the operation called Dynamic
+Feature Aggregation (DFA) in Section 3.1, and then the
+network structure is introduced in Section 3.2.
+3.1. Dynamic Feature Aggregation
+We
+define
+the
+input
+point
+cloud
+as
+X
+=
+{xi|i = 1, 2, ..., N}
+∈
+RN×3
+with
+the
+corresponding
+features defined as F = {fi|i = 1, 2, ...N} ∈ RN×D. Here
+xi represents the three-dimensional coordinates (x, y, z)
+of the i-th point. As the input point cloud only contain
+three-dimensional coordinates, the geometry coordinates
+can also be regarded as its initial feature.
+When extracting features at each layer, a local graph
+needs to be dynamically constructed, which is defined
+as G = (V, E), where V = {1, 2, ...n} and E ⊆ V × V
+are the vertices and edges, respectively. We construct a
+local graph structure by finding k-nearest neighbors in
+the feature domain, including self-loops.
+Suppose that
+xi is the center point of the graph structure, and then
+N(i) = {j : (i, j) ∈ E} is the neighboring point in the fea-
+ture domain. Specifically, the similarity of features is cal-
+culated and measured in the same way as Euclidean space
+4
+
+DFA
+(64)
+DFA
+(64)
+DFA
+(64)
+DFA
+(64)
+N , 3
+N , 64
+N , 64
+N , 64
+N , 64
+⊕
+N , 1024
+Pool
+1024
+⊕
+N , 1024
+1024
+Pool
+Pointnet(64)
+Pointnet(64)
+N , 1280
+⊕
+repeat
+⊕
+Classification
+Segmentation
+Model Architecture
+⊕
+Categorical 
+vector
+MLP
+MLP
+(N,192)
+(N,256)
+Spatial 
+transform
+(64)
+MLP
+MLP
+(512,256,c)
+(512,256,p)
+Figure 2: DFA based network architectures for classification and segmentation tasks. ⊕ stands for concatenated operations. The spatial
+transformation is designed to compute a 3 × 3 matrix to align the input point cloud to the canonical space. By concatenating local features
+and low-dimensional global features through MLP and max pooling, 1D global descriptors can be generated for classification tasks. For part
+segmentation, we generate 1024-dimensional global features, fuse the category feature vectors, and then concatenate the detailed local features
+again to output the category score of each point through MLP.
+distance in each feature dimension, and the k points with
+the smallest value are selected as the nearest neighbors.
+Then retrieve the 3D coordinates of each nearest neigh-
+bor. Given the input three-dimensional coordinates and
+D-dimensional features, our purpose is to learn and output
+M-dimensional features with the same number of points
+through the DFA layer.
+Because we establish the connection between the center
+point and the surrounding k-nearest neighbors by build-
+ing a local graph structure, so we define the feature of the
+edge as eij = hΘ(fi, fj) , where hΘ : RD × RD → RM
+is a nonlinear function with a set of learnable parameters
+Θ. Finally, we aggregate the edge features of the k near-
+est neighbors along each channel, and obtain the result
+for each center point fi that enters the DFA layer feature
+extraction, which is defined as follows:
+f
+′
+i =
+Π
+j∈N(i)hΘ(fi, fj)
+(1)
+Semantic Feature Encoding. We choose to find k-
+nearest neighbors in the feature domain, which means that
+the points sharing the same class will have high probabil-
+ity to be connected. Then we concatenate the feature of
+the center point and the feature differences with its neigh-
+bors as semantic feature information.
+Because this not
+only includes the features of all the original center points,
+but also transmits information to the surrounding points
+through the feature difference with the neighbors. And we
+define the encoding as follows:
+hfj = fi ⊕ (fi − fj), j ∈ N(i)
+(2)
+Here, ⊕ is the concatenate operation. We calculate and
+concatenate the feature differences and its own features
+along each dimension, aiming to encode semantically sim-
+ilar features and explore their latent information.
+Relative Position Encoding. We first need to store
+the original 3-dimensional position coordinate, and then
+find the latent position information of the corresponding
+nearest neighbors in the feature domain for each center
+point.
+We use the relative position information of the
+neighboring points to encode as follows:
+hxj = MLP(xi⊕xj⊕(xi−xj)⊕ ∥ xi−xj ∥), j ∈ N(i) (3)
+where xi and xj represent the original three-dimensional
+coordinates, (xi − xj) calculate the relative coordinates of
+the center point and the k-nearest neighbors of the fea-
+ture domain , ⊕ is the concatenate operation, and ∥ · ∥
+5
+
+calculates the Euclidean distance between the neighbours
+and center point. Unlike finding the nearest neighbors in
+the space restricted by geometry distance, we can discover
+more latent location information in the feature domain
+that may have similar semantic feature but with larger
+geometry distance.
+When obtaining the position and semantic embedding,
+we can concatenate these two parts first and then extract
+the edge features through the MLP operation:
+hij = MLP(hxj ⊕ hfj), j ∈ N(i)
+(4)
+Finally, we need to consider how to aggregate the fea-
+tures of the neighboring edges, that is Π in (1). We have
+three options for the over-aggregation Π. The first is to
+maximize the pool of edge features learned by all nearest
+neighbors to obtain the features of the center point. The
+second is to add all edge features. The third is to perform
+softmax on the neighbors to obtain a weight coefficient
+Wij, and then multiply it with each edge feature, that
+is, Wij × hij to obtain the attentive edge feature, and fi-
+nally add and update the features of the center point. The
+experimental results show that the first maximum pool-
+ing has the best performance, so we choose the maximum
+pooling to aggregate all edge features.
+3.2. Network Architecture
+We use the proposed DFA layer to design two network
+architectures for the point cloud classification and segmen-
+tation task as shown in Fig. 2. We send the initial point
+cloud into a spatial transformation network similar to the
+Pointnet [2] network. By learning the position information
+of the point cloud itself, we can learn a rotation matrix
+that is most conducive to the classification or segmenta-
+tion.
+The point clouds are multiplied and fed into our
+stacked DFA layer to extract features.
+Local and Global Information Aggregation. Fo-
+cusing only on the global features obtained by pooling on
+each point ignores the local interaction between points.
+Or only focusing on local features of surrounding points
+is one-sided. Therefore, we choose a combination of local
+features and global features to comprehensively learn the
+information contained in the point cloud, so that it can be
+better used in classification and segmentation tasks. Our
+local features are learned by several layers of DFA, and the
+lower-dimensional global features is obtained similarly to
+Pointnet [2] by using shared MLP and max pooling. Our
+ablation experiments have also confirmed that integration
+with global feature is beneficial. On the other hand, we
+set several local features and low-dimensional global fea-
+tures to the same dimension (64) because we think they
+are equally important, which is also confirmed in practice.
+Classification Network.
+Our classification network
+is shown in the upper part of Fig. 2, and the point cloud
+through the spatial transformation network is sequentially
+passed through four DFA to extract local features. The
+input of each layer is the output of the previous layer. We
+concatenate these four local features and the global fea-
+tures extracted from the initial point cloud, and then con-
+vert them to higher dimensions through MLP operations.
+Finally, global features are obtained by max pooling for
+classification prediction.
+Segmentation Network. Our segmentation network
+is similar to the classification network, as shown in the
+lower part of Fig.
+2.
+We pass the transformed point
+cloud through three DFA layers in sequence. The three
+local features and low-dimensional global features are also
+concatenated to obtain a 1024-dimensional global features
+through MLP and max pooling. If it is part segmenta-
+tion, then we add a category feature vector (64). If it is
+semantic segmentation, it will not be added. Finally we
+use the shared MLP to resize the features and predict the
+semantic label for each point.
+Dynamic Graph Update.
+Depending on the spa-
+tial interaction of the point cloud, locally adjacent parts
+can form subsets. However, considering the spatial neigh-
+bors for graph update sometimes leads to failure of fea-
+6
+
+ture aggregation. For example, for the point clouds of air
+plane, the aircraft wing and fuselage are adjacent in space,
+the mutually updated features are useless. So we use the
+point of finding k-nearest neighbors on the feature domain,
+which means that these points can constitute meaningful
+parts. Each time we find neighbors in the feature domain
+to reconstruct the local graph structure. It can be said
+that our graph is dynamically updated, so we can explore
+more latent location information, which is also a limitation
+that cannot be achieved by doing k-nearest neighbors in
+space.
+4. Experiments
+In this section, we evaluate our models using DFA for
+point cloud classification and part segmentation tasks.
+Methods
+Input
+point
+mAcc
+OA
+Pointnet[2]
+xyz
+1k
+86.0
+89.2
+Pointnet++[36]
+xyz
+1k
+-
+90.7
+Pointnet++[36]
+xyz,normal
+5k
+-
+91.9
+SpiderCNN[45]
+xyz,normal
+1k
+-
+92.4
+PointWeb[12]
+xyz,normal
+1k
+89.4
+92.3
+PointCNN[38]
+xyz
+1k
+88.1
+92.2
+DGCNN[5]
+xyz
+1k
+90.2
+92.2
+Point2Sequence[46]
+xyz
+1k
+90.4
+92.6
+FPConv[47]
+xyz,normal
+1k
+-
+92.5
+PointConv[15]
+xyz,normal
+1k
+-
+92.5
+KPConv[19]
+xyz
+6k
+-
+92.9
+Point2Node [48]
+xyz
+1k
+-
+93.0
+PointASNL[44]
+xyz
+1k
+-
+92.9
+PointASNL[44]
+xyz,normal
+1k
+-
+93.2
+PCT[41]
+xyz
+1k
+-
+93.2
+SO-Net[8]
+xyz,normal
+5k
+90.8
+93.4
+BL-Net[43]
+xyz
+1k
+-
+93.5
+AG-conv[49]
+xyz
+1k
+90.7
+93.4
+PointStack[50]
+xyz
+1k
+89.6
+93.3
+Ours(1024 points)
+xyz
+1k
+91.1
+93.6
+Ours(2048 points)
+xyz
+2k
+91.6
+94.0
+Table 1: Classification results on ModelNet40.
+4.1. Classification
+Data. We evaluate our point cloud classification model
+on the ModelNet40 [24] dataset.
+This dataset contains
+12311 mesh CAD models from 40 categories, where 9843
+models are used for training and 2468 models are used for
+testing. We follow the experimental setting of [2]. We uni-
+formly sample 1024 or 2048 points for each model, each
+using only 3D coordinates (x, y, z) as input.
+Data aug-
+mentation operations include point shifting, scaling and
+perturbing of the points.
+Network Configuration. The network architecture is
+shown in Fig. 2. At each layer we recompute the graph
+based on feature similarity. For the 1024 points we set the
+number of nearest neighbors k value to 20, and to maintain
+the same density, we set k to 40 for the 2048 points. We
+use four DFA layers to extract local geometric features and
+a Pointnet-like structure to extract low-dimensional global
+features. These are implemented using fully connected lay-
+ers (64). We connect the extracted multi-layer features to
+obtain 64×5 = 320-dimensional features. Then the global
+features are obtained, and then two fully connected layers
+are used to transform the global features for classification.
+All layers use LeakyReLU and batch normalization. We
+use the SGD optimizer with momentum of 0.9. The initial
+learning rate is 0.1, and the random drop rate of the fully
+connected layer is 0.5 to prevent overfitting. The batch size
+is set to 32. We use Pytorch implementation and train the
+network on two RTX 2080Ti GPUs.
+Results. Table 1 shows the results of the classification
+task, and the evaluation metrics we use on this dataset
+are the average class accuracy and overall accuracy. Our
+network only feeds 3D coordinates into training, which
+contains less raw information, but achieves the best re-
+sults on this dataset.
+The test result of 2048 sampling
+points is better than that of 1024 points, indicating that
+when more original information is included, our network
+can learn more features and have better performance.
+7
+
+PointNet
+DGCNN 
+ AG-conv 
+ ours 
+ground truth
+Figure 3: Visual comparison of four methods for part segmentation.
+Methods
+mIou
+air.
+bag
+cap
+car
+cha.
+ear.
+gui.
+kni.
+lam.
+lap.
+mot.
+mug
+pis.
+roc.
+ska.
+tab.
+NUM
+2690
+76
+55
+898
+3758
+69
+787
+392
+1547
+451
+202
+184
+283
+66
+152
+5271
+Pointnet[2]
+83.7
+83.4
+78.7
+82.5
+74.9
+89.6
+73.0
+91.5
+85.9
+80.8
+95.3
+65.2
+93.0
+81.2
+57.9
+72.8
+80.6
+Pointnet++[36]
+85.1
+82.4
+79.0
+87.7
+77.3
+90.8
+71.8
+91.0
+85.9
+83.7
+95.3
+71.6
+94.1
+81.3
+58.7
+76.4
+82.6
+SO-Net[8]
+84.9
+82.8
+77.8
+88.0
+77.3
+90.6
+73.5
+90.7
+83.9
+82.8
+94.8
+69.1
+94.2
+80.9
+53.1
+72.9
+83.0
+RGCNN[51]
+84.3
+80.2
+82.8
+92.6
+75.3
+89.2
+73.7
+91.3
+88.4
+83.3
+96.0
+63.9
+95.7
+60.9
+44.6
+72.9
+80.4
+DGCNN[5]
+85.2
+84.0
+83.4
+86.7
+77.8
+90.6
+74.7
+91.2
+87.5
+82.8
+95.7
+66.3
+94.9
+81.1
+63.5
+74.5
+82.6
+PCNN[37]
+85.1
+82.4
+80.1
+85.5
+79.5
+90.8
+73.2
+91.3
+86.0
+85.0
+96.7
+73.2
+94.8
+83.3
+51.0
+75.0
+81.8
+3D-GCN[39]
+85.1
+83.1
+84.0
+86.6
+77.5
+90.3
+74.1
+90.9
+86.4
+83.8
+95.3
+65.2
+93.0
+81.2
+59.6
+75.7
+82.8
+PointASNL[44]
+86.1
+84.1
+84.7
+87.9
+79.7
+92.2
+73.7
+91.0
+87.2
+84.2
+95.8
+74.4
+95.2
+81.0
+63.0
+76.3
+83.2
+PRA-Net[52]
+86.3
+84.4
+86.8
+89.5
+78.4
+91.4
+76.4
+91.5
+88.2
+85.3
+95.7
+73.4
+94.8
+82.1
+62.3
+75.5
+84.0
+Ours
+86.0
+85.4
+80.0
+85.8
+80.6
+92.4
+74.1
+92.0
+87.4
+84.6
+95.6
+73.5
+94.4
+83.9
+59.0
+74.0
+83.2
+Table 2: Part segmentation results on ShapeNet dataset. Metric is mIoU(%).
+4.2. Part Segmentation
+Data. We test our model on the ShapeNet dataset [25]
+for point cloud part segmentation. This dataset contains
+16881 shapes in 16 categories, of which 14006 are used
+for training and 2874 are used for testing. There are 50
+parts tags in total, and each model includes 2-6 parts.
+We follow the experimental setup of [2]. 2048 points are
+sampled from each shape, and the input consists only of
+the 3D coordinates.
+Network Configuration. We use three DFA layers to
+extract features, and operate the same as classification to
+obtain 1024-dimensional global features. Following [5], we
+also add a one-hot vector representing the category type
+to each point. Then we concatenate global features and
+category vectors as new global features with 1024 + 64 =
+1088-dimensions.
+We re-concatenate the previous three
+local features and convert them into the features of each
+point through three fully connected layers (512, 256, 128)
+for segmentation. The settings of our training parameters
+are the same as in the classification task, except that the
+batch size is changed to 16.
+Results. We evaluate the performance of part segmen-
+tation by the mIou metric. The Iou of a shape is computed
+by averaging of each part. The mean Iou (mIou) is calcu-
+lated by averaging the Ious of all testing instances. From
+the experimental results in table 2, it can be seen that
+8
+
+Methods
+mAcc
+mIou
+ceiling
+floor
+wall
+beam
+column
+windows
+door
+chair
+table
+bookcase
+sofa
+board
+clutter
+Pointnet[2]
+48.98
+41.09
+88.80
+97.33
+69.80
+0.05
+3.92
+46.26
+10.76
+58.93
+52.61
+5.85
+40.28
+26.38
+33.22
+SEGCloud[28]
+57.35
+48.92
+90.06
+96.05
+69.86
+0.00
+18.37
+38.35
+23.12
+70.40
+75.89
+40.88
+58.42
+12.96
+41.60
+PointCNN[38]
+63.86
+57.26
+92.31
+98.24
+79.41
+0.00
+17.60
+22.77
+62.09
+74.39
+80.59
+31.67
+66.67
+62.05
+56.74
+PointWeb[12]
+66.64
+60.28
+91.95
+98.48
+79.39
+0.00
+21.11
+59.72
+34.81
+76.33
+88.27
+46.89
+69.30
+64.91
+52.46
+SPG[53]
+66.50
+58.04
+89.35
+96.87
+78.12
+0.00
+42.81
+48.93
+61.58
+84.66
+75.41
+69.84
+52.60
+2.10
+52.22
+PCNN[37]
+67.01
+58.27
+92.26
+96.20
+75.89
+0.27
+5.98
+69.49
+63.45
+66.87
+65.63
+47.28
+68.91
+59.10
+46.22
+PCT[41]
+67.65
+61.33
+92.54
+98.42
+80.63
+0.00
+19.35
+61.64
+48.00
+76.58
+85.20
+46.22
+67.71
+67.93
+52.29
+Ours
+67.96
+62.18
+92.68
+98.50
+79.12
+0.05
+36.72
+67.45
+65.18
+75.36
+86.77
+71.52
+52.59
+65.02
+57.12
+Table 3: Semantic segmentation results on S3DIS dataset.
+Pointnet
+DGCNN 
+ ours 
+ ground truth
+Figure 4: Visual comparison of three methods for semantic segmentation.
+in some categories with a small number of samples, the
+segmentation effect is not good due to too few training
+samples. But overall, our method has better performance,
+especially with the highest mIou in many categories such
+as airplane, car, chair, etc. This benefits from these cat-
+egories having sufficient samples so that our network can
+learn rich features for part segmentation tasks.
+Fig.
+3
+shows the visual differences between us and several other
+mainstream methods on some categories. These methods
+are roughly capable of distinguishing different parts of an
+object, and the difference lies in the identification of de-
+tails. Looking closely at the tail section of the airplane,
+the fence section below the chair, the top of the car, and
+the connection between different parts in the guitar, our
+method is closer to the ground truth.
+4.3. Semantic Segmentation
+Data. We further test our model on the Stanford Large-
+Scale 3D Indoor Spaces Dataset (S3DIS) dataset [26] for
+point cloud semantic scene segmentation.
+This dataset
+is taken from 271 rooms in 6 different areas in 3 differ-
+ent buildings.
+The point cloud data of each scene has
+9-dimensional data including xyz three-dimensional coor-
+dinates, RGB color information, and the normalized posi-
+tion coordinates x′y′z′ of each point relative to the room
+where it is located. At the same time, each point cloud in
+the scene is assigned a semantic label from 13 categories
+9
+
+(such as ceiling, table, etc.).
+Network Configuration. Our semantic segmentation
+network configuration is the same as for part segmentation,
+the only difference is that no feature vector is added.
+Results. We divide each room into 1m × 1m blocks and
+sample 4096 points in each block during training. And we
+use area5 as the test set. For evaluation metrics, we use
+mean class accuracy (mAcc) and mean class intersection
+(mIou). The experimental results are shown in the table
+3, and the visualization is shown in the fig. 4.
+4.4. Ablation Studies
+In this subsection, we explore the effect of using different
+choices in the network. The effectiveness of our module
+and parameter selection is demonstrated in these ablation
+experiments.
+Number of neighbors. The k value of constructing
+the local graph structure has a great influence on the ex-
+tracted features. Therefore, it is very important to choose
+an appropriate value of k in the experiment. We conducted
+4 sets of experiments to explore the impact of choosing dif-
+ferent k values on the classification results of 2048 points,
+which is also shown in the table 4. When the value of k is
+10 and 20, the neighborhood of each center point is small
+and cannot fully interact with the neighbor points. Appro-
+priately increasing the value of k can also have room for
+improvement, which also shows that DFA can effectively
+use the features of neighborhood points to learn local fea-
+tures. By further increasing the value of k, it can be found
+that increasing the value of k all the time will not increase
+the accuracy of the model. Because when the value of k
+is too large, there will be many noise points that are very
+different from the center point features, which is useless or
+even burdensome for updating the center point features,
+and will also increase the amount of parameters and net-
+work training time. Choosing a neighbor k value of 40 can
+obtain the best average class accuracy and overall accu-
+racy.
+k
+mAcc
+OA
+10
+90.2
+93.3
+20
+90.8
+93.7
+40
+91.6
+94.0
+60
+91.5
+93.3
+Table 4: Number of neighbors(k)
+Selection of aggregate functions Π. It can be seen in
+many previous works[2][36][41] that some symmetric pool-
+ing functions such as max/sum/mean are often used to
+overcome the disordered characteristics of point clouds.
+In our DFA layer, we also need to aggregate edge features
+to update features for each center point. We experimented
+with different aggregation functions such as max, sum, or
+sum with attention weights which first do softmax on k-
+nearest neighbors dimension to get the attention weights
+and then multiply and accumulate them accordingly. The
+max function is to select the largest feature of points in
+the local neighborhood. The sum function is to add the
+features of all points in the neighborhood, and the mean
+function is to divide by the k value after the sum func-
+tion. Table 5 shows the results of our selection of differ-
+ent aggregation functions on a classification experiment of
+2048 points. Although the maximum pooling function will
+lose the non-largest part of the features, it will retain the
+largest part of the most significant features, and the ex-
+perimental results show that it is the most effective. We
+finally choose the best-performing max function to aggre-
+gate the edge features.
+Π
+mAcc
+OA
+max
+91.6
+94.0
+sum
+90.5
+93.4
+mean
+90.3
+93.2
+attention sum
+91.0
+93.5
+Table 5: Choice of different aggregation functions Π
+10
+
+Feature or space domains. Further, we explore in
+which domain is better to compute k-nearest neighbors,
+i.e., the feature domain or the spatial domain. If we choose
+to do k-nearest neighbors in the spatial domain, it means
+that the graph structure is fixed each time. On the one
+hand, the relative position coding will be the same, on
+the other hand, it is very limited to exchange information
+with fixed neighbor points each time. If we choose to do
+k-nearest neighbors on the feature domain, it means that
+the local graph structure is dynamically updated, and the
+neighbors of the graph are different each time but the fea-
+tures are similar. We can make better use of DFA layers to
+discover efficient features. We choose to compare the ex-
+perimental results in the classification task of 2048 points.
+As can be seen from the table 6, our way of exchanging
+information with neighbor updates in the feature domain
+is better. Because the k-nearest neighbors obtained in this
+way are more homogeneous. Especially for part segmen-
+tation, spatially adjacent points are not necessarily of the
+same class, so it is useless or even redundant to exchange
+information with these points.
+spatial or feature domain
+mAcc
+OA
+feature
+91.6
+94.0
+spatial
+91.1
+93.4
+Table 6: Comparison of k-nearest neighbors in feature domain and
+space.
+Relative position information. By computing the
+k-nearest neighbors of the feature domain, we are able to
+discover latent-location feature information that is not lim-
+ited by space. In this way, the relative position encoding
+in each DFA layer is different because the neighborhood
+points are changing. This allows us to connect points that
+may not be in close spatial locations. So we explore its ef-
+fectiveness by whether incorporating this part in the clas-
+sification task of 2048 points.
+The experimental results
+in table 7 show that adding location information encoding
+can have better performance. This also shows that the po-
+tential position information obtained by relative position
+encoding is crucial.
+Position information
+mAcc
+OA
+w
+91.6
+94.0
+w/o
+90.1
+93.3
+Table 7: Whether to add position information
+Low-dimensional
+global
+features.
+Inspired by
+Pointnet [2] and Pointnet++ [36], it is not advisable to
+only focus on global features or local features, so we adopt
+a fusion of both. Global features can provide overall direc-
+tion control, while local features can provide more detailed
+information. We believe that these are equally important
+in network learning, so after extracting local features of
+different depths, we concatenate these local features and
+low-dimensional global features together through MLP op-
+erations to upgrade to high-dimensional for subsequent
+tasks. To this end, we compare the classification results
+of 2048 points with or without adding low-dimensional
+global features. The table 8 confirms the effectiveness of
+our way of concatenating the learned local features and
+low-dimensional global features.
+Low-global features
+mAcc
+OA
+w
+91.6
+94.0
+w/o
+89.9
+93.1
+Table 8: Whether to add low-dimensional global features
+4.5. Model Complexity
+We use the stat package in pytorch to output some quan-
+titative results of the network model. It includes the total
+number of parameters of the network model, the number
+of floating-point operations required for network opera-
+tion, and the memory occupied by node inference. The
+experimental results are all tested based on the classifi-
+cation model on 1024 points. At the same time, we test
+11
+
+other mainstream methods for comparison as shown in the
+following table 9.
+It can be seen that our model has fewer parameters and
+does not occupy a large amount of memory, indicating that
+our network structure is lightweight, and not complicated
+and easy to implement. In networks based on graph meth-
+ods, the amount of computation is generally too large due
+to the need to interact with neighbors to update features.
+Compared with other methods of this type, our floating-
+point operations are also much less. At the same time the
+performance is still the best.
+Method
+Pparams
+Flops
+Memory
+OA
+Pointnet[2]
+0.7M
+0.5M
+10.5M
+89.2
+Pointnet++[36]
+2.2M
+3.1M
+231.5M
+91.9
+DGCNN[5]
+1.8M
+1.89G
+123.0M
+92.9
+AG-conv[49]
+1.9M
+2.9G
+202.0M
+93.4
+PCT[41]
+2.9M
+2.32G
+187.6M
+93.2
+ours
+1.1M
+2.17G
+154.5M
+93.6
+Table 9: Quantitative evaluation of classification on ModelNet40.
+5. Conclusion
+This paper proposes a new operation for point cloud
+learning and also demonstrates its performance in differ-
+ent tasks. The main contribution of our method is to ag-
+gregate local feature in the feature domain, explore the la-
+tent relative position information and semantic feature in-
+formation, and learn to obtain higher-dimensional features
+by concatenating local features and low-dimensional global
+features. Our DFA can dynamically construct graphs that
+are not spatially correlated and exchange information be-
+tween points with semantically similar features.
+Exper-
+imental results show that our network outperforms the
+state-of-the-art on several public datasets. Further, our
+DFA module is simple and efficient, and can be seamlessly
+integrated into other network models.
+References
+[1] S. Biasotti, A. Cerri, A. Bronstein, M. Bronstein, Recent trends,
+applications, and perspectives in 3d shape similarity assessment,
+in: Computer graphics forum, Vol. 35, Wiley Online Library,
+2016, pp. 87–119.
+[2] C. R. Qi, H. Su, K. Mo, L. J. Guibas, Pointnet: Deep learning on
+point sets for 3d classification and segmentation, in: Proceed-
+ings of the IEEE conference on computer vision and pattern
+recognition, 2017, pp. 652–660.
+[3] C. Wang, B. Samari, K. Siddiqi, Local spectral graph convolu-
+tion for point set feature learning, in: Proceedings of the Euro-
+pean conference on computer vision (ECCV), 2018, pp. 52–66.
+[4] Y. Shen, C. Feng, Y. Yang, D. Tian, Mining point cloud local
+structures by kernel correlation and graph pooling, in: Proceed-
+ings of the IEEE conference on computer vision and pattern
+recognition, 2018, pp. 4548–4557.
+[5] Y. Wang, Y. Sun, Z. Liu, S. E. Sarma, M. M. Bronstein, J. M.
+Solomon, Dynamic graph cnn for learning on point clouds, Acm
+Transactions On Graphics (tog) 38 (5) (2019) 1–12.
+[6] L. Wang, Y. Huang, Y. Hou, S. Zhang, J. Shan, Graph at-
+tention convolution for point cloud semantic segmentation, in:
+Proceedings of the IEEE/CVF conference on computer vision
+and pattern recognition, 2019, pp. 10296–10305.
+[7] J. Liu, B. Ni, C. Li, J. Yang, Q. Tian, Dynamic points agglom-
+eration for hierarchical point sets learning, in: Proceedings of
+the IEEE/CVF International Conference on Computer Vision,
+2019, pp. 7546–7555.
+[8] J. Li, B. M. Chen, G. H. Lee, So-net: Self-organizing network for
+point cloud analysis, in: Proceedings of the IEEE conference on
+computer vision and pattern recognition, 2018, pp. 9397–9406.
+[9] A. Mnih, K. Gregor, Neural variational inference and learn-
+ing in belief networks, in: International Conference on Machine
+Learning, PMLR, 2014, pp. 1791–1799.
+[10] Q. Huang, W. Wang, U. Neumann, Recurrent slice networks for
+3d segmentation of point clouds, in: Proceedings of the IEEE
+conference on computer vision and pattern recognition, 2018,
+pp. 2626–2635.
+[11] Z. Zhang, B.-S. Hua, S.-K. Yeung, Shellnet:
+Efficient point
+cloud convolutional neural networks using concentric shells
+statistics, in: Proceedings of the IEEE/CVF international con-
+ference on computer vision, 2019, pp. 1607–1616.
+[12] H. Zhao, L. Jiang, C.-W. Fu, J. Jia, Pointweb: Enhancing lo-
+cal neighborhood features for point cloud processing, in: Pro-
+ceedings of the IEEE/CVF conference on computer vision and
+pattern recognition, 2019, pp. 5565–5573.
+[13] H. Su, V. Jampani, D. Sun, S. Maji, E. Kalogerakis, M.-H.
+Yang, J. Kautz, Splatnet:
+Sparse lattice networks for point
+12
+
+cloud processing, in: Proceedings of the IEEE conference on
+computer vision and pattern recognition, 2018, pp. 2530–2539.
+[14] B.-S. Hua, M.-K. Tran, S.-K. Yeung, Pointwise convolutional
+neural networks, in: Proceedings of the IEEE conference on
+computer vision and pattern recognition, 2018, pp. 984–993.
+[15] W. Wu, Z. Qi, L. Fuxin, Pointconv: Deep convolutional net-
+works on 3d point clouds, in: Proceedings of the IEEE/CVF
+Conference on Computer Vision and Pattern Recognition, 2019,
+pp. 9621–9630.
+[16] S. Lan, R. Yu, G. Yu, L. S. Davis, Modeling local geometric
+structure of 3d point clouds using geo-cnn, in: Proceedings of
+the IEEE/cvf conference on computer vision and pattern recog-
+nition, 2019, pp. 998–1008.
+[17] A. Komarichev, Z. Zhong, J. Hua, A-cnn: Annularly convolu-
+tional neural networks on point clouds, in: Proceedings of the
+IEEE/CVF conference on computer vision and pattern recog-
+nition, 2019, pp. 7421–7430.
+[18] J. Mao, X. Wang, H. Li, Interpolated convolutional networks
+for 3d point cloud understanding, in:
+Proceedings of the
+IEEE/CVF international conference on computer vision, 2019,
+pp. 1578–1587.
+[19] H.
+Thomas,
+C.
+R.
+Qi,
+J.-E.
+Deschaud,
+B.
+Marcotegui,
+F. Goulette, L. J. Guibas, Kpconv: Flexible and deformable
+convolution for point clouds, in: Proceedings of the IEEE/CVF
+international conference on computer vision, 2019, pp. 6411–
+6420.
+[20] A. Paigwar, O. Erkent, C. Wolf, C. Laugier, Attentional point-
+net for 3d-object detection in point clouds, in: Proceedings of
+the IEEE/CVF Conference on Computer Vision and Pattern
+Recognition Workshops, 2019, pp. 0–0.
+[21] S. Xie, S. Liu, Z. Chen, Z. Tu, Attentional shapecontextnet for
+point cloud recognition, in: Proceedings of the IEEE conference
+on computer vision and pattern recognition, 2018, pp. 4606–
+4615.
+[22] W. Zhang, C. Xiao, Pcan:
+3d attention map learning using
+contextual information for point cloud based retrieval, in: Pro-
+ceedings of the IEEE/CVF Conference on Computer Vision and
+Pattern Recognition, 2019, pp. 12436–12445.
+[23] J. Yang, Q. Zhang, B. Ni, L. Li, J. Liu, M. Zhou, Q. Tian,
+Modeling point clouds with self-attention and gumbel subset
+sampling, in: Proceedings of the IEEE/CVF conference on com-
+puter vision and pattern recognition, 2019, pp. 3323–3332.
+[24] Z. Wu, S. Song, A. Khosla, F. Yu, L. Zhang, X. Tang, J. Xiao,
+3d shapenets: A deep representation for volumetric shapes, in:
+Proceedings of the IEEE conference on computer vision and
+pattern recognition, 2015, pp. 1912–1920.
+[25] L. Yi, V. G. Kim, D. Ceylan, I.-C. Shen, M. Yan, H. Su, C. Lu,
+Q. Huang, A. Sheffer, L. Guibas, A scalable active framework
+for region annotation in 3d shape collections, ACM Transactions
+on Graphics (ToG) 35 (6) (2016) 1–12.
+[26] I. Armeni, O. Sener, A. R. Zamir, H. Jiang, I. Brilakis, M. Fis-
+cher, S. Savarese, 3d semantic parsing of large-scale indoor
+spaces, in: Proceedings of the IEEE conference on computer
+vision and pattern recognition, 2016, pp. 1534–1543.
+[27] C. R. Qi, H. Su, M. Nießner, A. Dai, M. Yan, L. J. Guibas, Vol-
+umetric and multi-view cnns for object classification on 3d data,
+in: Proceedings of the IEEE conference on computer vision and
+pattern recognition, 2016, pp. 5648–5656.
+[28] L. Tchapmi, C. Choy, I. Armeni, J. Gwak, S. Savarese, Seg-
+cloud: Semantic segmentation of 3d point clouds, in: 2017 in-
+ternational conference on 3D vision (3DV), IEEE, 2017, pp.
+537–547.
+[29] P.-S. Wang, Y. Liu, Y.-X. Guo, C.-Y. Sun, X. Tong, O-cnn:
+Octree-based convolutional neural networks for 3d shape analy-
+sis, ACM Transactions On Graphics (TOG) 36 (4) (2017) 1–11.
+[30] H.-Y. Meng, L. Gao, Y.-K. Lai, D. Manocha, Vv-net: Voxel
+vae net with group convolutions for point cloud segmentation,
+in: Proceedings of the IEEE/CVF international conference on
+computer vision, 2019, pp. 8500–8508.
+[31] T. Shao, Y. Yang, Y. Weng, Q. Hou, K. Zhou, H-cnn: Spatial
+hashing based cnn for 3d shape analysis, IEEE transactions on
+visualization and computer graphics 26 (7) (2018) 2403–2416.
+[32] F. J. Lawin, M. Danelljan, P. Tosteberg, G. Bhat, F. S. Khan,
+M. Felsberg, Deep projective 3d semantic segmentation, in: In-
+ternational Conference on Computer Analysis of Images and
+Patterns, Springer, 2017, pp. 95–107.
+[33] J. Guerry, A. Boulch, B. Le Saux, J. Moras, A. Plyer, D. Fil-
+liat, Snapnet-r: Consistent 3d multi-view semantic labeling for
+robotics, in: Proceedings of the IEEE international conference
+on computer vision workshops, 2017, pp. 669–678.
+[34] M. Jaritz, J. Gu, H. Su, Multi-view pointnet for 3d scene un-
+derstanding, in: Proceedings of the IEEE/CVF International
+Conference on Computer Vision Workshops, 2019, pp. 0–0.
+[35] Z. Yang, L. Wang, Learning relationships for multi-view 3d ob-
+ject recognition, in:
+Proceedings of the IEEE/CVF Interna-
+tional Conference on Computer Vision, 2019, pp. 7505–7514.
+[36] C. R. Qi, L. Yi, H. Su, L. J. Guibas, Pointnet++: Deep hierar-
+chical feature learning on point sets in a metric space, Advances
+in neural information processing systems 30.
+[37] M.
+Atzmon,
+H.
+Maron,
+Y.
+Lipman,
+Point
+convolutional
+neural
+networks
+by
+extension
+operators,
+arXiv
+preprint
+arXiv:1803.10091.
+[38] Y. Li, R. Bu, M. Sun, W. Wu, X. Di, B. Chen, Pointcnn: Convo-
+lution on x-transformed points, Advances in neural information
+processing systems 31.
+[39] Z.-H. Lin, S.-Y. Huang, Y.-C. F. Wang, Convolution in the
+13
+
+cloud:
+Learning deformable kernels in 3d graph convolution
+networks for point cloud analysis, in:
+Proceedings of the
+IEEE/CVF conference on computer vision and pattern recog-
+nition, 2020, pp. 1800–1809.
+[40] G. Li, M. Muller, A. Thabet, B. Ghanem, Deepgcns: Can gcns
+go as deep as cnns?, in: Proceedings of the IEEE/CVF inter-
+national conference on computer vision, 2019, pp. 9267–9276.
+[41] M.-H. Guo, J.-X. Cai, Z.-N. Liu, T.-J. Mu, R. R. Martin, S.-M.
+Hu, Pct: Point cloud transformer, Computational Visual Media
+7 (2) (2021) 187–199.
+[42] H. Zhao, L. Jiang, J. Jia, P. H. Torr, V. Koltun, Point trans-
+former, in: Proceedings of the IEEE/CVF International Con-
+ference on Computer Vision, 2021, pp. 16259–16268.
+[43] W. Han, H. Wu, C. Wen, C. Wang, X. Li, Blnet: Bidirectional
+learning network for point clouds, Computational Visual Media
+(2022) 1–12.
+[44] X. Yan, C. Zheng, Z. Li, S. Wang, S. Cui, Pointasnl: Robust
+point clouds processing using nonlocal neural networks with
+adaptive sampling, in: Proceedings of the IEEE/CVF Confer-
+ence on Computer Vision and Pattern Recognition, 2020, pp.
+5589–5598.
+[45] Y. Xu, T. Fan, M. Xu, L. Zeng, Y. Qiao, Spidercnn: Deep
+learning on point sets with parameterized convolutional filters,
+in: Proceedings of the European Conference on Computer Vi-
+sion (ECCV), 2018, pp. 87–102.
+[46] X. Liu, Z. Han, Y.-S. Liu, M. Zwicker, Point2sequence: Learn-
+ing the shape representation of 3d point clouds with an
+attention-based sequence to sequence network, in: Proceedings
+of the AAAI Conference on Artificial Intelligence, Vol. 33, 2019,
+pp. 8778–8785.
+[47] Y. Lin, Z. Yan, H. Huang, D. Du, L. Liu, S. Cui, X. Han, Fp-
+conv: Learning local flattening for point convolution, in: Pro-
+ceedings of the IEEE/CVF Conference on Computer Vision and
+Pattern Recognition, 2020, pp. 4293–4302.
+[48] W. Han, C. Wen, C. Wang, X. Li, Q. Li, Point2node: Correla-
+tion learning of dynamic-node for point cloud feature modeling,
+in: Proceedings of the AAAI Conference on Artificial Intelli-
+gence, Vol. 34, 2020, pp. 10925–10932.
+[49] H. Zhou, Y. Feng, M. Fang, M. Wei, J. Qin, T. Lu, Adaptive
+graph convolution for point cloud analysis, in: Proceedings of
+the IEEE/CVF International Conference on Computer Vision,
+2021, pp. 4965–4974.
+[50] K. T. Wijaya, D.-H. Paek, S.-H. Kong, Advanced feature learn-
+ing on point clouds using multi-resolution features and learnable
+pooling, arXiv preprint arXiv:2205.09962.
+[51] G. Te, W. Hu, A. Zheng, Z. Guo, Rgcnn: Regularized graph cnn
+for point cloud segmentation, in: Proceedings of the 26th ACM
+international conference on Multimedia, 2018, pp. 746–754.
+[52] S. Cheng, X. Chen, X. He, Z. Liu, X. Bai, Pra-net:
+Point
+relation-aware network for 3d point cloud analysis, IEEE Trans-
+actions on Image Processing PP (99).
+[53] L. Landrieu, M. Simonovsky, Large-scale point cloud semantic
+segmentation with superpoint graphs, in: Proceedings of the
+IEEE conference on computer vision and pattern recognition,
+2018, pp. 4558–4567.
+14
+

3NAyT4oBgHgl3EQfo_gV/content/2301.00515v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:38f99e325325ec339db4458a18e9ad613c7b76c40ac403376024ab4f7e4464df
+size 3587107

3NAyT4oBgHgl3EQfo_gV/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:4ea2041feb817da058c01223a475070b60a9c3eb3f8355742636c288ef1c3ea5
+size 5505069

3NFST4oBgHgl3EQfYjhQ/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:37f989248bf0e7d7631889667f317461d99ee6d1f09c4d31ff451edb81b5236b
+size 3473453

3tAyT4oBgHgl3EQfb_f6/content/2301.00276v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:66a45f1bf1b8c0ce86f4c2698182a0db7fd8ea428831d4d3721b205967f1b3d6
+size 397459

3tAyT4oBgHgl3EQfb_f6/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ad3e59720d8a4c56546248e3fa00303564b64620f6fb76d6a3c812bffc166bf6
+size 5046317

3tAzT4oBgHgl3EQfuf3H/content/tmp_files/2301.01693v1.pdf.txt

@@ -0,0 +1,940 @@
+Mortality modeling at old-age: an mixture model approach
+Silvio C. Patricio*
+The Interdisciplinary Centre on Population Dynamics, University of Southern Denmark
+Fredy Castellares
+Departamento de Estat´ıstica, Universidade Federal de Minas Gerais
+Bernardo Queiroz
+Departamento de Demografia, Universidade Federal de Minas Gerais
+January 5, 2023
+Abstract
+In this paper, we propose a mixture-based model for mortality modeling above age 70. The proposed
+model is compared with 4 other widely used models: the Beard, Gompertz, Makeham, and Perks models.
+Our model captures well the mortality rate’s behavior at all the ages. We applied the method to a country
+with high quality data, Japan, and one with lower data quality, Brazil. In the comparative study for the
+Japanese population, the model presented a better fit to the data, obtaining an absolute mean percentage
+error of less than 7%, while the other models presented values greater than 30%.
+Keywords: mixture model, old-age, mortality modeling
+1
+Introduction
+In the past centuries, much has been done to model the process of mortality in populations and its con-
+sequences (Graunt, 1662; Gompertz, 1825a; Wilmoth, 2000; van Raalte, 2021). One of humanity’s most
+outstanding achievements in the last century, perhaps the last millennium, has been the four-decade increase
+in human life expectancy over the past 160 years (Vaupel et al., 2021; Wilmoth, 2000) and the improvement
+in human mortality. All these changes in human longevity directly affect pension, welfare, and health care
+systems (Cutler et al., 2006).
+*silca@sam.sdu.dk
+1
+arXiv:2301.01693v1  [stat.AP]  4 Jan 2023
+
+Despite pioneering work by Graunt and Gompertz, understanding of mortality for older ages remains
+a challenge, specially in developing countries with more defective data. In general, mortality estimates at
+older ages are limited by small numbers both in the exposure, death count and problems with age declaration
+(Feehan, 2018; Wrigley-Field, 2014; Nepomuceno et al., 2019). There is an important and ongoing debate
+about the levels of mortality at older ages. In general terms, the debate is whether mortality at older ages
+is declining or continues to increase (Gavrilov & Gavrilova, 2019; Feehan, 2018). In some settings, such
+as Brazil, there is also an important question on the crossover of mortality at older ages when comparing
+different population sub-groups (Nepomuceno et al., 2019; Pinheiro & Queiroz, 2019; Gomes & Turra,
+2009).
+In addition to the problem of the quality of the data, there is a debate on hypotheses of selectivity and
+of the biological limit of mortality in human populations that, in different ways, would impact the behavior
+of mortality taxes in more advanced ages. One of the consequences of the mortality selectivity hypothesis
+would be a greater rate of deceleration of the rates of mortality in more advanced ages. In this context,
+there are a series of models to explain mortality behavior at older ages. The choice of the appropriate model
+depends on the hypotheses assumed, whether in relation to the quality of the two data or in relation to the
+impacts produced by the selectivity.
+There are several possible explanations for the observed results and estimates. First one is related to
+data quality in different areas of a country, across sub-population groups and age. For instance, it could be a
+consequence of different age misreporting patterns or issues with quality of vital registration systems (Black
+et al., 2017). Preston et al (2000) investigated how different types of age misreporting can affect estimates
+of mortality rates at older ages, by analyzing the effects of three patterns of age misreporting: net age
+overstatement, net age understatement, and symmetric age misreporting.. It is also possible that mortality
+selection plays a role in the observed levels of mortality at older ages (Barbi et al., 2018; Wachter, 2018).
+In the context of higher mortality rates at young ages, survivors to older ages would be physiologically
+stronger and then live longer than others.
+Unfortunately, data quality at older ages limits the understanding of mortality and the evolution of
+survivorship at older ages. Feehan (2018) uses alternative methods to cohort mortality above age 80. He
+finds that no model can be universally applied to estimate old-age mortality, but he argues that Log-Quad
+(Wilmoth et al., 2012) provides a good fit. However, the log-quad method is based on standard mortality
+changes from the Human Mortality Database that is constructed from a series of countries in the Northern
+Hemisphere and might be limited to low and middle income countries.
+In this paper, we suggest a model that captures decline in mortality rates at older ages, which is a
+characteristic observed in some populations. Based on the proposed model, we perform a comparative study
+using establish mortality laws with our proposed approach. The analysis was split into two parts. First, to
+2
+
+compare the four widely used models with the proposed model: in this part we will study the behavior of
+these models in two databases: one with good quality data on mortality in Japan in 2015 (obtained from The
+Human Mortality Database of mortality), and the other database that has limited data regarding mortality
+in Brazil in 2010. In it the models will be evaluated from Mean Absolute Percentage Error (MAPE) of
+the log-hazard using the leave-one-out cross-validation method, and the model with the least MAPE will
+all be the best model. Moreover, as some models are complex, the genetic algorithm was used to obtain
+the estimates via maximum likelihood. Using this algorithm ensures convergence to the global maximum
+value. The second part applies the proposed model to different databases, and aims to understand the model
+behavior and also to verify its potential for application to real data.The model presented a better fit to the
+data, obtaining an absolute mean percentage error of less than 7%, while the other models presented values
+greater than 30%.
+2
+Models specification’s and parameter estimation
+Considering a non negative random variable (r.v.) T defined in a probability space (R+, B, Pθ), representing
+the individual life-spam, the r.v. T can be characterized by the survival function
+S(x|θ) = Pθ(T > x)
+which is associated with the density
+f(x|θ) = − ∂
+∂xS(x|θ).
+If S is a continuous survival function associated with a f density function, then the function µ defined in
+R+ by
+µ(x|θ) = lim
+ε↓0
+Pθ(x < T < x + ε|X > x)
+ε
+= f(x|θ)
+S(x|θ)
+it’s called the T mortality force. This function is usually used to describe the force of mortality for a group
+of people or population.
+The inferences in the model are based on the assumption that the number of death has a Poisson dis-
+tribution. Therefore, be D = (D0, D1, . . . , Dm)′ a random sample with Poisson distribution, with Dk
+representing the number of deaths between ages [k, k + 1), with k = 0, . . . , m, i.e. the number of death of
+people with k years old.
+For this approach it is considered that E(Dk) = µ(k|θ)Ek, with µ(k|θ) representing the mortality force
+at age k, where θ = (θ1, θ2, . . . , θp)′ is the parameter vector that characterizes the mortality rate, and Ek the
+population at age k exposed to risk, that are assumptions widely used by demographers (Brillinger et al.,
+3
+
+1986). Also, as it is the Poisson distribution, we have to V(Dk) = µ(k|θ)Ek, same value of expectation.
+Be D = (D0, . . . , Dm)′ e E = (E0, . . . , Em)′. The log-likelihood function from θ is given by
+ℓ(θ|D) =
+m
+�
+k=1
+Dk log λ(θ, k) − λ(θ, k),
+(1)
+with λ(θ, x) = µ(x|θ)E(x). The likelihood estimate �θ is obtained from maximizing the log-likelihood
+function with in equation 1, with respect to θ. Obtaining the partial derivative vector of the equation 1, with
+respect to θi, i = 1, . . . , p, we have
+∂ℓ(θ|D)
+∂θi
+=
+m
+�
+k=1
+�
+Dk
+µ(k|θ) − Ek
+� ∂µ(k|θ)
+∂θi
+.
+(2)
+The likelihood estimation can also be obtained by equating the partial derivative vector to zero and simul-
+taneously solving the system of equations. The explicit form of the gradient vector is explained for each of
+the models considered in this article. The Newton-Raphson method can be applied to solve the likelihood
+equation to obtain the estimate �θ.
+2.1
+Beard model
+In this model introduced in Beard (1959), we have that the force of mortality is given by
+µ(k|θ) =
+aebk
+1 + δebk
+with θ = (a, b, δ)′ ∈ R3
++. From which we calculate the partial derivative with respect to a and b. E Equation
+2 gives us a general equation for the gradient vector, where it depends only on the mortality rate and its
+partial derivative with respect to each parameter. Hence we get
+∂ℓ(θ|D)
+∂a
+=
+m
+�
+k=1
+�
+Dk
+�1 + δebk
+aebk
+�
+− Ek
+�
+ebk
+(1 + δebk)
+∂ℓ(θ|D)
+∂b
+=
+m
+�
+k=1
+�
+Dk
+�1 + δebk
+aebk
+�
+− Ek
+�
+akebk
+(1 + δebk)2
+∂ℓ(θ|D)
+∂δ
+=
+m
+�
+k=1
+�
+Dk
+�1 + δebk
+aebk
+�
+− Ek
+�
+ae2bk
+(1 + δebk)2
+4
+
+representing the gradient vector.
+2.2
+Gompertz model
+In this model introduced in Gompertz (1825b), we have that the force of mortality is given by
+µ(k|θ) = aebk,
+with θ = (a, b)′ ∈ R2
++. So for the gradient vector we have
+∂ℓ(θ|D)
+∂a
+=
+m
+�
+k=1
+� Dk
+aebk − Ek
+�
+ebk
+∂ℓ(θ|D)
+∂b
+=
+m
+�
+k=1
+� Dk
+aebk − Ek
+�
+akebk
+2.3
+Makeham model
+In this model introduced in Makeham (1860), we have that the force of mortality is given by
+µ(k|θ) = aebk + c,
+with θ = (a, b, c)′ ∈ R3
++. So for the gradient vector we have
+∂ℓ(θ|D)
+∂a
+=
+m
+�
+k=1
+�
+Dk
+aebk + c − Ek
+�
+ebk
+∂ℓ(θ|D)
+∂b
+=
+m
+�
+k=1
+�
+Dk
+aebk + c − Ek
+�
+akebk
+∂ℓ(θ|D)
+∂c
+=
+m
+�
+k=1
+�
+Dk
+aebk + c − Ek
+�
+5
+
+2.4
+Perks model
+In this model introduced in Perks (1932), we have that the force of mortality is given by
+µ(k|θ) = γ + aebk
+1 + δebk
+with θ = (a, b, γ, δ)′. So for the gradient vector we have
+∂ℓ(θ|D)
+∂a
+=
+m
+�
+k=1
+�
+Dk
+� 1 + δebk
+γ + aebk
+�
+− Ek
+�
+ebk
+1 + δebk
+∂ℓ(θ|D)
+∂b
+=
+m
+�
+k=1
+�
+Dk
+� 1 + δebk
+γ + aebk
+�
+− Ek
+� k(a − δγ)ebk
+(1 + δebk)2
+∂ℓ(θ|D)
+∂γ
+=
+m
+�
+k=1
+�
+Dk
+� 1 + δebk
+γ + aebk
+�
+− Ek
+�
+1
+1 + δebk
+∂ℓ(θ|D)
+∂δ
+=
+m
+�
+k=1
+�
+Dk
+� 1 + δebk
+γ + aebk
+�
+− Ek
+� ebk �
+aebk + γ
+�
+(1 + δebk)2
+2.5
+Mixture model
+As with Makeham, we will seek to decompose mortality into two components: premature and senescent
+mortality, respectively modeled by an exponential and a Gompertz component. However, Makeham dis-
+tinguishes these components through mortality force, and here we propose to distinguish them through
+distribution. Therefore, we are considering that the r.v. T introduced at the beginning of this session is
+associated with a probability density function f, which is define as:
+f(x|θ) = p
+�
+λe−λx�
++ (1 − p)
+�
+ab exp
+�
+a
+�
+ebx − 1
+�
++ bx
+��
+(3)
+with θ = (a, b, λ, p)′.
+The density f is a Gompertz and a exponential distribution a mixture. The Gompertz distribution will fit
+the senescence deaths count, and the exponential distribution will fit the premature deaths, such as accidents
+and disease. Briefly, this model considers the existence of two sub populations in the death count, one
+6
+
+Gompertz and the other Exponential, and the parameters p and q = 1 − p represent the proportions of each
+one.
+Since the random variable T is associated with a density function, we can also associate it with a hazard
+function. In this case the force of mortality is defined by:
+µ(x|θ) = f(x|θ)
+S(x|θ) = p
+�
+λe−λx�
++ (1 − p)
+�
+ab exp
+�
+a
+�
+ebx − 1
+�
++ bx
+��
+pe−λx + (1 − p) exp{−a (ebx − 1)}
+,
+(4)
+for which there is no straightforward interpretation. Which is lost due to the ease of deriving functions such
+as statistical moments and expected average residual life (for more details, see Finkelstein (2009)) .From
+7
+
+this we can get the gradient vector which, for this model, is given by
+∂ℓ(θ|D)
+∂a
+=
+m
+�
+k=1
+�
+Dk
+pe−λk + (1 − p) exp{−a
+�
+ebk − 1
+�
+}
+p (λe−λk) + (1 − p) (ab exp {a (ebk − 1) + bk}) − Ek
+�
+×
+× b(1 − p)ea(keb−1)+bλ + ab(1 − p)(keb − 1)ea(keb−1)+bx
+pe−λk + (1 − p) exp{−a (ebk − 1)}
++
++ (−1)(1 − p)(1 − ebk)e−a(ebk−1) �
+p
+�
+λe−λk�
++ (1 − p)
+�
+ab exp
+�
+a
+�
+ebk − 1
+�
++ bk
+���
+(pe−λk + (1 − p) exp{−a (ebk − 1)})2
+∂ℓ(θ|D)
+∂b
+=
+m
+�
+k=1
+�
+Dk
+pe−λk + (1 − p) exp{−a
+�
+ebk − 1
+�
+}
+p (λe−λk) + (1 − p) (ab exp {a (ebk − 1) + bk}) − Ek
+�
+×
+×
+a(1 − p)xebx−a(ebx−1) �
+ab(1 − p)ea(xebx−1)+bx + λpe−λx�
+�
+(1 − p)ea(ebx−1) + pe−λx�2
++
++ a(1 − p)ea(ebx−1)+bx + ab(1 − p)ea(ebx−1)+bx �
+axeb + x
+�
+(1 − p)ea(ebx−1) + pe−λx
+∂ℓ(θ|D)
+∂λ
+=
+m
+�
+k=1
+�
+Dk
+pe−λk + (1 − p) exp{−a
+�
+ebk − 1
+�
+}
+p (λe−λk) + (1 − p) (ab exp {a (ebk − 1) + bk}) − Ek
+�
+×
+×
+�
+��
+pe−λx − λpxe−λx
+(1 − p)e−a(ebx−1) + pe−λx +
+pxe−λx �
+ab(1 − p)ea(xeb−1)+bx + λpe−λx�
+�
+(1 − p)e−a(ebx−1) + pe−λx
+�2
+�
+��
+∂ℓ(θ|D)
+∂p
+=
+m
+�
+k=1
+�
+Dk
+pe−λk + (1 − p) exp{−a
+�
+ebk − 1
+�
+}
+p (λe−λk) + (1 − p) (ab exp {a (ebk − 1) + bk}) − Ek
+�
+×
+×
+�
+�� λe−λx − abea(xeb−1)+bx
+(1 − p)e−a(ebx−1) + pe−λx −
+�
+e−λx − e−a(ebx−1)� �
+ab(1 − p)ea(xeb−1)+bx + λpe−λx�
+�
+(1 − p)e−a(ebx−1) + pe−λx
+�2
+�
+��
+3
+Data and empirical results
+In order to evaluate the proposed model, we will compare its performance on high and low-quality data.
+For this, we will evaluate its performance against four other models, using the Mean Absolute Percentage
+Error (MAPE) combined with the leave-one-out cross-validation method, which will measure the average
+distance between the log-hazard and the log-mortality rate. Moreover, as some models are highly nonlinear,
+8
+
+the Genetic Algorithm (Scrucca, 2013; Mirjalili, 2019) will be used to maximize the likelihood function.
+This algorithm ensures convergence to the global maximum value.
+3.1
+Models comparison
+In a high quality data setting
+In this scenario, we will use mortality data from Japan in 2015 obtained from The Human Mortality
+Database (HMD). The observed value of log µ is linearly increasing to a certain age, and then has a sharp
+drop. This behavior was also noted this country in the last three decades. However this is not restricted
+to Japan, other countries like Sweden, Germany, USA and Korea also had the same mortality behavior.
+The Figure 1 shows the estimated log-hazard function. We can clearly see the models of Beard, Gompertz,
+Makeham and Perks were not able to fit properly the mortality rate after age 100.
+70
+80
+90
+100
+110
+−5
+−4
+−3
+−2
+−1
+0
+1
+Age
+log(µ)
+Beard
+Gompertz
+Makeham
+Perks
+Mixture model
+Figure 1: Japan 2015 modeling
+The Gompertz model consider force of mortality being log-linear, but clearly this behavior does not
+describe the entire observed curve. For this model the estimated parameter is �θ = (0.0179, 0.1094)′. And
+this model has a MAPE of 34.0127, i.e., this model’s predictions are on average 34.0127 % distant of the
+observed value. A similar result can be obtained from the Makeham model, which has estimated parameter
+�θ = (0.0174, 0.1103, 0.0008)′, and MAPE 33.0288.
+The Beard can be seen as the ratio of a Gompertz and a Makeham models with c = 1, with the pa-
+rameters estimated by ML �θ = (0.0165, 0.1216, 0.0073)′. Despite Beard’s combination of Makeham and
+Gompertz models, this model provided the worse fit, reaching a MAPE of 55.6189.
+The Perks model also has a similar construction to Beard. It is the ratio between two Makeham models.
+For this model we estimate �θ = (0.0135, 0.1313, 0.0040, 0.0075)′. And as expected, this model had a very
+9
+
+similar behavior to the previous model, including in MAPE of 51.3591, suggesting that this model does not
+fit well to the data.
+Finally, for the proposed mixture-based model, we estimated �θ = (0.1155, 0.0163, 0.2061, 0.0126)′, and
+a MAPE of 6.9193, the best of the models presented in this study. In addition, this model was the only one
+that was able to capture the sharp drop in the mortality rate. With the estimated parameters we can interpret
+that the non-senescence death represents 1.2599 % of the total death after age 70.
+In a low quality data setting
+We observed that the model works well on data that has good quality, and now we aim to understand how
+the model behaves when the data has limitations. In this case we are going to use data from Brazil from
+2010 (Queiroz et al., 2020; Gonzaga & Schmertmann, 2016) . Previous studies showa a mortality crossover
+above age 60 when comparing more and less developed states in Brazil using the Topals model (Queiroz
+et al., 2020; Gonzaga & Schmertmann, 2016). It is argued that the result is related to the level of complete-
+ness of death counts, age misreporting and mortality selection. Thus, it is an important and relevant case
+study for the application of our prosed mixture model. For this, as before, we will compare the performance
+of the 5 models presented through MAPE.
+70
+75
+80
+85
+90
+95
+100
+−3.0
+−2.5
+−2.0
+−1.5
+−1.0
+−0.5
+0.0
+Age
+log(µ)
+Beard
+Gompertz
+Makeham
+Perks
+Mixture model
+Figure 2: Brazil 2010 modeling
+For the first model (Beard) we estimated �θ = (0.0375, 0.0942, 5.5625 × 10−8)′, and a MAP of 20.4629,
+i.e., on average this model distanced by 20 % of the mortality rate. We also got a similar conclusion about
+the Gompertz model, estimating �θ = (0.0375, 0.0943)′ and MAPE about 20.4499.
+The Makeham and Perks models also obtained similar results. For Makeham it was estimated �θ =
+10
+
+(0.01481, 0.1338, 0.03131)′ resulting in a MAP of 14.5473, and for Perks model it was estimated �θ =
+(0.0163, 0.0129, 0.0290, 3.4272 × 10−7)′ which results in MAPE of 14.9002.
+Finally, for the proposed model was estimated �θ = (0.1036, 0.0315, 0.2389, 0.0692)′, and a MAPE of
+18.0038%, which indicates that the model is not able to capture mortality well in these data. Therefore,
+the results found in this application match the results discussed in Feehan (2018) on the power of models
+capturing mortality at advanced ages universally.
+3.2
+Model applications
+As we have seen, the proposed model has a high capacity to fit the mortality at older ages. Therefore, we
+will illustrate the power of this model by applying it to mortality data from Japan (1993 and 2002), Sweden
+(2011), Germany (2016), USA (1990 and 1992), Spain (2012) and Italy (2011). Table 1 represents the
+estimate for each dataset, and Figure 4 represents their respective decomposed distribution of death.
+Table 1: Parameters estimated.
+Country
+Year
+ˆa
+ˆb
+ˆc
+ˆp
+MAPE
+Japan
+1993
+0.10911
+0.02916
+0.21615
+0.00250
+8.86459
+Japan
+2002
+0.10897
+0.02425
+0.30152
+0.03276
+7.49451
+Sweden
+2011
+0.12390
+0.01520
+0.26448
+0.01559
+12.27019
+Germany
+2016
+0.11046
+0.02090
+0.22283
+0.00397
+10.68258
+USA
+1990
+0.08845
+0.03569
+0.20360
+0.02569
+3.80694
+USA
+1992
+0.09057
+0.03404
+0.20575
+0.03217
+2.91887
+Spain
+2012
+0.12372
+0.01544
+0.22751
+0.01307
+12.38755
+Italy
+2011
+0.11606
+0.01768
+0.21710
+0.01999
+13.24385
+In Table 1 it can be seen that the estimated values for p are small, less than 0.04, which indicates that the
+proportion of premature deaths above age 70 does not exceed 4%. This result was already expected, since
+by truncating the mortality data at age 70, we are excluding infant mortality and mortality hump (Remund
+et al., 2018), and we only observe the tail of the distribution of premature mortality. Furthermore, the our
+result is also in agreement with Horiuchi & Wilmoth’s results, that above age 75 mortality decelerates for
+most causes of death (Horiuchi & Wilmoth, 1997).
+The estimated values for the c parameter are similar, and concentrated around 0.23. This suggests that,
+despite having different proportions, the distributions of premature death are similar, as can be seen on the
+left in Figure 3. Such similarity was not observed in the senescent death distributions, which have a marked
+difference, as can be seen on the right in Figure 3. Despite this, it is clear that the modal age of death is
+between 80 and 90, which is consistent with previous studies and (Horiuchi et al., 2013).
+11
+
+70
+80
+90
+100
+110
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+Age
+prematur mortality distribution
+JPN − 1993
+JPN − 2002
+SWE − 2011
+DEUTNP − 2016
+USA − 1990
+USA − 1992
+ESP − 2012
+ITA − 2011
+70
+80
+90
+100
+110
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+Age
+senescence mortality distribution
+JPN − 1993
+JPN − 2002
+SWE − 2011
+DEUTNP − 2016
+USA − 1990
+USA − 1992
+ESP − 2012
+ITA − 2011
+Figure 3: Estimates of mortality components.
+70
+80
+90
+100
+110
+0
+10000
+20000
+30000
+JPN − 1993
+Age
+dx
+70
+80
+90
+100
+110
+0
+10000
+20000
+30000
+JPN − 2002
+Age
+dx
+70
+80
+90
+100
+110
+0
+1000
+2000
+3000
+4000
+SWE − 2011
+Age
+dx
+70
+80
+90
+100
+110
+0
+10000
+20000
+30000
+DEUTNP − 2016
+Age
+dx
+70
+80
+90
+100
+110
+0
+20000
+40000
+60000
+USA − 1990
+Age
+dx
+70
+80
+90
+100
+110
+0
+20000
+40000
+60000
+USA − 1992
+Age
+dx
+70
+80
+90
+100
+110
+0
+5000
+10000
+15000
+ESP − 2012
+Age
+dx
+70
+80
+90
+100
+110
+0
+5000
+15000
+25000
+ITA − 2011
+Age
+dx
+Senescence deaths
+Premature deaths
+Overall deaths
+Figure 4: Estimations
+The Figure 4 shows the distribution of death estimated and broken down into premature and senescent
+deaths. In it, we can observe the quality of fit of the estimated model (black line). In addition, it is possible
+12
+
+to see that for Japan in 1993 and Germany in 2016, there were practically no premature deaths after age 70,
+this could also be inferred from analyzing the Table 1, where the values of estimate for p are small.
+4
+Conclusions and future works
+Robust estimates of mortality rates in advanced ages are a challenge for demographers for various reasons.
+Even in populations with good records of deaths and population there are disturbances in the function of
+the low number of events and/or some limitation in the information on the age of death. In case of countries
+where the problems of data quality is present, the challenges are greater.
+For some centuries there has been an ambition to decompose mortality into interpretable components.
+The best known are those proposed by Makeham (1860) and Heligman & Pollard (1980). However, in recent
+years researchers have devoted to this problem (Remund et al., 2017; Mazzuco et al., 2021). Therefore,
+this paper aims to bring a contribution to this discussion, delivering a new parametric model capable of
+decomposing mortality through mixing models in a frequentest framework. Mazzuco et al. (2021) proposes
+an approach similar to the one proposed in this paper, however the authors use a Bayesian framework.
+As we have seen, the proposed model fits well the mortality curve, specially above age 100, and this
+model does it without overparametrization, as Heligman & Pollard (1980). Furthermore, as it is a mixture
+model, the model is flexible to become the Gompertz model (p = 0), or the Exponential model (p = 1).
+When 0 < p < 1, the model fits a mortality curve with inflexion point (mortality deceleration) and plateau
+(mortality plateau).
+The use of Brazilian mortality data shed light on the performance of the model in a low quality database.
+We could see that the mixture-based model captures the dynamics of mortality well only when there is a
+drop in mortality rates, serving as an alternative to models that do not have this characteristic.
+Although the present work presents a model capable of capturing the specific dynamics of the force of
+mortality in certain populations, it also sheds light on other problems to be solved. Since the model is based
+on mixtures of distributions, we are interested in deriving hypothesis tests on the estimated parameters. One
+of the main ones is to test if p = 0, i.e. whether the model can be reduced to a Gompertz model; similar
+interest to that studied in B¨ohnstedt & Gampe (2019), when a hypothesis test for Gamma heterogeneity is
+derived, and important statistical properties are studied.
+Finally, in the recently published paper Vaupel et al. (2022) point out that estimating senescence mor-
+tality is of fundamental importance to understand the pace of human aging, human longevity and how far
+we can live. In this sense, this work brought a method capable of identifying and estimating senescent mor-
+tality, without having a great computational cost, often seen in Bayesian analysis (See Barber et al. (2015)),
+or overparameterized models, as seen in Heligman & Pollard (1980).
+13
+
+Bibliography
+Barber, S., Voss, J., & Webster, M. (2015). The rate of convergence for approximate bayesian computation.
+Electronic Journal of Statistics, 9(1), 80–105.
+Barbi, E., Lagona, F., Marsili, M., Vaupel, J. W., & Wachter, K. W. (2018). The plateau of human mortality:
+Demography of longevity pioneers. Science, 360(6396), 1459–1461.
+Beard, R. E. (1959). Note on some mathematical mortality models. In Ciba Foundation Symposium-The
+Lifespan of Animals (Colloquia on Ageing), volume 5 (pp. 302–311).: Wiley Online Library.
+Black, D. A., Hsu, Y.-C., Sanders, S. G., Schofield, L. S., & Taylor, L. J. (2017). The methuselah effect: The
+pernicious impact of unreported deaths on old-age mortality estimates. Demography, 54(6), 2001–2024.
+B¨ohnstedt, M. & Gampe, J. (2019). Detecting mortality deceleration: Likelihood inference and model
+selection in the gamma-gompertz model. Statistics & Probability Letters, 150, 68–73.
+Brillinger, D. R. et al. (1986). The natural variability of vital rates and associated statistics. Biometrics,
+42(4), 693–734.
+Cutler, D., Deaton, A., & Lleras-Muney, A. (2006). The determinants of mortality. Journal of economic
+perspectives, 20(3), 97–120.
+Feehan, D. M. (2018). Separating the signal from the noise: evidence for deceleration in old-age death
+rates. Demography, 55(6), 2025–2044.
+Finkelstein, M. (2009). Understanding the shape of the mixture failure rate (with engineering and demo-
+graphic applications). Applied Stochastic Models in Business and Industry, 25(6), 643–663.
+Gavrilov, L. A. & Gavrilova, N. S. (2019). Late-life mortality is underestimated because of data errors.
+PLoS biology, 17(2), e3000148.
+Gomes, M. M. F. & Turra, C. M. (2009). The number of centenarians in brazil: indirect estimates based on
+death certificates. Demographic Research, 20, 495–502.
+Gompertz, B. (1825a). Xxiv. on the nature of the function expressive of the law of human mortality, and
+on a new mode of determining the value of life contingencies. in a letter to francis baily, esq. frs &c.
+Philosophical transactions of the Royal Society of London, (115), 513–583.
+14
+
+Gompertz, B. (1825b). Xxiv. on the nature of the function expressive of the law of human mortality, and
+on a new mode of determining the value of life contingencies. in a letter to francis baily, esq. frs &c.
+Philosophical transactions of the Royal Society of London, 0(115), 513–583.
+Gonzaga, M. R. & Schmertmann, C. P. (2016). Estimating age-and sex-specific mortality rates for small
+areas with topals regression: an application to brazil in 2010. Revista Brasileira de Estudos de Populac¸˜ao,
+33, 629–652.
+Graunt, J. (1662). Natural and political observations mentioned in a following index, and made upon the
+bills of mortality. In Mathematical Demography (pp. 11–20). Springer.
+Heligman, L. & Pollard, J. H. (1980). The age pattern of mortality. Journal of the Institute of Actuaries,
+107(1), 49–80.
+Horiuchi, S., Ouellette, N., Cheung, S. L. K., & Robine, J.-M. (2013). Modal age at death: lifespan indicator
+in the era of longevity extension. Vienna Yearbook of Population Research, (pp. 37–69).
+Horiuchi, S. & Wilmoth, J. R. (1997). Age patterns of the life table aging rate for major causes of death
+in japan, 1951–1990. The Journals of Gerontology Series A: Biological Sciences and Medical Sciences,
+52(1), B67–B77.
+Makeham, W. M. (1860). On the law of mortality and the construction of annuity tables. Journal of the
+Institute of Actuaries, 8(6), 301–310.
+Mazzuco, S. S., Suhrcke, M. M., & Zanotto, L. L. (2021). How to measure premature mortality? a proposal
+combining “relative” and “absolute” approaches. Population health metrics, 19(1), 1–14.
+Mirjalili, S. (2019). Genetic algorithm. In Evolutionary algorithms and neural networks (pp. 43–55).
+Springer.
+Nepomuceno, M., Turra, C., et al. (2019). The population of centenarians in Brazil: historical estimates
+from 1900 to 2000. Technical report, Max Planck Institute for Demographic Research, Rostock, Ger-
+many.
+Perks, W. (1932). On some experiments in the graduation of mortality statistics. Journal of the Institute of
+Actuaries, 63(1), 12–57.
+Pinheiro, P. C. & Queiroz, B. L. (2019). Regional disparities in brazilian adult mortality: an analysis using
+modal age at death (m) and compression of mortality (iqr). Anais, (pp. 1–20).
+15
+
+Queiroz, B. L., Gonzaga, M. R., Vasconcelos, A., Lopes, B. T., & Abreu, D. M. (2020). Comparative
+analysis of completeness of death registration, adult mortality and life expectancy at birth in brazil at the
+subnational level. Population health metrics, 18(1), 1–15.
+Remund, A., Camarda, C. G., & Riffe, T. (2017). Analyzing the young adult mortality hump in r with
+morthump. Rostock: Max Planck Institute for Demographic Research (MPIDR Technical Report TR-
+2018-003).
+Remund, A., Camarda, C. G., & Riffe, T. (2018). A cause-of-death decomposition of young adult excess
+mortality. Demography, 55(3), 957–978.
+Scrucca, L. (2013). GA: A package for genetic algorithms in R. Journal of Statistical Software, 53(4),
+1–37.
+van Raalte, A. A. (2021). What have we learned about mortality patterns over the past 25 years? Population
+Studies, 75(sup1), 105–132.
+Vaupel, J. W. et al. (2022). The Pull of the Plateau and the Sway of the Mode: Formal Relationships to
+Estimate the Pace of Senescence. Technical report, Center for Open Science.
+Vaupel, J. W., Villavicencio, F., & Bergeron-Boucher, M.-P. (2021). Demographic perspectives on the rise
+of longevity. Proceedings of the National Academy of Sciences, 118(9).
+Wachter, K. W. (2018). Hypothetical errors and plateaus: A response to newman. PLoS biology, 16(12),
+e3000076.
+Wilmoth, J., Zureick, S., Canudas-Romo, V., Inoue, M., & Sawyer, C. (2012). A flexible two-dimensional
+mortality model for use in indirect estimation. Population studies, 66(1), 1–28.
+Wilmoth, J. R. (2000). Demography of longevity: past, present, and future trends. Experimental gerontol-
+ogy, 35(9-10), 1111–1129.
+Wrigley-Field, E. (2014). Mortality deceleration and mortality selection: three unexpected implications of
+a simple model. Demography, 51(1), 51–71.
+16
+

5NFIT4oBgHgl3EQf7itm/content/2301.11398v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:f0af6ac12e85736f6ec662252891b06a672016d4ccd614c5506042172bf99333
+size 167907

5NFIT4oBgHgl3EQf7itm/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:58c273ab7d0f1f50c47cf0bded92781c6b77e1552f222e318279db3eeab0605f
+size 2162733

5NFIT4oBgHgl3EQf7itm/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:e61d1b452ae3b3789cbfe10d69aefd71423cc69edc5d32d107dc6f8671a99325
+size 76066

5NFKT4oBgHgl3EQf-C45/content/tmp_files/2301.11956v1.pdf.txt

@@ -0,0 +1,2309 @@
+On the Connection Between MPNN and Graph Transformer
+Chen Cai 1 Truong Son Hy 1 Rose Yu 1 Yusu Wang 1
+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 understand-
+ing, 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 Per-
+former/Linear Transformer (Choromanski et al.,
+2020; Katharopoulos et al., 2020b), then MPNN
++ VN with only O(1) depth and O(1) width
+can approximate a self-attention layer in Per-
+former/Linear Transformer. Next, via a connec-
+tion between MPNN + VN and DeepSets, we
+prove the MPNN + VN with O(nd) width and
+O(1) depth can approximate the self-attention
+layer arbitrarily well, where d is the input fea-
+ture dimension. Lastly, under some assumptions,
+we provide an explicit construction of MPNN +
+VN with O(1) width and O(n) depth approxi-
+mating 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.
+1University of California San Diego, San Diego, USA. Corre-
+spondence to: Chen Cai <c1cai@ucsd.edu>.
+Copyright 2023 by the author(s).
+VN
+Transformer
+(a)
+(b)
+Figure 1: MPNN + VN and Graph Transformers.
+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; Rampášek 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
+1GT in this paper refers to the practice of tokenizing graph
+nodes and applying standard transformers on top (Ying et al., 2021;
+Kim et al., 2022). There exists a more sophisticated GT (Kreuzer
+et al., 2021) that further conditions attention on edge types but it is
+not considered in this paper.
+arXiv:2301.11956v1  [cs.LG]  27 Jan 2023
+
+On the Connection Between MPNN and Graph Transformer
+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 equiv-
+ariant 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 O(1) depth and O(1) width, MPNN + VN
+can approximate one self-attention layer of Performer
+(Choromanski et al., 2020) and Linear Transformer
+(Katharopoulos et al., 2020b), a type of linear trans-
+formers (Tay et al., 2020).
+• Via a link between MPNN + VN with DeepSets (Za-
+heer et al., 2017), we prove MPNN + VN with O(1)
+depth and O(nd) width (d is the input feature dimen-
+sion) 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 O(n) depth O(1) width
+MPNN + VN approximating 1 self-attention layer ar-
+bitrarily 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 sur-
+prisingly 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 imple-
+mentation of MPNN + VN is able to further improve
+the early implementation of MPNN + VN on OGB
+datasets and 3) MPNN + VN outperforms Linear Trans-
+former (Katharopoulos et al., 2020b) 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 & Suzuki, 2019; Cai & Wang, 2020) and
+over-squashing in MPNN (Alon & Yahav, 2020; Topping
+et al., 2021).
+Fully-connected Graph transformer (Dwivedi & 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 dif-
+fusion 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 re-
+search in GT include combining MPNN and GT (Wu et al.,
+2021; Rampášek et al., 2022), encoding the substructures
+(Chen et al., 2022), and efficient graph transformers for
+large graphs (Wu et al., 2022).
+
+On the Connection Between MPNN and Graph Transformer
+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.
+Depth
+Width
+Self-Attention
+Note
+Theorem 4.1
+O(1)
+O(1)
+Approximate
+Approximate self attention in Performer (Choromanski et al., 2020)
+Theorem 5.5
+O(1)
+O(nd)
+Full
+Leverage the universality of equivariant DeepSets
+Theorem 6.3
+O(n)
+O(1)
+Full
+Explicit construction, strong assumption on X
+Proposition B.10
+O(n)
+O(1)
+Full
+Explicit construction, more relaxed (but still strong) assumption on X
+3. Preliminaries
+We denote X ∈ Rn×d the concatenation of graph node
+features and positional encodings, where node i has feature
+xi ∈ Rd. When necessary, we use x(l)
+j
+to denote the node
+j’s feature at depth l. Let M be the space of multisets of
+vectors in Rd. We use X ⊆ Rn×d to denote the space of
+node features and the Xi be the projection of X on i-th
+coordinate. ∥ · ∥ denotes the 2-norm. [x, y, z] denotes the
+concatenation of x, y, z. [n] stands for the set {1, 2, ..., n}.
+Definition 3.1 (attention). We denote key and query matrix
+as WK, WQ ∈ Rd×d′, and value matrix as WV ∈ Rd×d
+2. Attention score between two vectors u, v ∈ Rd×1 is de-
+fined as α(u, v) = softmax(uT WQ(WK)T v). We denote
+A as the space of attention α for different WQ, WK, WV .
+We also define unnormalized attention score α′(·, ·) to be
+α′(u, v) = uT WQ(WK)T v. Self attention layer is a ma-
+trix function L : Rn×d → Rn×d of the following form:
+L(X) = softmax(XWQ(XWK)T )XWV .
+3.1. MPNN Layer
+Definition 3.2 (MPNN layer (Gilmer et al., 2017)). An
+MPNN layer on a graph G with node features x(k) at k-th
+layer and edge features e is of the following form
+x(k)
+i
+= γ(k) �
+x(k−1)
+i
+, τj∈N (i)φ(k) �
+x(k−1)
+i
+, x(k−1)
+j
+, ej,i
+��
+Here γ : Rd × Rd′ → Rd is update function, φ : Rd ×
+Rd × Rde → Rd′ is message function where de is the edge
+feature dimension, τ : M → Rd is permutation invariant
+aggregation function and 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
+2For 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.
+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
+x(k)
+vn = γ(k)
+vn
+�
+x(k−1)
+i
+, τj∈[n]φ(k)
+vn-gn
+�
+x(k−1)
+i
+, x(k−1)
+j
+, ej,i
+��
+(1)
+for the virtual node, and
+x(k)
+i
+= γ(k)
+gn (x(k−1)
+i
+, τj∈N1(i)φ(k)
+gn-vn
+�
+x(k−1)
+i
+, x(k−1)
+j
+, ej,i
+�
++ τj∈N2(i)φ(k)
+gn-gn
+�
+x(k−1)
+i
+, x(k−1)
+j
+, ej,i)
+�
+(2)
+for graph node. Here N1(i) for graph node i is the virtual
+node and N2(i) is the set of neighboring graph nodes.
+Our proof of approximating self-attention layer 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 set-
+ting φ(k)
+gn-gn 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 Defini-
+tion 3.3 except we set θgn-gn to be 0. I.e., we have
+x(k)
+vn = γ(k)
+vn
+�
+x(k−1)
+i
+, τj∈[n]φ(k)
+vn-gn
+�
+x(k−1)
+i
+, x(k−1)
+j
+, ej,i
+��
+for the virtual node, and
+x(k)
+i
+= γ(k)
+gn
+�
+x(k−1)
+i
+, τj∈N1(i)φ(k)
+gn-vn
+�
+x(k−1)
+i
+, x(k−1)
+j
+, ej,i
+��
+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). Us-
+ing 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)
+
+On the Connection Between MPNN and Graph Transformer
+3.2. Assumptions
+We have two mild assumptions on feature space X ⊂ Rn×d
+and the regularity of target function L.
+AS1. ∀i ∈ [n], xi ∈ Xi, ∥xi∥ < C1. This implies X is
+compact.
+AS2. ∥WQ∥ < C2, ∥WK∥ < C2, ∥WV ∥ < C2 for target
+layer L. Combined with AS1 on X, this means α′(xi, xj)
+is both upper and lower bounded, which further implies
+�
+j eα′(xi,xj) be both upper bounded and lower bounded.
+4. O(1)-depth O(1)-width MPNN + VN for
+unbiased approximation of attention
+The standard self-attention takes O(n2) computational time,
+therefore not scalable for large graphs. Reducing the compu-
+tational 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, Per-
+former (Choromanski et al., 2020) and Linear Transformer
+(Katharopoulos et al., 2020b).
+One full self-attention layer L is of the following form
+x(l+1)
+i
+=
+n
+�
+j=1
+κ
+�
+W (l)
+Q x(l)
+i , W (l)
+K x(l)
+j
+�
+�n
+k=1 κ
+�
+W (l)
+Q x(l)
+i , W (l)
+K x(l)
+k
+�·
+�
+W (l)
+V x(l)
+j
+�
+(3)
+where κ
+:
+Rd × Rd
+→
+R is the softmax kernel
+κ(x, y) := exp(xT y). The kernel function can be ap-
+proximated via κ(x, y) = ⟨Φ(x), Φ(y)⟩V ≈ φ(x)T φ(y)
+where the first equation is by Mercer’s theorem and
+φ(·) : Rd → Rm is a low-dimensional feature map
+with random transformation. For Performer (Choroman-
+ski et al., 2020), the choice of φ is taken as φ(x) =
+exp
+�
+−∥x∥2
+2
+2
+�
+√m
+�
+exp
+�
+wT
+1 x
+�
+, · · · , exp
+�
+wT
+mx
+��
+where wk ∼
+N (0, Id) is i.i.d sampled random variable. For Linear Trans-
+former (Katharopoulos et al., 2020b), φ(x) = elu(x) + 1.
+By switching κ(x, y) to be φ(x)T φ(y), and denote qi =
+W (l)
+Q x(l)
+i , ki = W (l)
+K x(l)
+i
+and vi = W (l)
+V x(l)
+i , the approx-
+imated version of Equation (3) by Performer and Linear
+Transformer becomes
+x(l+1)
+i
+=
+n
+�
+j=1
+φ (qi)T φ (kj)
+�n
+k=1 φ (qi)T φ (kk)
+· vj
+=
+�
+φ (qi)T �n
+j=1 φ (kj) ⊗ vj
+�T
+φ (qi)T �n
+k=1 φ (kk)
+.
+(4)
+where we use the matrix multiplication association rule to
+derive the second equality.
+The key advantage of Equation (4) is that �n
+j=1 φ (kj) and
+�n
+j=1 φ(kj) ⊗ vj can be approximated by the virtual node,
+and shared for all graph nodes, using only O(1) layers of
+MPNNs. We denote the self-attention layer of this form
+in Equation (4) as LPerformer. Linear Transformer differs
+from Performer by choosing a different form of φ(x) =
+Relu(x) + 1 in its self-attention layer LLinear-Transformer.
+In particular, the VN will approximate �n
+j=1 φ (kj) and
+�n
+j=1 φ (kj) ⊗ vj, and represent it as its feature. Both
+φ (kj) and φ (kj)⊗vj can be approximated arbitrarily well
+by an MLP with constant width (constant in n but can be
+exponential in d) and depth. Note that φ(kj) ⊗ vj ∈ Rdm
+but can be reshaped to 1 dimensional feature vector.
+More specifically, the initial feature for the virtual node is
+1(d+1)m, where d is the dimension of node features and m
+is the number of random projections ωi. Message function
++ aggregation function for virtual node τφvn-gn : R(d+1)m ×
+M → R(d+1)m is
+τj∈[n]φ(k)
+vn-gn(·, {xi}i) = [
+n
+�
+j=1
+φ (kj) ,
+ReshapeTo1D(
+n
+�
+j=1
+φ (kj) ⊗ vj)]
+(5)
+where ReshapeTo1D(·) flattens a 2D matrix to a 1D vec-
+tor in raster order. This function can be arbitrarily approxi-
+mated by MLP. Note that the virtual node’s feature dimen-
+sion is (d + 1)m (where recall m is the dimension of the
+feature map φ 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 vir-
+tual node might be overloaded when passing information
+among nodes. The update function for virtual node γvn :
+R(d+1)m × R(d+1)m → 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 γgn
+:
+R(d+1)m × Rd → Rd of the form
+γgn(xi, [
+n
+�
+j=1
+φ (kj) , ReshapeTo1D(
+n
+�
+j=1
+φ (kj) ⊗ vj)])
+=
+�
+φ (qi) �n
+j=1 φ (kj) ⊗ vj
+�T
+φ (qi)T �n
+k=1 φ (kk)
+(6)
+to update the graph node feature.
+As the update function γgn can not be computed exactly in
+MLP, what is left is to show that error induced by using
+MLP to approximate τφvn-gn and γgn in Equation (5) and
+Equation (6) can be made arbitrarily small.
+Theorem 4.1. Under the AS1 and AS2, MPNN + VN of
+O(1) width and O(1) depth can approximate LPerformer and
+LLinear-Transformer arbitrarily well.
+
+On the Connection Between MPNN and Graph Transformer
+Proof. We first prove the case of LPerformer. We can decom-
+pose our target function as the composition of τj∈[n]φ(k)
+vn-gn,
+γgn and φ. By the uniform continuity of the functions,
+it suffices to show that 1) we can approximate φ, 2) we
+can approximate operations in γgn and τφvn-gn arbitrar-
+ily well on the compact domain, and 3) the denominator
+φ (qi)T �n
+k=1 φ (kk) is uniformly lower bounded by a pos-
+itive number for any node features in X.
+For 1), each component of φ is continuous and all inputs
+kj, qj lie in the compact domain so φ can be approximated
+arbitrarily well by MLP with O(1) width and O(1) depth
+(Cybenko, 1989).
+For 2), we need to approximate the operations in γgn and
+τφvn-gn, i.e., approximate multiplication, and vector-scalar
+division arbitrarily well. As all those operations are con-
+tinuous, it boils down to showing that all operands lie
+in a compact domain. By assumption AS1 and AS2 on
+WQ, WK, WV and input feature X, we know that qi, ki, vi
+lies in a compact domain for all graph nodes i. As φ is con-
+tinuous, this implies that φ(qi), �n
+j=1 φ(kj) ⊗ vj 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 φ (qi)T �n
+k=1 φ (kk) is always
+positive. We just need to show that the denominator is bound
+from below by a positive constant. For Performer, φ(x) =
+exp
+�
+−∥x∥2
+2
+2
+�
+√m
+�
+exp
+�
+wT
+1 x
+�
+, · · · , exp
+�
+wT
+mx
+��
+where wk ∼
+N (0, Id). As all norm of input x to φ is upper bounded
+by AS1, exp( −∥x∥2
+2
+2
+) is lower bounded. As m is fixed,
+we know that ∥wT
+i x∥ ≤ ∥wi∥∥x∥, which implies that
+wT
+i x is lower bounded by −∥wi∥∥x∥ which further im-
+plies that exp(wT
+i x) is lower bounded. This means that
+φ (qi)T �n
+k=1 φ (kk) is lower bounded.
+For Linear Transformer, the proof is essentially the same
+as above. We only need to show that φ(x) = elu(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.,
+2020b) and Sparse Transformer (Child et al., 2019).
+5. O(1) depth O(nd) width MPNN + VN
+We have shown that the MPNN + VN can approximate self-
+attention in Performer and Linear Transformer using only
+O(1) depth and 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
+O(nd) width, can approximate 1 self-attention layer (and
+full transformer) arbitrarily well.
+The main observation is that MPNN + VN is able to ex-
+actly 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 & 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 O(1) depth and O(nd) width fol-
+lowing a result on DeepSets from Segol & Lipman (2019).
+We first introduce the permutation equivariant map, equiv-
+ariant DeepSets, and permutation equivariant universality.
+Definition 5.1 (permutation equivariant map). A map F :
+Rn×k → Rn×l satisfying F (σ · X) = σ · F (X) for all
+σ ∈ Sn and X ∈ Rn×d is called permutation equivariant.
+Definition 5.2 (equivariant DeepSets of Zaheer et al.
+(2017)). Equivariant DeepSets has the following form
+F (X) = Lds
+m◦ν◦· · ·◦ν◦Lds
+1 (X), where Lds
+i is a linear per-
+mutation equivariant layer and ν is a nonlinear layer such as
+ReLU. The linear permutation equivariant layer in DeepSets
+has the following form Lds
+i (X) = XA+ 1
+n11T XB+1cT ,
+where A, B ∈ Rdi×di+1, c ∈ Rdi+1 is the weights and bias
+in layer i, and ν is ReLU.
+Definition 5.3 (permutation equivariant universality).
+Given a compact domain X of Rn×din, permutation equiv-
+ariant universality of a model F : Rn×din → Rn×dout means
+that for every permutation equivariant continuous function
+H : Rn×din → Rn×dout defined over X, and any ϵ > 0,
+there exists a choice of m (i.e., network depth), di (i.e., net-
+work width at layer i) and the trainable parameters of F so
+that ∥H(X) − F (X)∥∞ < ϵ for all X ∈ X.
+The universality of equivariant DeepSets is stated as follows.
+Theorem 5.4 (Segol & Lipman (2019)). DeepSets with con-
+stant layer is universal. Using ReLU activation the width
+ω := maxidi (di is the width for i-th layer of DeepSets)
+required for universal permutation equivariant network sat-
+isfies ω ≤ dout + din +
+� n + din
+din
+�
+= O(ndin).
+We are now ready to state our main theorem.
+Theorem 5.5. MPNN + VN can simulate (not just approx-
+imate) equivariant DeepSets: Rn×d → Rn×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 O(1) depth and O(nd) width
+is permutation equivariant universal, and can approximate
+self-attention layer and transformers arbitrarily well.
+Proof. Equivariant DeepSets has the following form
+F (X) = Lds
+m ◦ ν ◦ · · · ◦ ν ◦ Lds
+1 (X), where Lds
+i is the
+
+On the Connection Between MPNN and Graph Transformer
+Table 2: Baselines for Peptides-func (graph classification) and Peptides-struct (graph regression). The perfor-
+mance metric is Average Precision (AP) for classification and MAE for regression. Bold: Best score.
+Model
+# Params.
+Peptides-func
+Peptides-struct
+Test AP before VN
+Test AP after VN ↑ Test MAE before VN Test MAE after VN ↓
+GCN
+508k
+0.5930±0.0023
+0.6623±0.0038
+0.3496±0.0013
+0.2488±0.0021
+GINE
+476k
+0.5498±0.0079
+0.6346±0.0071
+0.3547±0.0045
+0.2584±0.0011
+GatedGCN
+509k
+0.5864±0.0077
+0.6635±0.0024
+0.3420±0.0013
+0.2523±0.0016
+GatedGCN+RWSE
+506k
+0.6069±0.0035
+0.6685±0.0062
+0.3357±0.0006
+0.2529±0.0009
+Transformer+LapPE
+488k
+0.6326±0.0126
+-
+0.2529±0.0016
+-
+SAN+LapPE
+493k
+0.6384±0.0121
+-
+0.2683±0.0043
+-
+SAN+RWSE
+500k
+0.6439±0.0075
+-
+0.2545±0.0012
+-
+linear permutation equivariant layer and ν is an entrywise
+nonlinear activation layer. Recall that the linear equivariant
+layer has the form Lds
+i (X) = XA+ 1
+n11T XB +1cT . As
+one can use the same nonlinear entrywise activation layer ν
+in MPNN + VN, it suffices to prove that MPNN + VN can
+compute linear permutation equivariant layer Lds. Now we
+show that 2 layers of MPNN + VN can exactly simulate any
+given linear permutation equivariant layer Lds.
+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 xi ∈ Rd.
+At layer 1: VN node feature is 1
+n11T X, average of node
+features. The collection of features over n graph node fea-
+ture is XA. 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 XA + 1
+n11T XB + 1cT . Here we only need to
+perform the matrix multiplication of the VN feature with B,
+as well as add a bias c. This can be done by implementing a
+linear function for γgn.
+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 Lds
+i , which implies that MPNN + VN can simulate
+1 layer of DeepSets exactly with constant depth and con-
+stant 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 O(nd) width
+is able to approximate any continuous equivariant maps.
+As the self-attention layer 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 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. O(n) depth O(1) width MPNN + VN
+The previous section shows that we can approximate a full at-
+tention layer in Transformer using MPNN with O(1) depth
+but O(nd) 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 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 O(1) width and
+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 (V , δ) separable node fea-
+tures. This is needed to ensure that VN can approximately
+select one node feature to process at each iteration with
+attention αvn, the self-attention in the virtual node.
+Definition 6.1 ((V , δ) separable by ¯α). Given a graph G
+of size n and a fixed V ∈ Rn×d = [v1, ..., vn] and ¯α ∈ A,
+we say node feature X ∈ Rn×d of G is (V , δ) separable
+by some ¯α if the following holds. For any node feature xi,
+there exist weights W ¯α
+K, W ¯α
+Q in attention score ¯α such that
+¯α(xi, vi) > maxj̸=i ¯α(xj, vi) + δ. We say set X is (V , δ)
+separable by ¯α if every element X ∈ X is (V , δ) separable
+by ¯α.
+The use of (V , δ) separability is to approximate hard se-
+lection function arbitrarily well, which is stated below and
+proved in Appendix B.1.
+Lemma 6.2 (approximate hard selection). Given X is
+(V , δ) separable by ¯α for some fixed V ∈ Rn×d, ¯α ∈ A
+
+On the Connection Between MPNN and Graph Transformer
+Table 3: Test performance in graph-level OGB benchmarks (Hu et al., 2020). Shown is the mean ± s.d. of 10 runs.
+Model
+ogbg-molhiv
+ogbg-molpcba
+ogbg-ppa
+ogbg-code2
+AUROC ↑
+Avg. Precision ↑
+Accuracy ↑
+F1 score ↑
+GCN
+0.7606 ± 0.0097
+0.2020 ± 0.0024
+0.6839 ± 0.0084
+0.1507 ± 0.0018
+GCN+virtual node
+0.7599 ± 0.0119
+0.2424 ± 0.0034
+0.6857 ± 0.0061
+0.1595 ± 0.0018
+GIN
+0.7558 ± 0.0140
+0.2266 ± 0.0028
+0.6892 ± 0.0100
+0.1495 ± 0.0023
+GIN+virtual node
+0.7707 ± 0.0149
+0.2703 ± 0.0023
+0.7037 ± 0.0107
+0.1581 ± 0.0026
+SAN
+0.7785 ± 0.2470
+0.2765 ± 0.0042
+–
+–
+GraphTrans (GCN-Virtual)
+–
+0.2761 ± 0.0029
+–
+0.1830 ± 0.0024
+K-Subtree SAT
+–
+–
+0.7522 ± 0.0056
+0.1937 ± 0.0028
+GPS
+0.7880 ± 0.0101
+0.2907 ± 0.0028
+0.8015 ± 0.0033
+0.1894 ± 0.0024
+MPNN + VN + NoPE
+0.7676 ± 0.0172
+0.2823 ± 0.0026
+0.8055 ± 0.0038
+0.1727 ± 0.0017
+MPNN + VN + PE
+0.7687 ± 0.0136
+0.2848 ± 0.0026
+0.8027 ± 0.0026
+0.1719 ± 0.0013
+and δ > 0, the following holds. For any ϵ > 0 and i ∈ [n],
+there exists a set of attention weights Wi,Q, Wi,K in i-th
+layer of MPNN + VN such that αvn(xi, vi) > 1 − ϵ for
+any xi ∈ Xi. In other words, we can approximate a hard
+selection function fi(x1, ..., xn) = xi arbitrarily well on
+X by setting αvn = ¯α.
+With the notation set up, We now state an extra assumption
+needed for deep MPNN + VN case and the main theorem.
+AS3. X is (V , δ) separable by ¯α for some fixed V ∈ Rn×d,
+¯α ∈ A and δ > 0.
+Theorem 6.3. Assume AS 1-3 hold for the compact set X
+and L. Given any graph G of size n with node features X ∈
+X, and a self-attention layer L on G (fix WK, WQ, WV
+in α), there exists a O(n) layer of heterogeneous MPNN
++ VN with the specific aggregate/update/message function
+that can approximate L on 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 ˜xi, an approximation of xi via attention mechanism,
+enabled by Lemma 6.2, and send ˜xi to the virtual node.
+Virtual node will then pass the ˜xi to all graph nodes and
+computes the approximation of eα(xi,xj), ∀j ∈ [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
+7.1. 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 signifi-
+cant improvement over MPNN on Peptides-func and
+Peptides-struct for all MPNNs. To test the effec-
+tiveness 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 outper-
+forms transformers on Peptides-struct.
+7.2. Stronger MPNN + VN Implementation
+Next, by leveraging the modularized implementation from
+GraphGPS (Rampášek et al., 2022), we implemented a ver-
+sion of MPNN + VN with/without extra positional embed-
+ding. 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 em-
+bedding. 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 fol-
+lowed (Rampášek et al., 2022) and used the original vali-
+dation 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 +
+
+On the Connection Between MPNN and Graph Transformer
+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.
+Model
+PCQM4Mv2
+Test-dev MAE ↓
+Validation MAE ↓
+Training MAE
+# Param.
+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
+GRPE (Park et al., 2022)
+0.0898
+0.0890
+n/a
+46.2M
+EGT (Hussain et al., 2022)
+0.0872
+0.0869
+n/a
+89.3M
+Graphormer (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
+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
+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.
+7.3. 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., 2020a)
+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 lon-
+gitudes from 180.125◦E to 269.875◦E and latitudes from
+−14.875◦N to 14.875◦N. Following the procedure from
+(de Bezenac et al., 2018; de Bézenac 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◦-degree reso-
+lution, 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 ar-
+chitectures, and training scheme in Appendix D.4.
+We compare our model to other baselines including TF-
+Net (Wang et al., 2020a), a SOTA method for spatiotempo-
+ral forecasting, Linear Transformer (Katharopoulos et al.,
+2020a; Wang et al., 2020b) with Laplacian positional en-
+coding (LapPE), and Multilayer Perceptron (MLP). We use
+Mean Square Error (MSE) as the metric and report the er-
+rors 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.
+Table 5: Results of SST prediction.
+Model
+4 weeks
+2 weeks
+1 week
+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
+MPNN + VN
+0.2806
+0.2130
+0.1540
+Furthermore, aligned with our theory in Section 4, MPNN +
+VN indeed achieves comparable results with Linear Trans-
+former 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
+O(1)-depth O(1) width for arbitrary approximation error.
+For self-attention in full transformer, we prove that hetero-
+geneous MPNN + VN of either O(1) depth O(nd) width or
+O(n) depth O(1) width (under some assumptions) can ap-
+proximate 1 self-attention layer arbitrarily well. Compared
+to early results (Kim et al., 2022) showing GT can approx-
+imate 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 ef-
+forts, 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 re-
+quire 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 ques-
+tions is an important future direction for the graph learning
+community.
+
+On the Connection Between MPNN and Graph Transformer
+References
+Alon, U. and Yahav, E. On the bottleneck of graph neural
+networks and its practical implications. arXiv preprint
+arXiv:2006.05205, 2020.
+Battaglia, P. W., Hamrick, J. B., Bapst, V., Sanchez-
+Gonzalez, A., Zambaldi, V., Malinowski, M., Tacchetti,
+A., Raposo, D., Santoro, A., Faulkner, R., et al. Rela-
+tional inductive biases, deep learning, and graph networks.
+arXiv preprint arXiv:1806.01261, 2018.
+Brody, S., Alon, U., and Yahav, E. How attentive are graph
+attention networks? arXiv preprint arXiv:2105.14491,
+2021.
+Cai, C. and Wang, Y. A note on over-smoothing for graph
+neural networks. arXiv preprint arXiv:2006.13318, 2020.
+Chen, D., O’Bray, L., and Borgwardt, K. Structure-aware
+transformer for graph representation learning. In Interna-
+tional Conference on Machine Learning, pp. 3469–3489.
+PMLR, 2022.
+Child, R., Gray, S., Radford, A., and Sutskever, I. Gen-
+erating long sequences with sparse transformers. arXiv
+preprint arXiv:1904.10509, 2019.
+Choromanski, K., Likhosherstov, V., Dohan, D., Song, X.,
+Gane, A., Sarlos, T., Hawkins, P., Davis, J., Mohiuddin,
+A., Kaiser, L., et al. Rethinking attention with performers.
+arXiv preprint arXiv:2009.14794, 2020.
+Cybenko, G. Approximation by superpositions of a sig-
+moidal function. Mathematics of control, signals and
+systems, 2(4):303–314, 1989.
+de Bezenac, E., Pajot, A., and Gallinari, P. Deep learn-
+ing for physical processes: Incorporating prior scientific
+knowledge. In International Conference on Learning
+Representations, 2018. URL https://openreview.
+net/forum?id=By4HsfWAZ.
+de Bézenac, E., Pajot, A., and Gallinari, P. Deep learn-
+ing for physical processes: incorporating prior scien-
+tific knowledge. Journal of Statistical Mechanics: The-
+ory and Experiment, 2019(12):124009, dec 2019. doi:
+10.1088/1742-5468/ab3195. URL https://dx.doi.
+org/10.1088/1742-5468/ab3195.
+Dosovitskiy, A., Beyer, L., Kolesnikov, A., Weissenborn,
+D., Zhai, X., Unterthiner, T., Dehghani, M., Minderer, M.,
+Heigold, G., Gelly, S., et al. An image is worth 16x16
+words: Transformers for image recognition at scale. arXiv
+preprint arXiv:2010.11929, 2020.
+Dwivedi, V. P. and Bresson, X.
+A generalization
+of transformer networks to graphs.
+arXiv preprint
+arXiv:2012.09699, 2020.
+Dwivedi, V. P., Rampášek, L., Galkin, M., Parviz, A., Wolf,
+G., Luu, A. T., and Beaini, D. Long range graph bench-
+mark. arXiv preprint arXiv:2206.08164, 2022.
+d’Ascoli, S., Touvron, H., Leavitt, M. L., Morcos, A. S.,
+Biroli, G., and Sagun, L. Convit: Improving vision trans-
+formers with soft convolutional inductive biases. In In-
+ternational Conference on Machine Learning, pp. 2286–
+2296. PMLR, 2021.
+Gilmer, J., Schoenholz, S. S., Riley, P. F., Vinyals, O., and
+Dahl, G. E. Neural message passing for quantum chem-
+istry. In International conference on machine learning,
+pp. 1263–1272. PMLR, 2017.
+Han, K., Wang, Y., Chen, H., Chen, X., Guo, J., Liu, Z.,
+Tang, Y., Xiao, A., Xu, C., Xu, Y., et al. A survey on
+vision transformer. IEEE transactions on pattern analysis
+and machine intelligence, 2022.
+Hu, W., Fey, M., Zitnik, M., Dong, Y., Ren, H., Liu, B.,
+Catasta, M., and Leskovec, J. Open graph benchmark:
+Datasets for machine learning on graphs. Advances in
+neural information processing systems, 33:22118–22133,
+2020.
+Hu, W., Fey, M., Ren, H., Nakata, M., Dong, Y.,
+and Leskovec, J.
+Ogb-lsc: A large-scale challenge
+for machine learning on graphs.
+arXiv preprint
+arXiv:2103.09430, 2021.
+Huang, B., Liu, C., Banzon, V., Freeman, E., Gra-
+ham,
+G.,
+Hankins,
+B.,
+Smith,
+T.,
+and Zhang,
+H.-M.
+Improvements of the daily optimum inter-
+polation sea surface temperature (doisst) version
+2.1.
+Journal of Climate, 34(8):2923 – 2939, 2021.
+doi:
+10.1175/JCLI-D-20-0166.1.
+URL https:
+//journals.ametsoc.org/view/journals/
+clim/34/8/JCLI-D-20-0166.1.xml.
+Hussain, M. S., Zaki, M. J., and Subramanian, D. Global
+self-attention as a replacement for graph convolution. In
+Proceedings of the 28th ACM SIGKDD Conference on
+Knowledge Discovery and Data Mining, pp. 655–665,
+2022.
+Hwang, E., Thost, V., Dasgupta, S. S., and Ma, T. An
+analysis of virtual nodes in graph neural networks for link
+prediction. In Learning on Graphs Conference, 2022.
+Kalyan, K. S., Rajasekharan, A., and Sangeetha, S. Am-
+mus: A survey of transformer-based pretrained mod-
+els in natural language processing.
+arXiv preprint
+arXiv:2108.05542, 2021.
+Katharopoulos, A., Vyas, A., Pappas, N., and Fleuret, F.
+Transformers are rnns: Fast autoregressive transformers
+with linear attention. In Proceedings of the International
+
+On the Connection Between MPNN and Graph Transformer
+Conference on Machine Learning (ICML), 2020a. URL
+https://arxiv.org/abs/2006.16236.
+Katharopoulos, A., Vyas, A., Pappas, N., and Fleuret, F.
+Transformers are rnns: Fast autoregressive transformers
+with linear attention. In International Conference on
+Machine Learning, pp. 5156–5165. PMLR, 2020b.
+Kim, J., Nguyen, T. D., Min, S., Cho, S., Lee, M., Lee,
+H., and Hong, S. Pure transformers are powerful graph
+learners. arXiv preprint arXiv:2207.02505, 2022.
+Kingma, D. and Ba, J. Adam: A method for stochastic
+optimization. International Conference on Learning Rep-
+resentations, 12 2014.
+Kreuzer, D., Beaini, D., Hamilton, W., Létourneau, V., and
+Tossou, P. Rethinking graph transformers with spectral
+attention. Advances in Neural Information Processing
+Systems, 34:21618–21629, 2021.
+Li, Q., Han, Z., and Wu, X.-M. Deeper insights into graph
+convolutional networks for semi-supervised learning. In
+Thirty-Second AAAI conference on artificial intelligence,
+2018.
+Lim, D., Robinson, J., Zhao, L., Smidt, T., Sra, S., Maron,
+H., and Jegelka, S. Sign and basis invariant networks
+for spectral graph representation learning. arXiv preprint
+arXiv:2202.13013, 2022.
+Liu, Z., Lin, Y., Cao, Y., Hu, H., Wei, Y., Zhang, Z., Lin,
+S., and Guo, B. Swin transformer: Hierarchical vision
+transformer using shifted windows. In Proceedings of the
+IEEE/CVF International Conference on Computer Vision,
+pp. 10012–10022, 2021.
+Maron, H., Ben-Hamu, H., Shamir, N., and Lipman, Y.
+Invariant and equivariant graph networks. arXiv preprint
+arXiv:1812.09902, 2018.
+Mialon, G., Chen, D., Selosse, M., and Mairal, J. Graphit:
+Encoding graph structure in transformers. arXiv preprint
+arXiv:2106.05667, 2021.
+Oono, K. and Suzuki, T. Graph neural networks exponen-
+tially lose expressive power for node classification. arXiv
+preprint arXiv:1905.10947, 2019.
+Park, W., Chang, W.-G., Lee, D., Kim, J., et al. Grpe:
+Relative positional encoding for graph transformer. In
+ICLR2022 Machine Learning for Drug Discovery, 2022.
+Rampášek, L., Galkin, M., Dwivedi, V. P., Luu, A. T., Wolf,
+G., and Beaini, D. Recipe for a general, powerful, scal-
+able graph transformer. arXiv preprint arXiv:2205.12454,
+2022.
+Reynolds, R. W., Smith, T. M., Liu, C., Chelton, D. B.,
+Casey, K. S., and Schlax, M. G. Daily high-resolution
+blended analyses for sea surface temperature. J. Climate,
+20:5473–5496, 2007.
+Santoro, A., Raposo, D., Barrett, D. G., Malinowski, M.,
+Pascanu, R., Battaglia, P., and Lillicrap, T. A simple
+neural network module for relational reasoning. Advances
+in neural information processing systems, 30, 2017.
+Segol, N. and Lipman, Y. On universal equivariant set
+networks. arXiv preprint arXiv:1910.02421, 2019.
+Shi, Y., Zheng, S., Ke, G., Shen, Y., You, J., He, J.,
+Luo, S., Liu, C., He, D., and Liu, T.-Y. Benchmarking
+graphormer on large-scale molecular modeling datasets.
+arXiv preprint arXiv:2203.04810, 2022.
+Tay, Y., Dehghani, M., Bahri, D., and Metzler, D. Effi-
+cient transformers: A survey. ACM Computing Surveys
+(CSUR), 2020.
+Topping, J., Di Giovanni, F., Chamberlain, B. P., Dong,
+X., and Bronstein, M. M. Understanding over-squashing
+and bottlenecks on graphs via curvature. arXiv preprint
+arXiv:2111.14522, 2021.
+Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones,
+L., Gomez, A. N., Kaiser, Ł., and Polosukhin, I. At-
+tention is all you need. Advances in neural information
+processing systems, 30, 2017.
+Veliˇckovi´c, P., Cucurull, G., Casanova, A., Romero, A.,
+Lio, P., and Bengio, Y. Graph attention networks. arXiv
+preprint arXiv:1710.10903, 2017.
+Wang, R., Kashinath, K., Mustafa, M., Albert, A., and Yu,
+R. Towards physics-informed deep learning for turbulent
+flow prediction. pp. 1457–1466, 08 2020a. doi: 10.1145/
+3394486.3403198.
+Wang, R., Walters, R., and Yu, R. Meta-learning dynamics
+forecasting using task inference. In Oh, A. H., Agarwal,
+A., Belgrave, D., and Cho, K. (eds.), Advances in Neural
+Information Processing Systems, 2022. URL https:
+//openreview.net/forum?id=BsSP7pZGFQO.
+Wang, S., Li, B. Z., Khabsa, M., Fang, H., and Ma, H.
+Linformer: Self-attention with linear complexity. arXiv
+preprint arXiv:2006.04768, 2020b.
+Wolf, T., Debut, L., Sanh, V., Chaumond, J., Delangue, C.,
+Moi, A., Cistac, P., Rault, T., Louf, R., Funtowicz, M.,
+et al. Transformers: State-of-the-art natural language
+processing. In Proceedings of the 2020 conference on em-
+pirical methods in natural language processing: system
+demonstrations, pp. 38–45, 2020.
+
+On the Connection Between MPNN and Graph Transformer
+Wu, Q., Zhao, W., Li, Z., Wipf, D., and Yan, J. Nodeformer:
+A scalable graph structure learning transformer for node
+classification. In Advances in Neural Information Pro-
+cessing Systems, 2022.
+Wu, Z., Jain, P., Wright, M., Mirhoseini, A., Gonzalez,
+J. E., and Stoica, I. Representing long-range context for
+graph neural networks with global attention. Advances
+in Neural Information Processing Systems, 34:13266–
+13279, 2021.
+Yang, C., Wang, R., Yao, S., Liu, S., and Abdelzaher, T.
+Revisiting over-smoothing in deep gcns. arXiv preprint
+arXiv:2003.13663, 2020.
+Ying, C., Cai, T., Luo, S., Zheng, S., Ke, G., He, D., Shen, Y.,
+and Liu, T.-Y. Do transformers really perform badly for
+graph representation? Advances in Neural Information
+Processing Systems, 34:28877–28888, 2021.
+Zaheer, M., Kottur, S., Ravanbakhsh, S., Poczos, B.,
+Salakhutdinov, R. R., and Smola, A. J. Deep sets. Ad-
+vances in neural information processing systems, 30,
+2017.
+Zhao, L. and Akoglu, L. Pairnorm: Tackling oversmoothing
+in gnns. arXiv preprint arXiv:1909.12223, 2019.
+Zweig, A. and Bruna, J. Exponential separations in symmet-
+ric neural networks. arXiv preprint arXiv:2206.01266,
+2022.
+
+On the Connection Between MPNN and Graph Transformer
+A. Notations
+We provide a notation table for references.
+Table 6: Summary of important notations.
+Symbol
+Meaning
+X ∈ X ⊂ Rn×d
+graph node features
+xi ∈ R1×d
+graph node i’s feature
+˜xi ∈ R1×d
+approximated graph node i’s feature via attention selection
+M
+A multiset of vectors in Rd
+W (l)
+Q , W (l)
+K , W (l)
+V
+∈ Rd×d′
+attention matrix of l-th self-attention layer in graph transformer
+X
+feature space
+Xi
+projection of feature space onto i-th coordinate
+Lds
+i
+i-th linear permutation equivariant layer in DeepSets
+L, L′
+full self attention layer; approximate self attention layer in Performer
+z(l)
+vn , z(l)
+i
+virtual/graph node feature at layer l of heterogeneous MPNN + VN
+αvn
+attention score in MPNN + VN
+α(·, ·)
+normalized attention score
+αGATv2(·, ·)
+normalized attention score with GATv2
+α′(·, ·)
+unnormalized attention score. α′(u, v) = uWQ(WK)T vT
+α′
+GATv2(·, ·)
+unnormalized attention score with GATv2. α′
+GATv2(u, v) := aT LeakyReLU (W · [u∥v] + b)
+A
+space of attentions, where each element α ∈ A is of form α(u, v) = softmax(uWQ(WK)T vT )
+C1
+upper bound on norm of all node features ∥xi∥
+C2
+upper bound on the norm of WQ, WK, WV in target L
+C3
+upper bound on the norm of attention weights of αvn when selecting xi
+γ(k)(·, ·)
+update function
+θ(k)(·, ·)
+message function
+τ(·)
+aggregation function
+B. O(n) Heterogeneous MPNN + VN Layer with O(1) Width Can Approximate 1 Self Attention
+Layer Arbitrarily Well
+B.1. Assumptions
+Definition B.1 ((V , δ) separable by ¯α). Given a graph G of size n and a fixed V ∈ Rn×d = [v1, ..., vn] and ¯α ∈ A, we
+say node feature X ∈ Rn×d of G is (V , δ) separable by some ¯α if the following holds. For any node feature xi, there exist
+weights W ¯α
+K, W ¯α
+Q in attention score ¯α such that ¯α(xi, vi) > maxj̸=i ¯α(xj, vi) + δ. We say set X is (V , δ) separable by ¯α
+if every element X ∈ X is (V , δ) separable by ¯α.
+A special case of (V , δ) separable is when δ = 0, i.e., ∀i, ¯α(xi, vi) > maxj̸=i ¯α(xj, vi). We provide a geometric
+characterization of X being (V , 0) separable.
+Lemma B.2. Given ¯α and V , X is (V , 0) separable by ¯α ⇐⇒ xi is not in the convex hull spanned by {xj}j̸=i. ⇐⇒ there
+are no points in the convex hull of {xi}i∈[n].
+Proof. The second equivalence is trivial so we only prove the first equivalence. By definition, X is (V , 0) separable by ¯α
+⇐⇒ ¯α(xi, vi) > maxj̸=i ¯α(xj, vi)∀i ∈ [n] ⇐⇒ ⟨xi, W ¯α
+QW ¯α,T
+K
+vi⟩ > maxj̸=i⟨xj, W ¯α
+QW ¯α,T
+K
+vi⟩∀i ∈ [n].
+By denoting the v′
+i := W ¯α
+QW ¯α,T
+K
+vi ∈ Rd, we know that ⟨xi, v′
+i⟩ > maxj̸=i⟨xj, v′
+i⟩∀i ∈ [n], which implies that
+∀i ∈ [n], xi can be linearly seprated from {xj}j̸=i ⇐⇒ xi is not in the convex hull spanned by {xj}j̸=i, which concludes
+the proof.
+Lemma B.3 (approximate hard selection). Given X is (V , δ) separable by ¯α for some fixed V ∈ Rn×d, ¯α ∈ A and
+δ > 0, the following holds. For any ϵ > 0 and i ∈ [n], there exists a set of attention weights Wi,Q, Wi,K in i-th layer of
+
+On the Connection Between MPNN and Graph Transformer
+MPNN + VN such that αvn(xi, vi) > 1 − ϵ for any xi ∈ Xi. In other words, we can approximate a hard selection function
+fi(x1, ..., xn) = xi arbitrarily well on X by setting αvn = ¯α.
+Proof. Denote ¯α′ as the unnormalized ¯α. As X is (V , δ) separable by ¯α, by definition we know that ¯α(xi, vi) >
+maxj̸=i ¯α(xj, vi) + δ holds for any i ∈ [n] and xi ∈ M. We can amplify this by multiple the weight matrix in ¯α by a
+constant factor c to make ¯α′(xi, vi) > maxj̸=i ¯α′(xj, vi) + cδ. This implies that e¯α′(xi,vi) > ecδ maxj̸=i e¯α′(xj,vi). This
+means after softmax, the attention score ¯α(xi, vi) will be at least
+ecδ
+ecδ+n−1. We can pick a large enough c(δ, ϵ) such that
+¯α(xi, vi) > 1 − ϵ for any xi ∈ Xi and ϵ > 0.
+Proof Intuition and Outline. On the high level, i-th MPNN + VN layer will select ˜xi, an approximation i-th node feature
+xi via attention mechanism, enabled by Lemma 6.2, and send ˜xi to the virtual node. Virtual node will then pass the ˜xi to all
+graph nodes and computes the approximation of eα(xi,xj), ∀j ∈ [n]. Repeat such procedures n times for all graph nodes,
+and finally, use the last layer for attention normalization.
+The main challenge of the proof is to 1) come up with message/update/aggregation functions for heterogeneous MPNN
++ VN layer, which is shown in Appendix B.2, and 2) ensure the approximation error, both from approximating Aggre-
+gate/Message/Update function with MLP and the noisy input, can be well controlled, which is proved in Appendix B.4.
+We will first instantiate the Aggregate/Message/Update function for virtual/graph nodes in Appendix B.2, and prove that
+each component can be either exactly computed or approximated to an arbitrary degree by MLP. Then we go through an
+example in Appendix B.3 of approximate self-attention layer L with O(n) MPNN + VN layers. The main proof is presented
+in Appendix B.4, where we show that the approximation error introduced during different steps is well controlled. Lastly, in
+Appendix B.5 we show assumption on node features can be relaxed if a more powerful attention mechanism GATv2 (Brody
+et al., 2021) is allowed in MPNN + VN.
+B.2. Aggregate/Message/Update Functions
+Let M be a multiset of vectors in Rd. The specific form of Aggregate/Message/Update for virtual and graph nodes are listed
+below. Note that ideal forms will be implemented as MLP, which will incur an approximation error that can be controlled to
+an arbitrary degree. We use z(k)
+vn denotes the virtual node’s feature at l-th layer, and z(k)
+i
+denotes the graph node i’s node
+feature. Iteration index k starts with 0 and the node index starts with 1.
+B.2.1. VIRTUAL NODE
+At k-th iteration, virtual node i’s feature z(k)
+i
+is a concatenation of three component [˜xi, vk+1, 0] where the first component
+is the approximately selected node features xi ∈ Rd, the second component is the vi ∈ Rd that is used to select the node
+feature in i-th iteration. The last component is just a placeholder to ensure the dimension of the virtual node and graph node
+are the same. It is introduced to simplify notation.
+Initial feature is z(0)
+vn = [0d, v1, 0].
+Message function + Aggregation function τj∈[n]φ(k)
+vn-gn : R2d+1 × M → R2d+1 has two cases to discuss depending on value
+of k. For k = 1, 2, ..., n,
+τj∈[n]φ(k)
+vn-gn(z(k−1)
+vn
+, {z(k−1)
+i
+}i) =
+��
+i αvn(z(k−1)
+vn
+, z(k−1)
+i
+)z(k−1)
+i
+k = 1, 2, ..., n
+12d+1
+k = n + 1, n + 2
+(7)
+where z(k−1)
+vn
+= [˜xk−1, vk, 0]. z(k−1)
+i
+= [
+2d+1 dim
+�
+��
+�
+xi
+����
+d dim
+, ..., ...] is the node i’s feature, where the first d coordinates remain fixed for
+different iteration k. τj∈[n]φ(k)
+vn-gn use attention αvn to approximately select k-th node feature [
+2d+1 dim
+�
+��
+�
+xk
+����
+d dim
+, ..., ...]. Note that the
+particular form of attention αvn needed for soft selection is not important as long as we can approximate hard selection
+
+On the Connection Between MPNN and Graph Transformer
+arbitrarily well. As the z(k−1)
+vn
+contains vk and z(k−1)
+i
+contains xi (see definition of graph node feature in Appendix B.2.2),
+this step can be made as close to hard selection as possible, according to Lemma B.7.
+In the case of k = n + 1, τj∈[n]φ(k)
+vn-gn : R2d+1
+� �� �
+vn
+× M
+����
+set of gn
+→ Rd simply returns 12d+1. This can be exactly implemented by
+an MLP.
+Update function γ(k)
+vn
+: R2d+1
+� �� �
+vn
+× R2d+1
+� �� �
+gn
+→ R2d+1: Given the virtual node’s feature in the last iteration, and the selected
+feature in virtual node y = [xk, ..., ...] with αvn,
+γ(k)
+vn (·, y) =
+�
+�
+�
+�
+�
+[y0:d, vk+1, 0]
+k = 1, ..., n − 1
+[y0:d, 0d, 0]
+k = n
+12d+1
+k = n + 1, n + 2
+(8)
+where y0:d denotes the first d channels of y ∈ R2d+1. y denotes the selected node zi’s feature in Message/Aggregation
+function. γ(k)
+vn can be exactly implemented by an MLP for any k = 1, ..., n + 2.
+B.2.2. GRAPH NODE
+Graph node i’s feature vi ∈ R2d+1 can be thought of as a concatenation of three components [ xi
+����
+d dim
+, tmp
+����
+d dim
+, partialsum
+�
+��
+�
+1 dim
+],
+where xi, ∈ Rd, tmp ∈ Rd 3, and partialsum ∈ R.
+In particular, xi is the initial node feature. The first d channel will stay the same until the layer n + 2. tmp =
+�
+j∈subset of[n] eα′
+ijxj stands for the unnormalized attention contribution up to the current iteration. partialsum ∈ R
+is a partial sum of the unnormalized attention score, which will be used for normalization in the n + 2-th iteration.
+Initial feature z(0)
+gn = [xi, 0d, 0].
+Message function + Aggregate function: τj∈[n]φ(k)
+gn-vn : R2d+1 × R2d+1 → R2d+1 is just “copying the second argument”
+since there is just one incoming message from the virtual node, i.e., τj∈[n]φ(k)
+gn-vn(x, {y}) = y. This function can be exactly
+implemented by an MLP.
+Update function γ(k)
+gn : R2d+1
+� �� �
+gn
+× R2d+1
+� �� �
+vn
+→ R2d+1 is of the following form.
+γ(k)
+gn ([x, tmp, partialsum], y) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+[x, tmp, partialsum]
+k = 1
+[x, tmp + eα′(x,y0:d)WV y0:d,
+partialsum + eα′(x,y0:d)]
+k = 2, ..., n + 1
+[
+tmp
+partialsum, 0d, 0]
+k = n + 2
+(9)
+where α′(x, y0:d) is the usual unnormalized attention score. Update function γ(k)
+gn can be arbitrarily approximated by an
+MLP, which is proved below.
+Lemma B.4. Update function γ(k)
+gn can be arbitrarily approximated by an MLP from R2d+1 × R2d+1 to R2d+1 for all
+k = 1, ..., n + 2.
+Proof. We will show that for any k = 1, ..., n + 2, the target function γ(k)
+gn : R2d+1 × R2d+1 → R2d+1 is continuous and
+the domain is compact. By the universality of MLP in approximating continuous function on the compact domain, we know
+γ(k)
+gn can be approximated to arbitrary precision by an MLP.
+3tmp technicially denotes the dimension of projected feature by WV and does not has to be in Rd. We use Rd here to reduce the
+notation clutter.
+
+On the Connection Between MPNN and Graph Transformer
+Recall that
+γ(k)
+gn ([x, tmp, partialsum], y) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+[x, tmp, partialsum]
+k = 1
+[x, tmp + eα′(x,y0:d)WV y0:d,
+partialsum + eα′(x,y0:d)]
+k = 2, ..., n + 1
+[
+tmp
+partialsum, 0d, 0]
+k = n + 2
+it is easy to see that k = 1, γ(1)
+gn is continuous. We next show for k = 2, ..., n + 2, γ(1)
+gn is also continuous and all arguments
+lie in a compact domain.
+γ(k)
+gn
+is continuous because to a) α′(x, y) is continuous b) scalar-vector multiplication, sum, and exponential are all
+continuous. Next, we show that four component x, tmp, partialsum, y0:d all lies in a compact domain.
+x is the initial node features, and by AS1 their norm is bounded so x is in a compact domain.
+tmp is an approximation of eα′
+i,1WV x1 + eα′
+i,2WV x2 + .... As α′(xi, xj) is both upper and lower bounded by AS2 for all
+i, j ∈ [n] and xi is bounded by AS1, eα′
+i,1WV x1 + eα′
+i,2WV x2 + ... is also bounded from below and above. tmp will also
+be bounded as we can control the error to any precision.
+partialsum is an approximation of eα′
+i,1 + eα′
+i,2 + .... For the same reason as the case above, partialsum is also bounded
+both below and above.
+y0:d will be ˜xi at i-th iteration so it will also be bounded by AS1.
+Therefore we conclude the proof.
+B.3. A Running Example
+We provide an example to illustrate how node features are updated in each iteration.
+Time 0: All nodes are initialized as indicated in Appendix B.2. Virtual node feature z(0)
+vn = [0d, v1, 0]. Graph node feature
+z(0)
+i
+= [xi, 0d, 0] for all i ∈ [n].
+Time 1:
+For virtual node, according to the definition of τj∈[n]φ(1)
+vn-gn in Equation (7), it will pick an approximation of x1, i.e. ˜x1.
+Note that the approximation error can be made arbitrarily small. VN’s node feature z(1)
+vn = [˜x1, v2, 0].
+For i-th graph node feature, z(0)
+vn = 1d, and z(0)
+i
+= [xi, 0d, 0]. According to γ(k)
+gn in Equation (9), z(1)
+i
+= [xi, 0d, 0].
+Time 2:
+For the virtual node feature: similar to the analysis in time 1, VN’s feature z(2)
+vn = [˜x2, v3, 0] now. Note that the weights and
+bias in τj∈[n]φ(2)
+vn-gn will be different from those in τj∈[n]φ(1)
+vn-gn.
+For i-th graph node feature, as z(1)
+vn
+= [˜x1, v2, 0] and z(1)
+i
+= [xi, 0d, 0], according to γ(k)
+gn
+in Equation (9), z(2)
+i
+=
+[xi, e
+�
+α′
+i,1WV ˜x1, e
+�
+α′
+i,1]. Here �
+α′
+i,1 := α′(xi, ˜x1). We will use similar notations in later iterations. 4
+Time 3:
+Similar to the analysis above, z(3)
+vn = [�
+x3, v4, 0].
+z(3)
+i
+= [xi, e
+�
+α′
+i,1WV ˜x1 + e
+�
+α′
+i,2WV ˜x2, e
+�
+α′
+i,1 + e
+�
+α′
+i,2].
+Time n:
+z(n)
+vn = [˜xn, 0d, 0].
+4To reduce the notation clutter and provide an intuition of the proof, we omit the approximation error introduced by using MLP to
+approximate aggregation/message/update function, and assume the aggregation/message/update can be exactly implemented by neural
+networks. In the proofs, approximation error by MLP is handled rigorously.
+
+On the Connection Between MPNN and Graph Transformer
+z(n)
+i
+= xi, e
+�
+α′
+i,1WV ˜x1 + ... + e
+�
+α′
+i,n−1WV �
+xn−1
+�
+��
+�
+n−1 terms
+,
+e
+�
+α′
+i,1 + e
+�
+α′
+i,2 + ... + e
+�
+α′
+i,n−1]
+�
+��
+�
+n−1 terms
+.
+Time n + 1:
+According to Appendix B.2.1, in n + 1 iteration, the virtual node’s feature will be 1d.
+z(n+1)
+i
+= [xi, �
+k∈[n] e
+�
+α′
+ikWV ˜xk, �
+k∈[n] e
+�
+α′
+ik]
+Time n + 2 (final layer):
+For the virtual node, its node feature will stay the same.
+For the graph node feature, the last layer will serve as a normalization of the attention score (use MLP to approximate vector-
+scalar multiplication), and set the last channel to be 0 (projection), resulting in an approximation of [xi,
+�
+k∈[n] e
+�
+α′
+ik WV ˜xk
+�
+k∈[n] e
+�
+α′
+ik
+, 0].
+Finally, we need one more linear transformation to make the node feature become [
+�
+k∈[n] e
+�
+α′
+ik WV ˜xk
+�
+k∈[n] e
+�
+α′
+ik
+, 0d, 0]. The first d
+channel is an approximation of the output of the self-attention layer for node i where the approximation error can be made
+as small as possible. This is proved in Appendix B, and we conclude that heterogeneous MPNN + VN can approximate the
+self-attention layer L to arbitrary precision with O(n) MPNN layers.
+B.4. Controlling Error
+On the high level, there are three major sources of approximation error: 1) approximate hard selection with self-attention and
+2) approximate equation γ(k)
+gn with MLPs, and 3) attention normalization in the last layer. In all cases, we aim to approximate
+the output of a continuous map Lc(x). However, our input is usually not exact x but an approximation of ˜x. We also cannot
+access the original map Lc but instead, an MLP approximation of Lc, denoted as LMLP. The following lemma allows to
+control the difference between Lc(x) and LMLP(˜x).
+Lemma B.5. Let Lc be a continuous map from compact set to compact set in Euclidean space. Let LMLP be the ap-
+proximation of Lc by MLP. If we can control ∥x − ˜x∥ to an arbitrarily small degree, we can then control the error
+∥Lc(x) − LMLP(˜x)∥ arbitrarily small.
+Proof. By triangle inequality ∥Lc(x) − LMLP(˜x)∥ ≤ ∥Lc(x) − LMLP(x))∥ + ∥LMLP(x) − LMLP(˜x)∥.
+For the first term ∥Lc(˜x)−LMLP(˜x)∥, by the universality of MLP, we can control the error ∥Lc(˜x)−LMLP(˜x)∥ in arbitrary
+degree.
+For the second term ∥LMLP(x) − LMLP(˜x)∥, as LMLP is continuous on a compact domain, it is uniformly continuous by
+Heine-Cantor theorem. This means that we can control the ∥LMLP(x) − LMLP(˜x)∥ as long as we can control ∥x − ˜x∥,
+independent from different x. By assumption, this is indeed the case so we conclude the proof.
+Remark B.6. The implication is that when we are trying to approximate the output of a continuous map Lc on the compact
+domain by an MLP LMLP, it suffices to show the input is 1) ∥Lc − LMLP∥∞ and 2) ∥˜x − x∥ can be made arbitrarily small.
+The first point is usually done by the universality of MLP on the compact domain (Cybenko, 1989). The second point needs
+to be shown case by case.
+In the Appendix B.3, to simplify the notations we omit the error introduced by using MLP to approximate aggrega-
+tion/message/update functions (continuous functions on the compact domain of Rd.) in MPNN + VN. Lemma B.5 justify
+such reasoning.
+Lemma B.7 (˜xi approximates xi. �
+α′
+i,j approximates α′
+i,j.). For any ϵ > 0 and x ∈ X, there exist a set of weights for
+message/aggregate functions of the virtual node such that ||xi − ˜xi|| < ϵ and |α′
+i,j − �
+α′
+i,j| < ϵ.
+
+On the Connection Between MPNN and Graph Transformer
+Proof. By Lemma 6.2 We know that �
+αi,j := �α(xi, xj) → δ(i − j) as C3(ϵ) goes to infinity. Therefore we have
+||˜xi − xi|| = ||
+�
+j
+�
+αi,jxj − xi|| = ||
+�
+(�αi,j − δ(i − j))xj|| < ϵ
+�
+||xj|| < nC1ϵ
+(10)
+As n and C1 are fixed, we can make the upper bound as small as we want by increasing C3.
+|α′
+i,j−�
+α′
+i,j| = |α′(xi, xj)−α′
+MLP(˜xi, xj)| = |α′(xi, xj)−α′(˜xi, xj)|+|α′(˜xi, xj)−α′
+MLP(˜xi, xj)| = |α′(xi−˜xi, xj)| =
+(xi − ˜xi)T xjC2
+2 + ϵ < nC1ϵC1C2
+2 + ϵ = (nC2
+1C2
+2 + 1)ϵ. As α′
+i,j, �
+α′
+i,j is bounded from above and below, it’s easy to see
+that |eα′
+i,j − e
+�
+α′
+i,j| = |eα′
+i,j(1 − eα′
+i,j− �
+α′
+i,j)| < C(1 − eα′
+i,j− �
+α′
+i,j) can be controlled to arbitrarily degree.
+Theorem 6.3. Assume AS 1-3 hold for the compact set X and L. Given any graph G of size n with node features X ∈ X,
+and a self-attention layer L on G (fix WK, WQ, WV in α), there exists a O(n) layer of heterogeneous MPNN + VN with
+the specific aggregate/update/message function that can approximate L on X arbitrarily well.
+Proof. i-th MPNN + VN layer will select ˜xi, an arbitrary approximation i-th node feature xi via attention mechanism. This
+is detailed in the message/aggregation function of the virtual node in Appendix B.2.1. Assuming the regularity condition on
+feature space X, detailed in AS3, the approximation error can be made as small as needed, as shown in Lemmas 6.2 and B.7.
+Virtual node will then pass the ˜xi to all graph nodes, which computes an approximation of eα′(˜xi,xj), ∀j ∈ [n]. This step
+is detailed in the update function γ(k)
+gn of graph nodes, which can also be approximated arbitrarily well by MLP, proved
+in Lemma B.4. By Lemma B.5, we have an arbitrary approximation of eα′(˜xi,xj), ∀j ∈ [n], which itself is an arbitrary
+approximation of eα′(xi,xj), ∀j ∈ [n].
+Repeat such procedures n times for all graph nodes, we have an arbitrary approximation of �
+k∈[n] eα′
+ikWV xk ∈ Rd and
+�
+k∈[n] eα′
+ik ∈ R. Finally, we use the last layer to approximate attention normalization Lc(x, y) = x
+y , where x ∈ Rd, y ∈ R.
+As inputs for attention normalization are arbitrary approximation of �
+k∈[n] eα′
+ikWV xk and �
+k∈[n] eα′
+ik, both of them
+are lower/upper bounded according to AS1 and AS2. Since the denominator is upper bounded by a positive number, this
+implies that the target function Lc is continuous in both arguments. By evoking Lemma B.5 again, we conclude that we can
+approximate its output
+�
+k∈[n] eα′
+ik WV xk
+�
+k∈[n] eα′
+ik
+arbitrarily well. This concludes the proof.
+B.5. Relaxing Assumptions with More Powerful Attention
+One limitation of Theorem 6.3 are assumptions on node features space X: we need to 1) restrict the variability of node
+feature so that we can select one node feature to process each iteration. 2) The space of the node feature also need to satisfy
+certain configuration in order for VN to select it. For 2), we now consider a different attention function for αvn in MPNN +
+VN that can relax the assumptions AS3 on X.
+More powerful attention mechanism. From proof of Theorem 6.3, we just need α(·, ·) uniformly select every node in
+X ∈ X. The unnormalized bilinear attention α′ is weak in the sense that f(·) = ⟨xiWQW T
+K, ·⟩ has a linear level set. Such
+a constraint can be relaxed via an improved attention module GATv2. Observing the ranking of the attention scores given by
+GAT (Veliˇckovi´c et al., 2017) is unconditioned on the query node, Brody et al. (2021) proposed GATv2, a more expressive
+attention mechanism. In particular, the unnormalized attention score α′
+GATv2(u, v) := aT LeakyReLU (W · [u∥v] + b),
+where [·||·] is concatenation. We will let αvn = αGATv2 to select features in τj∈[n]φ(k)
+vn-gn.
+Lemma B.8. α′
+GATv2(·, ·) can approximate any continuous function from Rd × Rd → R. For any v ∈ Rd, a restriction of
+α′
+GATv2(·, v) can approximate any continuous function from Rd → R.
+Proof. Any function continuous in both arguments of α′
+GATv2 is also continuous in the concatenation of both arguments. As
+any continuous functions in R2d can be approximated by α′
+GATv2 on a compact domain according to the universality of MLP
+(Cybenko, 1989), we finish the proof for the first statement.
+
+On the Connection Between MPNN and Graph Transformer
+(a)
+(b)
+Figure 2: In the left figure, we have one example of X being (V , δ) separable, for which α can uniformly select any point
+(marked as red) xi ∈ Xi. In the right figure, we change αvn in MPNN + VN to αGATv2, which allows us to select more
+diverse feature configurations. The cluster in the middle cannot be selected by any α ∈ A but can be selected by αGATv2
+according to Proposition B.10.
+For the second statement, we can write W as 2 × 2 block matrix and restrict it to cases where only W11 is non-zero. Then
+we have
+α′
+GATv2(u, v) = aT LeakyReLU
+�� W11
+W12
+W21
+W22
+�
+·
+� u
+v
+�
++ b
+�
+= aT LeakyReLU (W11u + b)
+(11)
+which gives us an MLP on the first argument u. By the universality of MLP, we conclude the proof for the second statement.
+Definition B.9. Given δ > 0, We call X is δ nonlinearly separable if and only if mini̸=j d(Xi, Xj) > δ.
+AS 3’. X is δ nonlinearly separable for some δ > 0.
+Proposition B.10. If X ⊂ Rn×d satisfies that Xi is δ-separated from Xj for any i, j ∈ [n], the following holds. For any
+X ∈ X and i ∈ [n], there exist a αGATv2 to select any xi ∈ Xi. This implies that we can arbitrarily approximate the
+self-attention layer L after relaxing AS3 to AS3’.
+Proof. For any i ∈ [n], as Xi is δ-separated from other Xj, ∀j ̸= i, we can draw a region Ωi ⊂ Rd that contains Xi and
+separate Xi from other Xj(j ̸= i), where the distance from Xi from other Xj is at least δ according to the definition of
+Definition B.9. Next, we show how to construct a continuous function f whose value in Xi is at least 1 larger than its values
+in any other Xj ∀j ̸= i.
+We set the values of f in Xi to be 1.5 and values of f in Xj, ∀j ̸= i to be 0. We can then interpolate f in areas outside
+of ∪Xi (one way is to set the values of f(x) based on d(x, Xi), which results in a continuous function that satisfies our
+requirement. By the universality of αGATv2, we can approximate f to arbitrary precision, and this will let us select any
+Xi.
+C. On the Limitation of MPNN + VN
+Although we showed that in the main paper, MPNN + VN of varying depth/width can approximate the self-attention of
+full/linear transformers, this does not imply that there is no difference in practice between MPNN + VN and GT. Our
+theoretical analysis mainly focuses on approximating self-attention without considering computational efficiency. In this
+section, we mention a few limitations of MPNN + VN compared to GT.
+C.1. Representation Gap
+The main limitation of deep MPNN + VN approximating full self-attention is that we require a quite strong assumption:
+we restrict the variability of node features in order to select one node feature to process each iteration. Such assumption is
+relaxed by employing stronger attention in MPNN + VN but is still quite strong.
+For the large width case, the main limitation is the computational complexity: even though the self-attention layer requires
+O(n2) complexity, to approximate it in wide MPNN + VN framework, the complexity will become O(nd) where d is the
+dimension of node features.
+
+On the Connection Between MPNN and Graph Transformer
+We think such limitation shares a similarity with research in universal permutational invariant functions. Both DeepSets
+(Zaheer et al., 2017) and Relational Network (Santoro et al., 2017) are universal permutational invariant architecture but
+there is still a representation gap between the two (Zweig & Bruna, 2022). Under the restriction to analytic activation
+functions, one can construct a symmetric function acting on sets of size n with elements in dimension d, which can be
+efficiently approximated by the Relational Network, but provably requires width exponential in n and d for the DeepSets.
+We believe a similar representation gap also exists between GT and MPNN + VN and leave the characterization of functions
+lying in such gap as the future work.
+C.2. On The Difficulty of Approximating Other Linear Transformers
+In Section 4, we showed MPNN + VN of O(1) width and depth can approximate the self-attention layer of one type of
+linear transformer, Performer. The literature on efficient transformers is vast (Tay et al., 2020) and we do not expect MPNN
++ VN can approximate many other efficient transformers. Here we sketch a few other linear transformers that are hard to
+approximate by MPNN + VN of constant depth and width.
+Linformer (Wang et al., 2020b) projects the n×d dimension keys and values to k×d suing additional projection layers, which
+in graph setting is equivalent to graph coarsening. As MPNN + VN still operates on the original graph, it fundamentally
+lacks the key component to approximate Linformer.
+We consider various types of efficient transformers effectively generalize the virtual node trick. By first switching to a more
+expansive model and reducing the computational complexity later on, efficient transformers effectively explore a larger
+model design space than MPNN + VN, which always sticks to the linear complexity.
+C.3. Difficulty of Representing SAN Type Attention
+In SAN (Kreuzer et al., 2021), different attentions are used conditional on whether an edge is presented in the graph or not,
+detailed below. One may wonder whether we can approximate such a framework in MPNN + VN.
+In our proof of using MPNN + VN to approximate regular GT, we mainly work with Definition 3.4 where we do not use any
+gn-gn edges and therefore not leverage the graph topology. It is straightforward to use gn-gn edges and obtain the different
+message/update/aggregate functions for gn-gn edges non-gn-gn edges. Although we still achieve the similar goal of SAN to
+condition on the edge types, it turns out that we can not arbitrarily approximate SAN.
+Without loss of generality, SAN uses two types of attention depending on whether two nodes are connected by the edge.
+Specifically,
+ˆwk,l
+ij =
+�
+�
+�
+Q1,k,lhl
+i◦K1,k,lhl
+j◦E1,k,leij
+√dk
+if i and j are connected in sparse graph
+Q2,k,lhl
+i◦K2,k,lhl
+j◦E2,k,leij
+√dk
+otherwise
+�
+�
+�
+wk,l
+ij =
+�
+�
+�
+1
+1+γ · softmax
+��
+dk ˆwk,l
+ij
+�
+if i and j are connected in sparse graph
+γ
+1+γ · softmax
+��
+dk ˆwk,l
+ij
+�
+otherwise
+�
+�
+�
+(12)
+where ◦ denotes element-wise multiplication and Q1,k,l, Q2,k,l, K1,k,l, K2,k,l, E1,k,l, E2,k,l ∈ Rdk×d. γ ∈ R+is a
+hyperparameter that tunes the amount of bias towards full-graph attention, allowing flexibility of the model to different
+datasets and tasks where the necessity to capture long-range dependencies may vary.
+To reduce the notation clutter, we remove the layer index l, and edge features, and also consider only one-attention head
+case (remove attention index k). The equation is then simplified to
+ˆwij =
+�
+�
+�
+Q1hl
+i◦K1hl
+j
+√dk
+if i and j are connected in sparse graph
+Q2hl
+i◦K2hl
+j
+√dk
+otherwise
+�
+�
+�
+wij =
+�
+1
+1+γ · softmax (�
+d ˆwij)
+if i and j are connected in sparse graph
+γ
+1+γ · softmax (�
+d ˆwij)
+otherwise
+�
+(13)
+We will then show that Equation (13) can not be expressed (up to an arbitrary approximation error) in MPNN + VN
+framework. To simulate SAN type attention, our MPNN + VN framework will have to first simulate one type of attention
+for all edges, as we did in the main paper, and then simulate the second type of attention between gn-gn edges by properly
+
+On the Connection Between MPNN and Graph Transformer
+offset the contribution from the first attention. This turns out to be impossible as we cannot express the difference between
+two attention in the new attention mechanism.
+D. Experimental Details
+D.1. Dataset Description
+ogbg-molhiv and ogbg-molpcba (Hu et al., 2020) are molecular property prediction datasets adopted by OGB from
+MoleculeNet. These datasets use a common node (atom) and edge (bond) featurization that represent chemophysical
+properties. The prediction task of ogbg-molhiv is a binary classification of molecule?s fitness to inhibit HIV replication. The
+ogbg-molpcba, derived from PubChem BioAssay, targets to predict the results of 128 bioassays in the multi-task binary
+classification setting.
+ogbg-ppa (Wu et al., 2021) consists of protein-protein association (PPA) networks derived from 1581 species categorized
+into 37 taxonomic groups. Nodes represent proteins and edges encode the normalized level of 7 different associations
+between two proteins. The task is to classify which of the 37 groups does a PPA network originate from.
+ogbg-code2 (Wu et al., 2021) consists of abstract syntax trees (ASTs) derived from the source code of functions written in
+Python. The task is to predict the first 5 subtokens of the original function?s name.
+OGB-LSC PCQM4Mv2 (Hu et al., 2021) is a large-scale molecular dataset that shares the same featurization as ogbg-mol*
+datasets. It consists of 529,434 molecule graphs. The task is to predict the HOMO-LUMO gap, a quantum physical property
+originally calculated using Density Functional Theory. True labels for original ?test-dev? and ?test-challange? dataset
+splits are kept private by the OGB-LSC challenge organizers. Therefore for the purpose of this paper, we used the original
+validation set as the test set, while we left out random 150K molecules for our validation set.
+D.2. Reproducibility
+For LRGB results in Section 7.1, we reproduce the original results up to very small differences.
+Table 7: Reproduce the original results up to small differences. No VN is used.
+Model
+# Params.
+Peptides-func
+Peptides-struct
+Test AP (reproduce)
+Test AP ↑
+Test MAE (reproduce)
+Test MAE ↓
+GCN
+508k
+0.5918±0.0065
+0.5930±0.0023
+0.3468±0.0009
+0.3496±0.0013
+GINE
+476k
+0.5595±0.0126
+0.5498±0.0079
+0.3532±0.0024
+0.3547±0.0045
+GatedGCN
+509k
+0.5886±0.0027
+0.5864±0.0077
+0.3409±0.0011
+0.3420±0.0013
+GatedGCN+RWSE
+506k
+0.6083±0.0032
+0.6069±0.0035
+0.3377±0.0025
+0.3357±0.0006
+D.3. Additional Experiments
+We tested MPNN + VN on PascalVOC-SP datasets and also observe improvement, shown in Table 8, although the
+improvement is not as large as that of Peptides-func and Peptides-struct datasets. The best MPNN + VN model
+is GatedGCN + LapPE where the performance gap to the best GT model is rather small.
+D.4. Predicting Sea Surface Temperature
+In this experiment, we consider a specific physical modeling problem: forecasting sea surface temperature (SST), that
+is the water temperature close to the ocean’s surface. SST is an essential climate indicator and plays a significant role
+in analyzing and monitoring the dynamics of weather, climate, and other biological systems for several applications in
+environmental protection, agriculture, and industry. We use the NOAA/NESDIS/NCEI Daily Optimum Interpolation Sea
+Surface Temperature (DOISST) version 2.1 proposed by (Huang et al., 2021) as an improvement upon version 2.0 from
+(Reynolds et al., 2007). We consider the daily SST data of the Pacific Ocean from 1982 to 2021, in the region of longitudes
+from 180.125◦E to 269.875◦E and latitudes from −14.875◦N to 14.875◦N. We reduce the resolution of the original data
+from 0.25◦-degree to 0.5◦-degree. Following the procedure from (de Bezenac et al., 2018), (de Bézenac et al., 2019) and
+(Wang et al., 2022), we divide the region into 11 square batches of equal size (see Table 10), each contains exactly 30
+
+On the Connection Between MPNN and Graph Transformer
+Table 8: Baseline experiments for PascalVOC-SP and COCO-SP with rag-boundary graph on SLIC compactness
+30 for the node classification task. The performance metric is macro F1 on the respective splits (Higher is better). All
+experiments are run 4 times with 4 different seeds. The MP-GNN models are 8 layers deep, while the transformer-based
+models have 4 layers in order to maintain comparable hidden representation size at the fixed parameter budget of 500k.
+Bold: Best score.
+Model
+# Params
+PascalVOC-SP
+Before VN + Test F1
+After VN + Test F1 ↑
+GCN
+496k
+0.1268±0.0060
+0.1901±0.0040
+GINE
+505k
+0.1265±0.0076
+0.1198±0.0073
+GatedGCN
+502k
+0.2873±0.0219
+0.2874±0.0178
+GatedGCN+LapPE
+502k
+0.2860±0.0085
+0.3103±0.0068
+Transformer+LapPE
+501k
+0.2694±0.0098
+-
+SAN+LapPE
+531k
+0.3230±0.0039
+-
+SAN+RWSE
+468k
+0.3216±0.0027
+-
+Table 9: Number of training, validation and testing examples for each setting in the task of SST prediction.
+History window
+Prediction window
+Train size
+Validation size
+Test size
+6 weeks
+4 weeks
+147, 884
+3, 245
+7, 271
+2 weeks
+148, 038
+3, 399
+7, 425
+1 week
+148, 115
+3, 476
+7, 502
+longitudes and 30 latitudes that can be represented as a grid graph of 900 nodes in which we connect each node to its nearest
+8 neighbors. We take time series from 1982 to 2018 as our training set, data in 2019 as our validation set, and data from 2020
+to 2021 as our testing set. In our experiments, we set the history window wh as 6 weeks (i.e. 42 days) and the prediction
+window wp as 4 weeks (i.e. 28 days), 2 weeks (i.e. 14 days) or 1 week (i.e. 7 days). For each example, each node of the
+graph is associated with an input time series capturing the temperatures at the corresponding (longitude, latitude) for the
+last wh days, and the task is to predict the output time series of temperatures for the next wp days. We represent each time
+series as a long vector and the learning task is fundamentally a node-level regression task. We make sure that there is no
+overlapping among training, validation and testing sets (e.g., the output of a training example will not appear in any input
+of another validation example). The number of training, validation, and testing examples are roughly 150K, 3K and 7K,
+respectively for each setting (see Table 9). We compare our MPNN + VN model with:
+• Multilayer Perceptron (MLP) which treats both the input and output as long vectors and has 512 hidden neurons.
+• TF-Net (Wang et al., 2020a) with the setting as in the original paper.
+• Linear Transformer (Katharopoulos et al., 2020a) (Wang et al., 2020b)5 with Laplacian positional encoding (LapPE).
+We compute the first 16 eigenvectors as positions for LapPE.
+Both MPNN and MPNN + VN have 3 layers of message passing with 256 hidden dimensions. We apply an MLP with one
+hidden layer of 512 neurons on top of the network to make the final prediction.
+We train all our models with 100 epochs with batch size 20, initial learning rate 10−3, and Adam optimizer (Kingma & Ba,
+2014).
+5The
+Linear
+Transformer
+implementation
+is
+publicly
+available
+at
+https://github.com/lucidrains/
+linear-attention-transformer
+
+On the Connection Between MPNN and Graph Transformer
+Table 10: These are 11 regions of the Pacific in our experiment.
+Index
+Longitudes
+Latitues
+1
+[180.125◦E, 194.875◦E]
+[-14.875◦N, -0.125◦N]
+2
+[195.125◦E, 209.875◦E]
+[-14.875◦N, -0.125◦N]
+3
+[210.125◦E, 224.875◦E]
+[-14.875◦N, -0.125◦N]
+4
+[225.125◦E, 239.875◦E]
+[-14.875◦N, -0.125◦N]
+5
+[240.125◦E, 254.875◦E]
+[-14.875◦N, -0.125◦N]
+6
+[255.125◦E, 269.875◦E]
+[-14.875◦N, -0.125◦N]
+7
+[180.125◦E, 194.875◦E]
+[0.125◦N, 14.875◦N]
+8
+[195.125◦E, 209.875◦E]
+[0.125◦N, 14.875◦N]
+9
+[210.125◦E, 224.875◦E]
+[0.125◦N, 14.875◦N]
+10
+[225.125◦E, 239.875◦E]
+[0.125◦N, 14.875◦N]
+11
+[240.125◦E, 254.875◦E]
+[0.125◦N, 14.875◦N]
+

5dE0T4oBgHgl3EQfegDz/content/tmp_files/2301.02393v1.pdf.txt

@@ -0,0 +1,2304 @@
+IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. XX, XXXX
+1
+Graph Convolution Based Cross-Network
+Multi-Scale Feature Fusion for Deep Vessel
+Segmentation
+Gangming Zhao, Kongming Liang, Chengwei Pan, Fandong Zhang, Xianpeng Wu,
+Xinyang Hu, and Yizhou Yu, Fellow, IEEE
+Abstract— Vessel segmentation is widely used to help
+with vascular disease diagnosis. Vessels reconstructed
+using existing methods are often not sufficiently accurate
+to meet clinical use standards. This is because 3D vessel
+structures are highly complicated and exhibit unique char-
+acteristics, including sparsity and anisotropy. In this paper,
+we propose a novel hybrid deep neural network for ves-
+sel segmentation. Our network consists of two cascaded
+subnetworks performing initial and refined segmentation
+respectively. The second subnetwork further has two tightly
+coupled components, a traditional CNN-based U-Net and
+a graph U-Net. Cross-network multi-scale feature fusion is
+performed between these two U-shaped networks to effec-
+tively support high-quality vessel segmentation. The entire
+cascaded network can be trained from end to end. The
+graph in the second subnetwork is constructed according
+to a vessel probability map as well as appearance and
+semantic similarities in the original CT volume. To tackle
+the challenges caused by the sparsity and anisotropy of
+vessels, a higher percentage of graph nodes are distributed
+in areas that potentially contain vessels while a higher per-
+centage of edges follow the orientation of potential nearby
+vessels. Extensive experiments demonstrate our deep net-
+work achieves state-of-the-art 3D vessel segmentation per-
+formance on multiple public and in-house datasets.
+Index Terms— Vessel Segmentation, Graph Convolu-
+tional Networks, Deep Learning
+This work was funded in part by National Key Research and Develop-
+ment Program of China (No. 2019YFC0118101), National Natural Sci-
+ence Foundation of China (Grant Nos. 62141605 and 82072005), Key
+Program of Beijing Municipal Natural Science Foundation (No.7191003),
+and Zhejiang Province Key Research & Development Program (No.
+2020C03073). (Corresponding authors: Yizhou Yu and Xinyang Hu.)
+Gangming Zhao and Yizhou Yu are with the Department of Com-
+puter Science, The University of Hong Kong, Hong Kong (e-mail:
+gmzhao@connect.hku.hk, yizhouy@acm.org).
+Kongming Liang is with Pattern Recognition and Intelligent Sys-
+tem Laboratory, School of Artificial Intelligence, Beijing University
+of Posts and Telecommunications, Beijing, China (e-mail: liangkong-
+ming@bupt.edu.cn).
+Chengwei Pan is with Institute of Artificial Intelligence, Beihang Uni-
+versity, Beijing, China (e-mail: pancw@buaa.edu.cn).
+Fandong Zhang is with the AI Lab, Deepwise Healthcare, Beijing,
+China (e-mail: zhangfandong@deepwise.com).
+Xinyang Hu and Xianpeng Wu are with Department of Cardiol-
+ogy of the Second Affiliated Hospital, School of Medicine, Zhejiang
+University, Hangzhou, China, and Key Laboratory of Cardiovascular
+of Zhejiang Province, Hangzhou, China (e-mail: hxy0507@zju.edu.cn,
+wxpzju123@163.com)
+G. Zhao, K. Liang and C. Pan have equal contribution.
+I. INTRODUCTION
+V
+ESSEL segmentation is widely used in daily practice
+to characterize many vascular diseases [1], [2]. For
+example, the obstructed vessels may lead to coronary heart
+disease, which is the worldwide leading cause of death [3],
+[4]. Since clinicians mainly rely on interactive tracing and
+segmentation, vessel reconstruction is traditionally a very
+time-consuming process and affects the efficiency of diagnosis
+and intervention. Thus, automatic vessel segmentation can
+facilitate the reviewing process and plays an important role
+in medical image analysis.
+Over the years, numerous methods have been proposed for
+automatic vessel segmentation. Due to the state-of-the-art per-
+formance of convolutional neural networks (CNNs) on a wide
+range of pixel-level labelling tasks [5]–[7], CNNs has also
+been applied to vessel segmentation [8]–[10]. Nonetheless,
+the reconstructed vessels are often not sufficiently accurate to
+meet clinical use standards. This is because vessel structures
+in 3D CT volumes are highly complicated and exhibit unique
+characteristics. First, since vessels are thin structures, they
+only occupy a sparse subset of pixels. Thus, there exists
+a severe imbalance between vessel and non-vessel pixels.
+Second, vessel segments are elongated tubular structures that
+are highly directional and anisotropic. Conventional CNNs
+adopt uniform spatial sampling, and therefore, are inept at
+modeling such sparse and anisotropic structures, giving rise
+to broken or incomplete results. Thus it becomes critical to
+design deep neural networks that can effectively exploit the
+aforementioned characteristics of vessels.
+In this paper, we propose a novel hybrid deep neural
+network for vessel segmentation. Our network consists of two
+cascaded subnetworks performing initial and refined segmen-
+tation respectively. The second subnetwork further consists of
+two tightly coupled components, a traditional CNN-based U-
+shaped network and a graph-based U-shaped network based on
+graph convolutions. Cross-network multi-scale feature fusion
+is performed between these two U-shaped networks to ef-
+fectively support high-quality vessel segmentation. The entire
+cascaded network can be trained from end to end.
+As shown in previous work [11]–[13], graph convolutional
+networks naturally possess a complex shape modeling ability
+which is well suited for structured data. By setting local
+arXiv:2301.02393v1  [eess.IV]  6 Jan 2023
+
+EMB
+NPS
+UFFC
+SignalProcessing Society
+0222
+IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. XX, XXXX
+regions (supervoxels) in a CT volume as nodes and con-
+nections among nearby local regions as edges, the whole
+CT volume can be regarded as a graph. Specifically, the
+graph in the second subnetwork is constructed according to
+a vessel probability map as well as appearance and semantic
+similarities in the original CT volume. To tackle the challenges
+brought up by the aforementioned characteristics of vessels,
+a higher percentage of graph nodes are distributed in areas
+that potentially contain vessels while a higher percentage of
+edges follow the orientation of potential nearby vessels. In
+addition, the CNN-based U-shaped network is first utilized to
+extract multi-scale features from the original CT volume. Then
+at every scale, the features from the CNN are mapped into
+node features at the corresponding scale of the graph-based
+U-shaped network and propagated by the GCN at that scale
+to counteract sparsity and anisotropy. Finally, the enhanced
+features are reversely mapped into the spatial domain and
+fused with the original features extracted by the CNN-based
+U-shaped network.
+In summary, our contributions in this paper are three-fold:
+• We propose a cascaded deep neural network for vessel
+segmentation. The two subnetworks in the cascade are
+respectively responsible for initial and refined segmen-
+tation. There are a pair of tightly coupled U-shaped
+networks in the second subnetwork of the cascade, one
+based on CNN and the other based on GCN. Cross-
+network multi-scale feature fusion is performed between
+these two U-shaped networks to effectively support high-
+quality vessel segmentation.
+• We propose a novel way to transform a dense 3D CT
+volume to a sparse graph format, which can efficiently
+represent sparse and anisotropic vessel structures. More-
+over, our method integrates both appearance and semantic
+similarities for graph construction.
+• Extensive
+experiments
+indicate
+our
+deep
+network
+achieves state-of-the-art 3D vessel segmentation perfor-
+mance on multiple public and in-house datasets for coro-
+nary vessels as well as head and neck vessels, including
+the public ASOCA dataset.
+II. RELATED WORK
+A. Graph Convolutional Networks
+Though CNNs achieve impressive performance in many
+computer vision tasks, they can not efficiently handle graph-
+structured data. To operate directly on graphs, GCN [11] is
+proposed by using layer-wise propagation rule for neural net-
+work models. Li et al. [13] further adapted the residual/dense
+connections and dilated convolutions from CNNs into GCN
+which can solve vanishing gradient problem and increase the
+depth of GCN. Gao et al. [14] proposed graph pooling and
+unpooling operations to develop an encoder-decoder model
+on graph for node classification. The above methods show
+that GCNs can achieve promising results on modeling graph
+structure. However, it is still challenging to integrate GCNs
+into an existing image segmentation framework which is
+dominated by CNNs.
+B. Multi-scale feature modeling
+Multi-scale feature modeling can efficiently capture the
+global contextual dependencies which plays an important role
+in image segmentation. Kamnitsas et al. [15] proposed a dual
+pathway deep convolutional neural network. The proposed
+dual pathway network incorporates both local and larger con-
+textual information by processing the input images at multiple
+scales simultaneously. Chen et al. [16] proposed to use several
+parallel atrous convolution with different rates to model the
+contextual dependencies at multiple scales. Zhao et al. [17]
+proposed a pyramid pooling module to generate feature maps
+in different levels for scene parsing. Recently, Tao et al. [18]
+proposed to combine multi-scale predictions with attention
+mechanism and achieved the state-of-the-art on Cityscapes
+and Mapillary Vistas. However, all the above methods adopt
+uniform spatial sampling for multi-scale feature learning and
+fail to model the sparsity and anisotropy of vessel.
+C. Medical Image Segmentation
+Deep learning has become a methodology of choice for
+medical image segmentation. Ronneberger et al. proposed
+UNET [19], which has an encoder-decoder architecture. To
+avoid missing spatial information, the decoder features from
+the previous level are up-sampled and combined with the
+encoder features at the corresponding level through skip con-
+nections. The 3D version of UNET [20] was further proposed
+by replacing all 2D operations with their 3D counterparts. In
+addition, a hybrid densely connected UNET [21] was proposed
+to extract intra-slice features with a 2D DenseUNET and
+aggregate volumetric contexts with its 3D counterpart. Dou et
+al. [22] presented a 3D fully convolutional network equipped
+with a 3D deep supervision mechanism to combat potential
+optimization difficulties. Likewise, Zhu et al. [23] proposed to
+use eight additional deeply supervised layers in their architec-
+ture. Jiang et al. [24] developed two multi-resolution residually
+connected networks to simultaneously combine features across
+multiple image resolutions and feature levels. ACSNet [25]
+combines global contexts and local details to deal with the
+shape and size variations of segmented regions. Similarly,
+PraNet [26] aggregates multi-scale features and successively
+refines the segmentation map through boundary extraction.
+Recently, Isensee et al. proposed nnUNET [27], which auto-
+matically adapts its architecture according to the geometry of
+input images. Zhou et al. [28] introduced nnFormer, which is
+an encoder-decoder architecture for volumetric medical image
+segmentation through the combination of convolution layers
+and Transformer blocks. In addition, the gated axial-attention
+model in [29] extends the existing architectures and introduces
+an additional control mechanism with a Local-Global training
+strategy.
+D. Vessel Segmentation
+Vessel segmentation plays an important role in medical
+image analysis. Kong et al. [9] proposed to use a tree-
+structured convolutional gated recurrent unit (ConvGRU) layer
+for modeling the anatomical structure of vessels. Since the
+
+ZHAO et al.: GRAPH CONVOLUTION BASED CROSS-NETWORK MULTI-SCALE FEATURE FUSION FOR DEEP VESSEL SEGMENTATION
+3
+Stage1 Segmentation
+Loss
+UNET-1
+Encoder
+UNET-1
+Decoder
+𝑨𝟎
+UNET-0
+Encoder
+UNET-0
+Decoder
+𝒀𝟎
+Weight Initialized From UNET-0
+Loss
+Trained From Scratch
+UNET-G
+Encoder
+UNET-2
+Encoder
+UNET-G
+Decoder
+UNET-2
+Decoder
+Stage0 Segmentation
+Graph Construction
+Preliminary Segmentation
+Cross-network Multi-scale Feature Fusion
+Forward  
+Backward
+×
+Vessel Structure Modeling
+Forward  
+Backward
+Fig. 1.
+Our proposed pipeline for vessel segmentation consists of three stages, preliminary segmentation using a U-Net (UNET-0), graph
+construction, and final segmentation with a cascaded network, which further consists of two subnetworks with the first subnetwork being a U-
+Net (UNET-1) and the second subnetwork being a pair of tightly coupled U-shaped networks, a CNN-based U-Net (UNET-2) and a graph U-Net
+(UNET-G). The preliminary segmentation in the first stage is used by the second stage to construct a graph, whose topology becomes the first level
+graph in UNET-G.
+input of the ConvGRU layer is a uniform local patch, their
+method cannot well exploit the anisotropy of vessels. Wang
+et al. [10] proposed a multi-task network to predict a vessel
+segmentation mask and a distance map. Values in the map
+represent distances from the center to the surface of every
+vessel. However, the global structure of vessels is not consid-
+ered, which limits contextual dependency modeling. There is
+much work [8], [30]–[32] on the utilization of graph neural
+networks for vessel segmentation. Shin et al. [8] incorporated
+a GCN into a CNN architecture to exploit the global structure
+of vessel shape. However, only the pixel with maximum
+vessel probability within every rectangular patch is sampled
+as a graph node, which limits the representation ability of
+the graph. In addition, GCN features are only calculated at
+a single scale and do not interact with CNN features. In
+contrast, our framework exhibits a very different way to learn
+the structural information of vessels. Specifically, we exploit
+superpixel generation algorithms such as SLIC [33] to better
+model the sparsity and anisotropy of vessels, and tightly couple
+a graph U-Net and a traditional CNN-based U-Net through
+multi-scale feature fusion across these two networks to better
+support high-quality vessel segmentation.
+III. OUR FRAMEWORK
+A. Overview
+Consider an input 3D image volume X ∈ RD×H×W ,
+where D, H and W are the spatial depth, height and width
+respectively. The pipeline of our proposed method for vessel
+segmentation can be decomposed into three stages as shown
+in Fig. 1.
+Preliminary Segmentation. An U-shaped network, UNET-
+0, is first utilized to create a probability map of the input
+image volume. This probability map is used for discovering
+local image regions that have a relatively high probability
+to contain vessels. Since the probability map may not be
+very accurate, to reduce the chance of missing regions that
+actually contain vessels, we apply the dilation operator, a
+type of image morphological operators, to the probability
+map to increase the size of image areas with relatively high
+probability values. The result is a preliminary probability map
+denoted as A0 ∈ (0, 1)D×H×W , which is further thresholded
+to produce a preliminary segmentation mask denoted as Y 0 ∈
+{0, 1}D×H×W . The preliminary segmentation mask is used
+for indicating vessel orientations in regions where vessels are
+likely to occur. In our experiments, we use a 7 × 7 square as
+the kernel of the dilation operator.
+Graph Construction. On the basis of the preliminary segmen-
+tation mask Y 0 and probability map A0, a graph G = (V, E) is
+constructed with a node set V, and an edge set E. To counteract
+the characteristics of vessel structures including sparsity and
+anisotropy, a higher percentage of graph nodes are distributed
+in regions where the preliminary probability map has relatively
+large values while a higher percentage of edges follow the
+orientation of the preliminary vessel segmentation mask.
+Final Segmentation with a Cascaded Network. Instead
+of using a network to refine the preliminary segmentation
+result obtained in the first stage, we start from scratch and
+train a cascaded network that takes the original 3D image
+volume as the input, and performs end-to-end segmentation
+to produce the final segmentation result. This network con-
+sists of two cascaded subnetworks performing initial and
+refined segmentation respectively. The first subnetwork is an
+U-shaped network, UNET-1, that shares the same network
+
+4
+IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. XX, XXXX
+architecture with UNET-0 in the first stage, but have different
+network weights because it is trained together with the second
+subnetwork. The second subnetwork further consists of two
+tightly coupled components, a traditional CNN-based U-Net
+(UNET-2) and a graph U-Net (UNET-G) [14]. The graph G
+constructed in the second stage becomes the graph with the
+highest spatial resolution in UNET-G. Cross-network multi-
+scale feature fusion is performed between UNET-2 and UNET-
+G to effectively support high-quality vessel segmentation.
+UNET-1 and UNET-2 are cascaded. The output from UNET-1
+includes a hard mask and a soft probability map P. Since the
+input to UNET-2 is the product of P and the original input
+image I, the cascaded network is differentiable. Note that
+UNET-0 is used to construct graphs as a pre-process. Once
+the graphs for all training samples have been precomputed,
+the entire cascaded network can be trained from end to end
+through gradient backpropagation.
+Now let us focus on the second subnetwork. For UNET-2,
+we represent its convolutional encoder and decoder features
+as Ec
+1:L = {Ec
+l } and Dc
+1:L = {Dc
+l } respectively with L
+being the number of feature levels. The Lth decoder and
+encoder stages have the lowest spatial resolution. UNET-
+G has the same number of feature levels as UNET-2. The
+encoder and decoder stages in UNET-2 and UNET-G have
+one-to-one correspondence. For the lth encoder in UNET-G,
+its initial graph feature is created as Eg
+l = f(Ec
+l , G) through
+a forward mapping function f(·) proposed in
+[34] aiming
+to transform the features between spatial domain and node
+domain. The forward mapping function f(·) is called KNN-
+map, which utilizes the K nearest neighborhoods to create the
+corresponding node feature. Once graph convolutions have
+been performed on Eg
+l , the resulting graph convolutional
+feature is mapped back to the original convolutional feature
+space of UNET-2 through a backward mapping function g(·)
+also proposed in [34] and fused with its initial encoder feature
+Ec
+l .
+B. Graph Construction
+Graph Nodes. Since graph neural networks cannot process
+dense 3D images directly due to high computational cost, we
+first group all the pixels from a 3D image into super-pixels
+and then represent each super-pixel as a graph node. Here
+we use the SLIC algorithm [33] for super-pixel generation.
+In order to capture the 3D structure of vessels, the local
+region (super-pixel) represented by a graph node should satisfy
+the following properties: 1) the summation of the vessel
+probabilities in the region is high; 2) the pixels in the region
+have similar appearance; 3) the shape of the region follows
+the local shape of the vessels. The SLIC algorithm is based
+on a distance measure, which originally consists of two terms,
+grayscale difference and Euclidean distance. To satisfy the
+aforementioned properties, we add a third term based on
+geodesic distance. The updated distance measure for SLIC and
+its three terms are formulated as follows.
+d(i, j) = dgray(i, j) + ddis(i, j) + dgeo(i, j),
+(1)
+where
+dgray(i, j) = |Xi − Xj|,
+(2)
+Preliminary 
+Segmentation
+Y
+A
+SP Images
+10
+1     2
+3               4        5
+6
+7
+8              
+9
+10
+11
+1
+2
+3
+4
+1
+2
+3
+5
+4
+6
+7
+8
+9    10
+11
+13
+12
+1    2              
+4         5
+3 
+6                  
+7
+8 9                  
+10                  
+11                    13
+12                 
+Edge:
+1:   1-2           2:   2-3 2-4 2-5
+3:   3-4 3-6     4:   4-6
+5:   5-9           6:   6-7 6-11
+7:   7-8           8:   8-12
+9:   9-10         11: 11-12     
+12: 12-13
+Edge:
+1: 1-2           2: 2-3 
+3: 3-4           4: None
+Edge:
+1: 1-2                2:   2-3 2-4 2-5
+3: 3-4 3-6 3-7   4:   4-5 4-10
+5: 5-9                6:   6-7
+7: 7-8                10: 10-11
+Edge:
+1:   1-2 1-3     2: 2-3
+3:   3-4           4: 4-5 4-6 4-11
+5:   5-11         6: 6-7 6-8
+7:   7-8 7-9     8: 8-9 8-10
+10: 11-13
+Nodes and Edges
+CT Images
+1
+3
+2
+Fig. 2. A simple example illustrating the graph construction process.
+ddis(i, j) =
+�
+(xi − xj)2 + (yi − yj)2 + (zi − zj)2,
+(3)
+dgeo(i, j) = min
+Q∈Pi,j
+�
+q∈Q
+A0
+q∥∇(Xq + X0
+q ) · uq∥,
+(4)
+where Xi denotes the gray scale of the ith pixel, [xi, yi, zi]T
+denotes its spatial coordinates, Pi,j represents the complete
+set of paths from pixel i to pixel j, Q denotes one path in
+Pi,j, q denotes any pixel on Q, and uq represents the unit
+tangent vector of path Q at q. The geodesic distance between
+two points is defined as the minimum of the integration of
+X, X0 and A0 as in (4) among all the paths in Pi,j, where
+X0 = X ◦ Y 0 and ◦ stands for element-wise multiplication.
+∇(Xq + X0
+q ) means the gradient of Xq + X0
+q . Xq + X0
+q
+doubles the value in vessel areas, therefore, it will create more
+graph nodes in vessels because of a larger distance between
+different nodes in these areas. In practice, we use the Dijkstra’s
+algorithm to calculate the geodesic distance. The definition
+of geodesic distance in (4) ensures that regions potentially
+containing vessels have a higher density of graph nodes. Note
+that the three distance terms have been individually normalized
+before added together in the overall distance measure.
+Graph Edges. As there typically exist a large number of
+graph nodes in a 3D image volume, in this paper, we only
+consider locally connected graphs to reduce computational
+cost. Each node i is only connected to other nearby nodes
+whose geodesic distance is below a predefined threshold tgeo.
+That is, there exists an edge between nodes i and j if and only
+if d′
+geo(i, j) < tgeo, where d′
+geo(i, j) is a modified version
+of the geodesic distance in (4) where A0 is replaced with
+(1−A0). Since our geodesic distance is affected by the vessel
+mask and probability map, this connection rule implies that the
+Euclidean distance between two connected nodes has a larger
+threshold when the nodes are near potential vessels and the
+
+ZHAO et al.: GRAPH CONVOLUTION BASED CROSS-NETWORK MULTI-SCALE FEATURE FUSION FOR DEEP VESSEL SEGMENTATION
+5
+Layers
+Output size
+UNET-0,1,2
+en-conv0
+256 × 256 × 256
+conv(3 × 3 × 3, 16)
+en-conv1
+128 × 128 × 128
+2 × BuildBlock(3 × 3 × 3, 32)
+en-conv2
+64 × 64 × 64
+2 × BuildBlock(3 × 3 × 3, 64))
+en-conv3
+32 × 32 × 32
+2 × BuildBlock(3 × 3 × 3, 128)
+en-conv4
+16 × 16 × 16
+2 × BuildBlock(3 × 3 × 3, 256)
+de-conv4
+16 × 16 × 16
+2 × BuildBlock(3 × 3 × 3, 256)
+de-conv3
+32 × 32 × 32
+2 × BuildBlock(3 × 3 × 3, 128)
+de-conv2
+64 × 64 × 64
+2 × BuildBlock(3 × 3 × 3, 64)
+de-conv1
+128 × 128 × 128
+2 × BuildBlock(3 × 3 × 3, 32)
+de-conv0
+256 × 256 × 256
+2 × BuildBlock(3 × 3 × 3, 16)
+classifier
+256 × 256 × 256
+conv(1 × 1 × 1, 2)
+TABLE I
+NETWORK ARCHITECTURE OF UNET-0,1,2 USED IN THE PROPOSED
+PIPELINE. CONVOLUTION LAYERS IN THE ENCODER OF THE ORIGINAL
+U-NET ARE REPLACED WITH RESIDUAL BLOCKS. INSIDE THE
+BRACKETS ARE THE SHAPE OF THE RESIDUAL BLOCKS, AND OUTSIDE
+THE BRACKETS IS THE NUMBER OF STACKED BLOCKS IN A STAGE.
+DOWNSAMPLING (MAX POOLING) IS PERFORMED AFTER EN-CONV0,
+EN-CONV1, EN-CONV2, EN-CONV3 WITH STRIDE 2, RESPECTIVELY.
+UPSAMPLING IS PERFORMED AFTER EACH DE-CONV STAGE, AND THE
+NUMBER OF INPUT CHANNELS OF EACH LAYER CAN BE FOUND FROM
+THE PRECEDING LAYER.
+orientation of the edge between the nodes roughly follows the
+local orientation of the preliminary vessel mask. As a result,
+the constructed graph has denser and longer connections in
+regions potentially containing vessels.
+In our constructed graph, every edge is associated with an
+edge weight, which is a product of two components, ew =
+es
+wea
+w, where es
+w and ea
+w represent semantic consistency and
+appearance similarity respectively. For a convolutional feature
+map F in UNET-2, we first create its node representation FV ∈
+R|V |×C through the forward mapping function f(·) in [34] on
+the feature map F. Then we define the semantic consistency
+of the edge between nodes i and j as
+es
+w(i, j) = σ([F i
+V , F j
+V ]ws),
+(5)
+where F i
+V , F j
+V
+represent the i-th and j-th node features,
+ws ∈ R2C is a trainable weight vector fusing the two node
+features, and σ(·) is the sigmoid activation function. [] means
+a concatenation.
+We use the gray-scale information associated with graph
+nodes to define the appearance similarity of an edge as
+ea
+w(i, j) = σ([F i
+X, F j
+X]wg)
+(6)
+where FX = f(X◦Y 0◦A0, G), wg ∈ R2C is another trainable
+weight vector fusing the mapped features at the two nodes.
+Instead of using the gray-scale information X from the input
+image volume only, we also include the semantic information
+Y 0 and A0 from the preliminary segmentation to focus on
+potential vessel regions.
+A simple example illustrating the above graph construction
+process is given in Fig. 2.
+C. Cross-Network Multi-Scale Feature Fusion
+The features from UNET-2 need to be mapped into the
+node domain of UNET-G and further enhanced through graph
+convolutions over the constructed graph structure to better
+Dataset
+Avg #Nodes per Image
+Avg #Edges per Node
+ASOCA
+12000
+8.12
+ACA
+9600
+7.11
+HNA
+13000
+7.03
+TABLE II
+STATISTICS OF CONSTRUCTED GRAPHS. AVERAGE NUMBER OF NODES
+PER IMAGE IS CALCULATED USING ALL IMAGES IN A DATASET. AND
+AVERAGE NUMBER OF EDGES PER NODE IS CALCULATED USING ALL
+NODES IN A DATASET.
+Dataset
+Avg #Nodes per Image
+Avg #Edges per Node
+Set1 1: n segmetns is 28000
+ASOCA
+19010
+8.67
+ACA
+14300
+7.40
+HNA
+22420
+8.10
+Set1 2: n segmetns is 14000
+ASOCA
+12000
+8.12
+ACA
+9600
+7.11
+HNA
+13000
+7.03
+Set1 3: n segmetns is 7000
+ASOCA
+6020
+6.12
+ACA
+3110
+4.11
+HNA
+5200
+4.03
+Set1 4: n segmetns is 3500
+ASOCA
+3210
+4.12
+ACA
+2930
+3.03
+HNA
+2122
+2.14
+TABLE III
+AVERAGE NUMBER OF GRAPH NODES AND EDGES FOR DIFFERENT
+VALUES OF N_SEGMENTS WHEN MIN_SIZE_FACTOR IS FIXED TO 0.5.
+observe global priors of vessel connectivity. Afterwards we
+reversely map the enhanced features to the spatial domain of
+UNET-2 and fuse them with the original features there through
+a residual connection.
+Encoder Feature Fusion. The encoder feature map Ec
+l from
+the lth stage of UNET-2 is transformed into node features
+at the corresponding level of UNET-G through the forward
+mapping function f(·) defined in [34]. Then the mapped fea-
+tures are fused with the down-sampled encoder features from
+the previous stage in UNET-G. A residual graph convolution
+module Ω(·) is utilized to enhance the fused features for
+more accurately modeling complex vessel structures and better
+observing global priors of vessel connectivity. Therefore, the
+graph convolutional encoder features at the lth stage of UNET-
+G are created as
+Eg
+l = Ω(f(Ec
+l , G) + down(Eg
+l−1)).
+(7)
+Then the graph convolutional features Eg
+l
+are reversely
+mapped to the original convolutional feature space of UNET-
+2 through the backward mapping function g(·) defined in
+[34] and fused with its initial encoder feature to produce the
+enhanced encoder feature at the lth level,
+El = g(Eg
+l ) + Ec
+l .
+(8)
+Decoder Feature Fusion. The decoder feature Dc
+l from the
+lth stage of UNET-2 is transformed into node features at the
+corresponding level of UNET-G through the same forward
+
+6
+IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. XX, XXXX
+Ground Truth                              UNet
+nnUNet
+DVS
+DDT     
+Ours
+Ground Truth                              UNet
+nnUNet
+DVS
+DDT     
+Ours
+Ground Truth                              UNet
+nnUNet
+DVS
+DDT     
+Ours
+Fig. 3.
+From left to right, it is the ground truth, the results of UNET, nnUNET, DVS, DDT and our model, respectively. The aorta and the coronary
+vessels are marked with red and green. Although DDT achieves the best performance compared with other previous state-of-the-art methods, it
+may generate incomplete vessel masks when the structure of vessels is complicated.
+Dataset
+Avg #Nodes per Image
+Avg #Edges per Node
+Set2 1: min size factor is 0.3
+ASOCA
+8900
+8.01
+ACA
+7230
+6.89
+HNA
+9600
+7.01
+Set2 2: min size factor is 0.4
+ASOCA
+10300
+8.02
+ACA
+8410
+7.12
+HNA
+11200
+7.13
+Set2 3: min size factor is 0.5
+ASOCA
+12000
+8.12
+ACA
+9600
+7.11
+HNA
+13000
+7.03
+Set2 4 min size factor is 0.6
+ASOCA
+12100
+8.13
+ACA
+9870
+7.21
+HNA
+13210
+7.13
+TABLE IV
+AVERAGE NUMBER OF GRAPH NODES AND EDGES FOR DIFFERENT
+VALUES OF MIN_SIZE_FACTOR. N_SEGMENTS IS FIXED TO 14000.
+mapping function f(·). Then the mapped features are fused
+with the up-sampled decoder features from the previous stage
+in UNET-G, and the fused features are enhanced with the same
+residual graph convolution module Ω(·) before further fused
+with the graph encoder feature Eg
+l through the skip connection
+at the lth stage of UNET-G. Thus the graph convolutional
+decoder features at the lth stage of UNET-G are defined as
+Dg
+l = Ω(f(Dc
+l , G) + up(Dg
+l+1)) + Eg
+l .
+(9)
+Then the graph convolutional decoder features Dg
+l are re-
+versely mapped to the original feature space of UNET-2
+through the same backward mapping function g(·). We further
+fuse the reversely mapped features with both the initial decoder
+feature of UNET-2 and the skip-connected enhanced encoder
+feature El to produce the enhanced decoder feature at the lth
+level,
+Dl = g(Dg
+l ) + Dc
+l + El.
+(10)
+The last enhanced decoder feature is used to produce the final
+segmentation of vessels with a pixel-wise softmax classifier.
+Forward and Backward Mappings We adopt the forward
+and backward mapping functions defined in [34] to map pixel-
+level features in a CNN-based U-Net to node features in a
+graph U-Net and vice versa. The key consideration during
+feature mapping design lies in revealing the relations between
+node and pixel-level features. As illustrated in the following
+equations, the kNN (k Nearest Neighbor) based forward map-
+ping φk with its auxiliary matrix A aggregates pixel-level
+features over irregular regions to obtain corresponding node
+features adaptively according to their spatial relations.
+φk(F, N) = (Qf)T F,
+(11)
+
+ZHAO et al.: GRAPH CONVOLUTION BASED CROSS-NETWORK MULTI-SCALE FEATURE FUSION FOR DEEP VESSEL SEGMENTATION
+7
+Ground Truth              UNet
+nnUNet
+DVS
+DDT     
+Ours
+Fig. 4.
+Sample visual results on the ACA dataset. From left to right, it is the ground truth, the results of UNET, nnUNET, DVS, DDT and our model,
+respectively. The aorta and the coronary vessels are marked with red and green. Although DDT achieves the best performance compared with
+other previous state-of-the-art methods, it may generate incomplete vessel masks when the structure of vessels is complicated.
+Qf = A(Λf)−1,
+(12)
+Aij =
+�
+1
+if j th node is kNN of i th pixel
+0
+Otherwise
+,
+(13)
+where N ∈ {V, U} represents the node set corresponding
+to pixel-level spatial visual features F
+∈ RHW ×C, A ∈
+RHW ×|N | is an auxiliary matrix that assigns spatial features
+to k nearest graph nodes, Λf ∈ R|N |×|N | is a diagonal matrix,
+Λf
+jj =
+HW
+�
+i=1
+Aij, and Qf ∈ RHW ×|N | is a normalized form of
+A and serves as the forward mapping matrix.
+The backward mapping function ψk projects each graph
+node feature back to the spatial domain. The backward map-
+ping follows similar design principles as the forward mapping
+and makes use of the same number of nearest neighbors.
+Formally, ψk is formulated as follows.
+ψk(Z, N) = Qr[Z]e,
+(14)
+Qr = (Λr)−1A,
+(15)
+where N ∈ {V, U} represents the node set of the graph,
+A ∈ RHW ×|N | is similar to the definition in Equation 13,
+[·]e indicates the indexing operator which selects nodes in the
+graph, Λr ∈ RHW ×HW is a diagonal matrix, Λr
+ii =
+|N |
+�
+j=1
+Aij,
+and Qr ∈ RHW ×|N | is the backward mapping matrix, which
+is also a normalized form of A.
+IV. EXPERIMENTS
+A. Datasets
+ASOCA Automated Segmentation of Coronary Arteries
+Dataset (ASOCA) is a public dataset in MICCAI-2020 chal-
+lenge 1 which aims to segment the coronary artery lumen.
+The dataset consists of a training set of 40 Cardiac Computed
+Tomography Angiography (CCTA) images and a test set of
+20 CCTA images. The images in the testing set were anno-
+tated and verified by experts we invited. The original image
+resolution of the ASOCA dataset is 512×512×N, where N
+is between 168 and 334.
+ACA Aorta and Coronary Artery Dataset (ACA) is an in-
+house dataset which contains 1000 CCTA images. The dataset
+is utilized to segment both aorta and coronary arteries. Each
+image is annotated by one expert annotator and verified by a
+second expert. We split the dataset into a training set of 800
+images, a validation set of 100 images and a test set of 100
+images. The original image resolution of the ACA dataset is
+512×512×N, where N is between 192 and 600.
+HNA Head and Neck Artery Dataset (HNA) is an in-house
+dataset which contains 800 CT angiography (CTA) images of
+head and neck. The images are annotated in the same way as
+ACA. Cerebral, vertebral and carotid arteries are annotated as
+the target vessel mask. The dataset is split into a training set
+of 640 images, a validation set of 80 images and a test set of
+1https://asoca.grand-challenge.org
+
+8
+IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. XX, XXXX
+GT
+UNet
+nnUNet
+DVS
+DDT
+Ours
+Fig. 5.
+Sample visual results on the HNA dataset. From left to right, it is the ground truth, the results of UNET, nnUNET, DVS, DDT and our model,
+respectively.
+80 images. The original image resolution of the HNA dataset
+is also 512×512×N, where N is between 192 and 600.
+B. Experimental Setup
+Evaluation Metrics. Dice coefficient (DICE) and average
+symmetric surface distance (ASSD) (ASSD is measured in
+millimeters) are adopted as the evaluation metrics since they
+are commonly used in medical image segmentation [35]. In
+addition, to evaluate unique characteristics of tubular structure,
+another two metrics called skeleton recall (SR) and skeleton
+precision (SP) are defined as follows:
+SR(S, G) = |S � Q(G)|
+|Q(G)|
+,
+(16)
+SP(S, G) = |Q(S) � G|
+|Q(S)|
+,
+(17)
+where S and G are the segmentation result and the ground
+truth annotation respectively. The function Q(·) is used to
+acquire the skeleton of a tubular mask, which can preserve
+original vascular topology and connectivity. Here we use
+skeletonization function [36] as the implementation of Q(·).
+Network Structure and Training. Each sub-network of the
+proposed method is a U-shaped network. All CNN-based U-
+shaped networks, including UNET-0, UNET-1 and UNET-2
+are based on the original U-Net [19] except that the original
+convolution layers in its encoder are replaced with residual
+blocks [37]. The network architecture of UNET-0,1,2 used in
+the proposed pipeline is given in Table I. UNET-G is a graph
+U-Net [14]. Each downsampling operation in UNET-G halves
+the number of graph nodes, and each upsampling operation
+doubles the number of nodes. The feature dimension of every
+graph node is always set to 256 in all the experiments reported
+in this paper. The input image is always resized to 256×256×
+256, and the batch size on a single GPU is 2. The proposed
+cascaded network is trained by jointly optimizing the weighted
+cross-entropy loss, Lwbce = −βy·log(p)−(1−y)·log(1−p),
+and the dice loss, LDice = 1 −
+2y·p
+∥y∥1+∥p∥1 , where y and p
+are the ground-truth and predicted masks, respectively. We set
+β = 5 to increase the vessel recall. All models are trained
+for 100 epochs from scratch using PyTorch [38] on NVIDIA
+Titan Xp pascal GPUs. We set the weight decay to 1e-4 and
+use Adam [39] as the optimizer with the initial learning rate
+set to 1e-4. The learning rate is reduced by a factor of 10 after
+every 40 epochs.
+Graph Hyperparameter Setting. We use a 3D version of the
+SLIC algorithm [33] to generate superpixels. Two parameters
+of the algorithm control the total number of superpixels in an
+image. One of them is ‘n segments’, which is the maximum
+
+ZHAO et al.: GRAPH CONVOLUTION BASED CROSS-NETWORK MULTI-SCALE FEATURE FUSION FOR DEEP VESSEL SEGMENTATION
+9
+Fig. 6.
+Sample visual results on the HNA dataset. From left to right are the ground truth, the results of UNET, nnUNET, DVS, DDT and our model,
+respectively.
+number of superpixels, and the other is ‘min size factor’,
+which defines the ratio between the minimum size of a super-
+pixel and the average size of a superpixel. In the experiments
+reported in this paper, ‘n segments’ is always set to 14000,
+and ‘min size factor’ is set between 0.3 and 0.65.
+In a graph, each node i is only connected to other nearby
+nodes whose geodesic distance is below a predefined threshold
+tgeo. That is, there exists an edge between nodes i and j if
+and only if d′
+geo(i, j) < tgeo, where d′
+geo(i, j) is the geodesic
+distance between nodes i and j. tgeo is a hyperparameter that
+needs to be empirically set only once for each dataset. For the
+ASOCA dataset, tgeo is set to 0.30. For the ACA dataset, tgeo
+is set to 0.35. For the HNA dataset, tgeo is set to 0.40.
+Table II shows the statistics of graph nodes and edges.
+C. Comparison with the State of the Art
+We compared our proposed model with existing state-
+of-the-art algorithms for vessel segmentation on the three
+datasets. The methods in these comparisons include DDT [10],
+DVS [8], UNET3d [20], nnUNET [27], ResUNET [40],
+DenseUNET [21], PSP-Net [17] and HMSA [18]. DDT
+performs tubular structure modeling and is specifically de-
+signed for vessel segmentation. For medical image analysis,
+nnUNET is considered a strong baseline as it achieves state-
+of-the-art performance on many well-established segmentation
+challenges. PSP-Net [17] and HMSA [18] are included for
+comparison since they are state-of-the-art methods for generic
+semantic segmentation. In addition, we include DVS for com-
+parison since it also uses a GCN for structure modeling. Since
+the proposed framework is not limited to a specific backbone
+network, we integrate it with more powerful backbones, e.g.
+ResUNET, DenseUNET and H-DenseUNET. As shown in
+Table V, VI and VII, the performance can be improved.
+As shown in Table V, the proposed method achieves the
+state-of-the-art performance in terms of four evaluation metrics
+on the ASOCA dataset, and outperforms the top-6 methods in
+the challenge leaderboard. Specifically, our method achieves
+89.91% DICE, 0.530 ASSD 95.8% SP and 96.0% SR. The
+DICE of our method is higher than that of DDT and the top-1
+method in the leaderboard by around 1.5%.
+On the ACA and HNA datasets, the proposed method
+also achieves the best performance among all the methods
+considered in the comparisons. Specifically, the proposed
+method outperforms DVS by 4.7% and 3.2% on ACA and
+HNA respectively in terms of DICE. This demonstrates that
+multi-scale feature interaction between CNNs and GCNs is
+important for vessel structure modeling.
+The above experiments demonstrate the superiority of the
+proposed method on three vessel segmentation tasks. Com-
+pared to other methods [17], [18], [20], [27], [40], the main
+advantage of our approach is that it constructs a vessel graph
+to capture the 3D structure of vessels. On the basis of the
+constructed vessel graph, our proposed method uses GCNs
+to enhance feature propagation along vessel structures, and
+improve the interconnection between isolated vessel predic-
+
+GT
+UNet
+nnUNet
+DVS
+DDT
+Ours10
+IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. XX, XXXX
+Method
+DICE (%)
+ASSD
+SP (%)
+SR (%)
+LB-1st
+88.56
+-
+-
+-
+LB-2nd
+88.00
+-
+-
+-
+LB-3rd
+87.94
+-
+-
+-
+LB-4th
+87.36
+-
+-
+-
+LB-5th
+87.17
+-
+-
+-
+LB-6th
+87.11
+-
+-
+-
+DDT [10]
+88.21
+0.571
+95.0
+94.5
+DVS [8]
+87.32
+0.582
+94.1
+93.2
+UNET3d [20]
+83.20
+0.644
+93.0
+91.0
+ResUNET [40]
+83.20
+0.644
+93.0
+91.0
+DenseUNET [21]
+83.20
+0.644
+93.0
+91.0
+H-DenseUNET [21]
+83.20
+0.644
+93.0
+91.0
+nnUNET [27]
+85.11
+0.572
+94.3
+92.4
+PSP-Net [17]
+84.12
+0.593
+94.2
+92.1
+HMSA [18]
+86.23
+0.561
+95.2
+93.3
+Ours
+89.89
+0.544
+95.6
+95.9
+Ours+ResUNET
+89.90
+0.541
+95.7
+95.9
+Ours+DenseUNET
+89.89
+0.540
+95.7
+95.9
+Ours+H-DenseUNET
+89.91
+0.530
+95.8
+96.0
+TABLE V
+PERFORMANCE COMPARISON ON THE ASOCA DATASET AMONG
+STATE-OF-THE-ART SEGMENTATION ALGORITHMS. THE RESULTS OF
+MICCAI LEADERBOARD ARE SHOWN IN HTTPS://ASOCA.
+GRAND-CHALLENGE.ORG/EVALUATION/CHALLENGE/LEADERBOARD/,
+WHICH ONLY SHOWS THE PERFORMANCE IN TERMS OF DICE. FOR
+OTHER METHODS, WE EVALUATE THEM IN TERMS OF FOUR
+PERFORMANCE METRICS INCLUDING DICE, ASSD, SP AND SR.
+tions. Although DVS [8] and our method both exploit GCNs,
+the major distinction is that we use a super-pixel algorithm to
+generate graph nodes from a preliminary segmentation and the
+pixel values of the input image. Leveraging super-pixels makes
+our constructed graph more completely cover potential vessel
+regions, and therefore, improve the skeleton recall. In addition,
+we make use of forward and backward feature mappings to
+perform more thorough feature fusion between the CNN-based
+UNET and the graph UNET.
+To further validate the robustness of the proposed method,
+we collect two subsets of 35 hard samples from the test
+sets of ACA and HNA, respectively. Arteries in the chosen
+samples have calcifications, stents or tortuous segments, which
+significantly increase the difficulty of vessel segmentation
+in clinical practice. Experimental results in Table VIII and
+Table IX show that the proposed method performs the best on
+these two subsets, which demonstrates the robustness of the
+proposed method on hard samples.
+Furthermore, we compare the inference time complexity of
+state-of-the-art networks in Table X. As shown in the table, the
+inference time of our method for a computed tomography an-
+giography image is 0.190/0.193/0.198 second on the ASOCA,
+ACA and HNA datasets, respectively. Since we use a GPU-
+based implementation [41] of the SLIC algorithm to generate
+super-pixels, the graph construction step of our method is
+very efficient, and the overall inference time of our method
+is comparable to that of other methods.
+D. Ablation Study
+Ablation of graph node construction. We investigate the ef-
+fectiveness of the three components of Eqn. (1) for graph node
+construction on the ACA dataset. As shown in Table XIV, all
+Method
+DICE (%)
+ASSD
+SP (%)
+SR (%)
+DDT [10]
+91.2
+0.497
+96.0
+89.2
+DVS [8]
+90.1
+0.503
+95.1
+88.3
+UNET3d [20]
+87.3
+0.711
+94.0
+89.4
+ResUNET [40]
+88.4
+0.612
+95.1
+89.6
+DenseUNET [21]
+88.9
+0.568
+95.2
+89.4
+H-DenseUNET [21]
+89.9
+0.528
+95.3
+90.1
+nnUNET [27]
+88.3
+0.630
+95.5
+90.6
+PSP-Net [17]
+89.0
+0.642
+95.0
+89.6
+HMSA [18]
+90.2
+0.592
+96.7
+90.1
+Ours
+94.2
+0.448
+97.1
+95.1
+Ours+ResUNET
+94.6
+0.445
+97.3
+95.2
+Ours+DenseUNET
+94.8
+0.444
+97.3
+95.3
+Ours+H-DenseUNET
+94.8
+0.443
+97.3
+95.2
+TABLE VI
+PERFORMANCE COMPARISON ON THE ACA DATASET AMONG
+STATE-OF-THE-ART SEGMENTATION ALGORITHMS.
+Method
+DICE (%)
+ASSD
+SP (%)
+SR (%)
+DDT [10]
+92.4
+0.401
+96.1
+93.3
+DVS [8]
+91.3
+0.472
+97.2
+92.4
+UNET3d [20]
+87.3
+0.664
+95.1
+90.5
+ResUNET [40]
+87.7
+0.661
+95.2
+90.7
+DenseUNET [21]
+88.1
+0.618
+95.3
+90.8
+H-DenseUNET [21]
+89.1
+0.588
+95.7
+92.0
+nnUNET [27]
+89.9
+0.600
+94.3
+91.2
+PSP-Net [17]
+90.1
+0.593
+94.7
+90.0
+HMSA [18]
+91.4
+0.543
+95.6
+91.1
+Ours
+94.3
+0.379
+97.1
+96.3
+Ours+ResUNET
+94.4
+0.376
+97.1
+96.6
+Ours+DenseUNET
+94.4
+0.375
+97.2
+96.6
+Ours+H-DenseUNET
+94.5
+0.374
+97.2
+96.5
+TABLE VII
+PERFORMANCE COMPARISON ON THE HNA DATASET AMONG
+STATE-OF-THE-ART SEGMENTATION ALGORITHMS.
+the components play important roles in the node construction
+process, and dgeo is the most important for the segmentation
+performance. Removing A0 in dgeo leads to 0.6% performance
+drop and removing X0 leads to about 1.2% performance
+drop in terms of DICE. We further investigate how the hy-
+perparameter ‘n segments’ and ‘min size factor’ of the SLIC
+algorithm affect the performance of our method. For the
+ablation study on ‘n segments’, we first fix ‘min size factor’
+to 0.5 and change the value of ‘n segments’ to 28000
+(Set1 1), 14000 (Set1 2), 7000 (Set1 3), and 3500 (Set1 4).
+Then we fix ‘n segments’ to 14000 and change the value of
+‘min size factor’ to 0.3 (Set2 1), 0.4 (Set2 2), 0.5 (Set2 3),
+and 0.6 (Set2 4). For the above eight settings, we demonstrate
+how the number of nodes and edges changes in Table III and
+Table IV. Then we conduct an ablation study on all three
+datasets and report the results in Table XV. From the exper-
+imental results, we can see that our model achieves the best
+performance by setting ‘n segments’ and ‘min size factor’ to
+14000 and 0.5, respectively. The corresponding number of
+nodes per image is 12000, 9600, and 13000 on the ASOCA,
+ACA and HNA datasets, respectively.
+Ablation of graph edge construction. Next, we investi-
+gate the effectiveness of the two components of graph edge
+construction. As shown in Table XV, both es
+w and ea
+w are
+important for vessel segmentation. Furthermore, if we discard
+
+ZHAO et al.: GRAPH CONVOLUTION BASED CROSS-NETWORK MULTI-SCALE FEATURE FUSION FOR DEEP VESSEL SEGMENTATION
+11
+Method
+DICE (%)
+ASSD
+SP (%)
+SR (%)
+DDT [10]
+87.1
+0.511
+92.0
+88.2
+DVS [8]
+86.2
+0.544
+91.1
+87.3
+UNET3d [20]
+86.0
+0.722
+91.0
+87.4
+ResUNET [40]
+86.4
+0.712
+91.2
+87.6
+DenseUNET [21]
+86.5
+0.701
+91.4
+87.6
+H-DenseUNET [21]
+87.1
+0.690
+91.6
+87.9
+nnUNET [27]
+87.2
+0.631
+92.5
+86.6
+PSP-Net [17]
+84.0
+0.742
+91.0
+84.6
+HMSA [18]
+85.2
+0.762
+89.7
+85.2
+Ours
+92.1
+0.453
+96.4
+93.3
+Ours+ResUNET
+92.2
+0.451
+96.5
+93.5
+Ours+DenseUNET
+92.4
+0.450
+96.5
+93.6
+Ours+H-DenseUNET
+92.5
+0.448
+96.6
+93.6
+TABLE VIII
+PERFORMANCE COMPARISON ON A SUBSET OF HARD SAMPLES FROM
+THE ACA DATASET AMONG STATE-OF-THE-ART SEGMENTATION
+ALGORITHMS. ARTERIES IN THESE HARD SAMPLES HAVE
+CALCIFICATIONS, STENTS OR TORTUOUS SEGMENTS.
+Method
+DICE (%)
+ASSD
+SP (%)
+SR (%)
+DDT [10]
+88.4
+0.504
+94.1
+91.3
+DVS [8]
+87.3
+0.584
+93.2
+90.4
+UNET3d [20]
+86.4
+0.654
+92.1
+89.5
+ResUNET [40]
+86.6
+0.631
+92.6
+89.9
+DenseUNET [21]
+86.7
+0.598
+92.7
+89.9
+H-DenseUNET [21]
+86.9
+0.552
+92.9
+89.9
+nnUNET [27]
+87.9
+0.554
+93.3
+90.5
+PSP-Net [17]
+87.1
+0.567
+92.7
+89.0
+HMSA [18]
+88.4
+0.542
+92.1
+91.3
+Ours
+90.1
+0.449
+96.0
+94.2
+Ours+ResUNet
+90.6
+0.441
+96.2
+94.3
+Ours+DenseUNet
+90.6
+0.440
+96.1
+94.5
+Ours+H-DenseUNet
+90.8
+0.432
+96.3
+94.6
+TABLE IX
+PERFORMANCE COMPARISON ON A SUBSET OF HARD SAMPLES FROM
+THE HNA DATASET AMONG STATE-OF-THE-ART SEGMENTATION
+ALGORITHMS. ARTERIES IN THESE HARD SAMPLES HAVE
+CALCIFICATIONS OR TORTUOUS SEGMENTS.
+the two edge terms and use the traditional binary edge, the
+performance drops by 1.7% in terms of DICE.
+Ablation of cross-network feature fusion. To show the
+effectiveness of cross-network feature fusion, we first discard
+UNET-G of our proposed framework and only keep the
+cascaded UNET. As shown in Tables XI, XII and XIII, the
+performance drops by 2.0%, 1.9% and 2.77% on ACA, HNA
+and ASOCA respectively, which further validates the impor-
+tance of vessel structure modeling. In addition, we find that the
+improvement of UNET-G on ASOCA dataset is much more
+significant than ACA and HNA. As the training set of ASOCA
+only contains 40 CT images, this demonstrates that CNNs
+cannot well exploit the characteristics of vessels when the size
+of training data is small. Then we evaluate the effectiveness of
+using multi-scale fusion and graph convolution. Experimental
+results show that both components are important for vessel
+segmentation.
+E. Visualization
+As shown in Fig. 3, the proposed method generates higher
+quality vessel masks than other state-of-the-art algorithms,
+including DDT, in most of the cases. Specifically, the proposed
+Method
+ASOCA
+ACA
+HNA
+DDT [10]
+0.182
+0.184
+0.187
+DVS [8]
+0.186
+0.187
+0.190
+UNET3d [20]
+0.132
+0.134
+0.136
+nnUNET [27]
+0.201
+0.204
+0.206
+ResUNET [40]
+0.136
+0.137
+0.139
+DenseUNET [21]
+0.141
+0.144
+0.147
+H-DenseUNET [21]
+0.139
+0.142
+0.146
+PSP-Net [17]
+0.142
+0.144
+0.145
+HMSA [18]
+0.341
+0.344
+0.347
+Ours
+0.190
+0.193
+0.198
+Ours+ResUNET
+0.192
+0.195
+0.201
+Ours+DenseUNET
+0.196
+0.198
+0.204
+Ours+H-DenseUNET
+0.195
+0.197
+0.202
+TABLE X
+COMPARISON OF INFERENCE TIME AMONG STATE-OF-THE-ART
+SEGMENTATION ALGORITHMS ON THE ASOCA, ACA AND HNA
+DATASETS. THE AVERAGE INFERENCE TIME OF EACH ALGORITHM ON
+EACH DATASET IS SHOWN. THE UNIT IS SECOND PER SAMPLE.
+method can well exploit vessel structures and generate more
+complete vessel masks. In comparison to the proposed method,
+DDT may generate isolated segmentation masks since it is
+incapable of modeling the global structure of vessels. Fig. 4
+and Fig. 6 further visualize vessel segmentation results from
+different methods on the ACA and HNA datasets respectively.
+We add more examples for qualitative comparison in Fig. 7.
+The good cases show that our GCN-based cascaded network
+can improve vessel connectivity among individual vessel pre-
+dictions and achieve a higher skeleton recall. In the meantime,
+most of the false positive predictions can be removed. From
+the bad case, we find that the proposed method is limited when
+the initial segmentation is far from the ground truth. In such
+cases, vessel segmentation errors in the initial segmentation
+Fig. 7.
+Sample visual results with and without the graph module on the
+ACA dataset. From left to right, it is the ground truth, the result without
+the graph module and the result with the graph module, respectively.
+
+12
+IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. XX, XXXX
+UNET-0
+UNET-G
+Cfeature in UNET-G
+Graph Convolution Ω
+DICE (%)
+ASSD
+SP (%)
+SR (%)
+✓
+✓
+✓
+✓
+94.2
+0.448
+97.1
+95.1
+✓
+✓
+✓
+⊠
+93.1
+0.469
+95.3
+94.2
+✓
+✓
+⊠
+✓
+92.8
+0.487
+95.1
+94.0
+-
+⊠
+-
+-
+92.2
+0.469
+95.3
+94.2
+⊠
+✓
+✓
+✓
+92.4
+0.470
+95.9
+92.7
+TABLE XI
+EFFECTIVENESS OF DIFFERENT COMPONENTS ON THE ACA DATASET. ‘UNET-G’ MEANS THE GRAPH UNET STRUCTURE ON OUR MODEL. IF IT IS
+REMOVED, OUR FRAMEWORK IS DEGENERATED INTO A CASCADED MODEL WITH TWO CNN-UNET STRUCTURES. ‘CFEATURE IN UNET-G’ MEANS
+WE FUSE CNN FEATURES OF DIFFERENT STAGES INTO UNET-G . IF IT IS DISCARDED, THE FEATURES OF UNET-G ARE ONLY ACQUIRED FROM ITS
+FIRST GRAPH FEATURES Eg
+1 THAT IS ACQUIRED BY CONDUCTING FORWARD MAPPING f ON Ec
+1. ‘GRAPH CONVOLUTION Ω’ AIMS TO PROPAGATE
+MESSAGE AND FUSE THE CNN FEATURES INTO UNET-G. WE UTILIZE IT TO COMPARE THE IMPORTANCE OF THE VESSEL GRAPH MODELLING
+ABILITY.
+UNET-0
+UNET-G
+Cfeature in UNET-G
+Graph Convolution Ω
+DICE (%)
+ASSD
+SP (%)
+SR (%)
+✓
+✓
+✓
+✓
+94.3
+0.379
+97.1
+96.3
+✓
+✓
+✓
+⊠
+93.1
+0.412
+96.3
+95.4
+✓
+✓
+⊠
+✓
+93.0
+0.434
+96.1
+95.2
+-
+⊠
+-
+-
+92.4
+0.471
+95.9
+94.8
+⊠
+✓
+✓
+✓
+92.6
+0.462
+96.1
+94.9
+TABLE XII
+EFFECTIVENESS OF DIFFERENT COMPONENTS ON THE HNA DATASET.
+UNET-0
+UNET-G
+Cfeature in UNET-G
+Graph Convolution Ω
+DICE (%)
+ASSD
+SP (%)
+SR (%)
+✓
+✓
+✓
+✓
+89.89
+0.544
+95.6
+95.9
+✓
+✓
+✓
+⊠
+88.03
+0.567
+95.0
+94.7
+✓
+✓
+⊠
+✓
+88.01
+0.568
+94.8
+94.4
+-
+⊠
+-
+-
+87.12
+0.579
+94.8
+93.4
+⊠
+✓
+✓
+✓
+87.91
+0.573
+94.9
+93.8
+TABLE XIII
+EFFECTIVENESS OF DIFFERENT COMPONENTS ON THE ASOCA DATASET.
+cannot be completely corrected by our cascaded network.
+V. CONCLUSIONS AND FUTURE WORK
+In this paper, we have presented a cascaded deep neural
+network for vessel segmentation on CTA images. Our ap-
+proach represents a new paradigm for modeling the structural
+information of 3D vessels using deep neural networks through
+the interaction between a pair of CNN-based U-Net and
+graph U-Net. By fusing the features across these two types
+of networks, our method successfully tackles the challenges
+brought up by the sparsity and anisotropy of vessel structures.
+Extensive experiments on both public and in-house datasets
+verify the superiority and effectiveness of our method. By
+constructing a vessel graph to complement CNNs, our method
+not only outperforms baseline methods but also achieves the
+state-of-the-art performance with DICE 89.91/94.8/94.5 on the
+ASOCA/ACA/HNA datasets, respectively.
+Our proposed framework provides a stronger spatial struc-
+ture representation by learning 3D vessel connectivity priors.
+Our future work includes 1) building a more powerful graph
+neural network to enhance message passing in our cross-
+network feature fusion module, 2) investigating better graph
+construction methods by exploiting more domain knowledge
+from medical experts, and 3) building a high-quality annotated
+dataset and a friendly open-source code base for 3D vessel
+segmentation tasks.
+Acknowledgment
+The retrospective study on our in-house datasets has been
+approved by the institutional review board of the Second
+Affiliated Hospital of Zhejiang University School of Medicine,
+and was carried out following the principles of the Declaration
+of Helsinki.
+REFERENCES
+[1] F. Gr´elard, F. Baldacci, A. Vialard, and J.-P. Domenger, “New methods
+for the geometrical analysis of tubular organs,” Medical image analysis,
+vol. 42, pp. 89–101, 2017.
+[2] J. Leipsic, S. Abbara, S. Achenbach, R. Cury, J. P. Earls, G. J. Mancini,
+K. Nieman, G. Pontone, and G. L. Raff, “Scct guidelines for the
+interpretation and reporting of coronary ct angiography: a report of the
+society of cardiovascular computed tomography guidelines committee,”
+Journal of cardiovascular computed tomography, vol. 8, no. 5, pp. 342–
+358, 2014.
+[3] W. H. Organization et al., “Fact sheet: the top ten causes of death,” Fact
+sheet, no. 310, 2008.
+[4] V. L. Roger, A. S. Go, D. M. Lloyd-Jones, R. J. Adams, J. D. Berry,
+T. M. Brown, M. R. Carnethon, S. Dai, G. De Simone, E. S. Ford et al.,
+“Heart disease and stroke statistics—2011 update: a report from the
+american heart association,” Circulation, vol. 123, no. 4, pp. e18–e209,
+2011.
+[5] J. Long, E. Shelhamer, and T. Darrell, “Fully convolutional networks
+for semantic segmentation,” in Proceedings of the IEEE conference on
+computer vision and pattern recognition, 2015, pp. 3431–3440.
+[6] Z. Huang, X. Wang, L. Huang, C. Huang, Y. Wei, and W. Liu, “Ccnet:
+Criss-cross attention for semantic segmentation,” in Proceedings of the
+IEEE International Conference on Computer Vision, 2019, pp. 603–612.
+
+ZHAO et al.: GRAPH CONVOLUTION BASED CROSS-NETWORK MULTI-SCALE FEATURE FUSION FOR DEEP VESSEL SEGMENTATION
+13
+dgray
+ddis
+dgeo
+DICE
+ASSD
+SP
+SR
+-
+-
+X0
+A0
+-
+-
+-
+-
+✓
+✓
+✓
+✓
+94.2
+0.448
+97.1
+95.1
+⊠
+✓
+✓
+✓
+93.2
+0.462
+95.3
+94.2
+✓
+⊠
+✓
+✓
+93.3
+0.465
+95.2
+93.1
+✓
+✓
+⊠
+✓
+93.0
+0.478
+94.1
+92.2
+✓
+✓
+✓
+⊠
+93.6
+0.487
+95.1
+94.2
+⊠
+⊠
+✓
+✓
+93.2
+0.488
+94.2
+93.3
+⊠
+✓
+⊠
+✓
+93.7
+0.481
+95.0
+94.3
+⊠
+✓
+✓
+⊠
+93.8
+0.472
+95.3
+94.0
+✓
+⊠
+⊠
+✓
+92.9
+0.512
+94.0
+94.7
+✓
+⊠
+✓
+⊠
+92.8
+0.522
+93.7
+94.2
+✓
+✓
+���
+⊠
+92.8
+0.513
+94.1
+94.0
+⊠
+⊠
+⊠
+✓
+92.1
+0.552
+93.2
+93.5
+⊠
+⊠
+✓
+⊠
+92.0
+0.561
+93.1
+93.2
+⊠
+✓
+⊠
+⊠
+92.3
+0.557
+93.0
+93.1
+✓
+⊠
+⊠
+⊠
+92.1
+0.562
+93.4
+93.3
+⊠
+⊠
+⊠
+⊠
+92.1
+0.541
+95.1
+92.1
+TABLE XIV
+EFFECTIVENESS OF GRAPH NODE SET CONSTRUCTION ON THE ACA
+DATASET. WE REMOVE DIFFERENT COMPONENTS OF GRAPH NODES TO
+EXPLORE THEIR INFLUENCE ON OUR FRAMEWORK. NOTE THAT DICE,
+SP AND SR ARE PRESENTED AS PERCENTAGE.
+es
+w
+ea
+w
+DICE (%)
+ASSD
+SP (%)
+SR (%)
+-
+Y 0
+A0
+-
+-
+-
+-
+✓
+✓
+✓
+94.2
+0.448
+97.1
+95.1
+✓
+✓
+⊠
+93.6
+0.465
+91.3
+90.7
+✓
+⊠
+✓
+93.9
+0.462
+91.6
+90.1
+⊠
+✓
+✓
+93.7
+0.461
+91.0
+89.8
+✓
+⊠
+⊠
+92.4
+0.466
+90.2
+88.8
+⊠
+✓
+⊠
+92.3
+0.472
+90.1
+87.4
+⊠
+⊠
+✓
+92.2
+0.486
+90.3
+88.1
+⊠
+⊠
+⊠
+92.5
+0.484
+86.1
+85.2
+TABLE XV
+EFFECTIVENESS OF GRAPH EDGE SET CONSTRUCTION ON THE ACA
+DATASET. WE REMOVE DIFFERENT COMPONENTS OF GRAPH EDGES TO
+EXPLORE THEIR INFLUENCE ON OUR FRAMEWORK. IF ALL
+COMPONENTS ARE REMOVED, GRAPH EDGES BECOME THE
+TRADITIONAL BINARY EDGES.
+[7] J. Fu, J. Liu, H. Tian, Y. Li, Y. Bao, Z. Fang, and H. Lu, “Dual attention
+network for scene segmentation,” in Proceedings of the IEEE Conference
+on Computer Vision and Pattern Recognition, 2019, pp. 3146–3154.
+[8] S. Y. Shin, S. Lee, I. D. Yun, and K. M. Lee, “Deep vessel segmentation
+by learning graphical connectivity,” Medical image analysis, vol. 58, p.
+101556, 2019.
+[9] B. Kong, X. Wang, J. Bai, Y. Lu, F. Gao, K. Cao, J. Xia, Q. Song, and
+Y. Yin, “Learning tree-structured representation for 3d coronary artery
+segmentation,” Computerized Medical Imaging and Graphics, vol. 80,
+p. 101688, 2020.
+[10] Y. Wang, X. Wei, F. Liu, J. Chen, Y. Zhou, W. Shen, E. K. Fishman, and
+A. L. Yuille, “Deep distance transform for tubular structure segmentation
+in ct scans,” in Proceedings of the IEEE/CVF Conference on Computer
+Vision and Pattern Recognition, 2020, pp. 3833–3842.
+[11] T. N. Kipf and M. Welling, “Semi-supervised classification with graph
+convolutional networks,” arXiv preprint arXiv:1609.02907, 2016.
+[12] P. Veliˇckovi´c, G. Cucurull, A. Casanova, A. Romero, P. Lio, and Y. Ben-
+gio, “Graph attention networks,” arXiv preprint arXiv:1710.10903, 2017.
+[13] G. Li, M. Muller, A. Thabet, and B. Ghanem, “Deepgcns: Can gcns go
+as deep as cnns?” in Proceedings of the IEEE International Conference
+on Computer Vision, 2019, pp. 9267–9276.
+[14] H. Gao and S. Ji, “Graph u-nets,” in Proceedings of the 36th Interna-
+tional Conference on Machine Learning, 2019.
+[15] K. Kamnitsas, C. Ledig, V. F. Newcombe, J. P. Simpson, A. D. Kane,
+D. K. Menon, D. Rueckert, and B. Glocker, “Efficient multi-scale 3d
+cnn with fully connected crf for accurate brain lesion segmentation,”
+Medical Image Analysis, vol. 36, pp. 61–78, 2017.
+[16] L.-C. Chen, G. Papandreou, F. Schroff, and H. Adam, “Rethinking
+Method
+DICE (%)
+ASSD
+SP (%)
+SR (%)
+ASOCA Dataset
+Set1 1
+88.12
+0.593
+94.1
+95.1
+Set1 2
+89.89
+0.544
+95.6
+95.9
+Set1 3
+89.01
+0.566
+93.1
+94.2
+Set1 4
+88.76
+0.612
+93.0
+93.8
+Set2 1
+88.78
+0.641
+93.3
+94.0
+Set2 2
+88.64
+0.633
+93.1
+93.9
+Set2 3
+89.89
+0.544
+95.6
+95.9
+Set2 4
+87.19
+0.646
+92.9
+94.1
+ACA Dataset
+Set1 1
+93.46
+0.510
+95.4
+95.2
+Set1 2
+94.20
+0.448
+97.1
+95.1
+Set1 3
+93.12
+0.534
+95.1
+93.2
+Set1 4
+93.22
+0.512
+96.1
+94.3
+Set2 1
+93.80
+0.476
+96.2
+94.3
+Set2 2
+92.91
+0.487
+94.4
+93.2
+Set2 3
+94.20
+0.448
+97.1
+95.1
+Set2 4
+93.12
+0.493
+96.2
+93.1
+HNA Dataset
+Set1 1
+89.12
+0.498
+95.1
+94.0
+Set1 2
+90.10
+0.449
+96.0
+94.2
+Set1 3
+88.81
+0.564
+93.2
+92.9
+Set1 4
+89.11
+0.571
+92.6
+91.7
+Set2 1
+88.24
+0.464
+95.0
+94.0
+Set2 2
+89.98
+0.541
+95.1
+93.9
+Set2 3
+90.10
+0.449
+96.0
+94.2
+Set2 4
+90.01
+0.448
+95.7
+93.7
+TABLE XVI
+PERFORMANCE COMPARISON ON THE ASOCA, ACA, AND HNA
+DATASETS AMONG DIFFERENT SETTINGS OF N_SEGMENTS AND
+MIN_SIZE_FACTOR. PERFORMANCE IS MEASURED IN TERMS OF FOUR
+METRICS INCLUDING DICE, ASSD, SP AND SR.
+atrous convolution for semantic image segmentation,” arXiv preprint
+arXiv:1706.05587, 2017.
+[17] H. Zhao, J. Shi, X. Qi, X. Wang, and J. Jia, “Pyramid scene parsing
+network,” in Proceedings of the IEEE conference on computer vision
+and pattern recognition, 2017, pp. 2881–2890.
+[18] A. Tao, K. Sapra, and B. Catanzaro, “Hierarchical multi-scale attention
+for semantic segmentation,” arXiv preprint arXiv:2005.10821, 2020.
+[19] O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional networks
+for biomedical image segmentation,” in International Conference on
+Medical image computing and computer-assisted intervention. Springer,
+2015, pp. 234–241.
+[20]
+¨Ozg¨un C¸ ic¸ek, A. Abdulkadir, S. S. Lienkamp, T. Brox, and O. Ron-
+neberger, “3d u-net: Learning dense volumetric segmentation from
+sparse annotation,” in International Conference on Medical Image
+Computing and Computer-Assisted Intervention, 2016, pp. 424–432.
+[21] X. Li, H. Chen, X. Qi, Q. Dou, C.-W. Fu, and P.-A. Heng, “H-denseunet:
+hybrid densely connected unet for liver and tumor segmentation from ct
+volumes,” IEEE transactions on medical imaging, vol. 37, no. 12, pp.
+2663–2674, 2018.
+[22] Q. Dou, L. Yu, H. Chen, Y. Jin, X. Yang, J. Qin, and P.-A.
+Heng, “3d deeply supervised network for automated segmentation
+of volumetric medical images,” Medical Image Analysis, vol. 41,
+pp. 40–54, 2017, special Issue on the 2016 Conference on Medical
+Image Computing and Computer Assisted Intervention (Analog to
+MICCAI 2015). [Online]. Available: https://www.sciencedirect.com/
+science/article/pii/S1361841517300725
+[23] Q. Zhu, B. Du, B. Turkbey, P. L. Choyke, and P. Yan, “Deeply-supervised
+cnn for prostate segmentation,” in 2017 International Joint Conference
+on Neural Networks (IJCNN), 2017, pp. 178–184.
+[24] J. Jiang, Y.-C. Hu, C.-J. Liu, D. Halpenny, M. D. Hellmann, J. O. Deasy,
+G. Mageras, and H. Veeraraghavan, “Multiple resolution residually
+connected feature streams for automatic lung tumor segmentation from
+ct images,” IEEE Transactions on Medical Imaging, vol. 38, no. 1, pp.
+134–144, 2019.
+[25] R. Zhang, G. Li, Z. Li, S. Cui, D. Qian, and Y. Yu, “Adaptive context
+selection for polyp segmentation,” in International Conference on Med-
+ical Image Computing and Computer-Assisted Intervention.
+Springer,
+2020, pp. 253–262.
+
+14
+IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. XX, NO. XX, XXXX
+[26] D.-P. Fan, G.-P. Ji, T. Zhou, G. Chen, H. Fu, J. Shen, and L. Shao,
+“Pranet: Parallel reverse attention network for polyp segmentation,” in
+International Conference on Medical Image Computing and Computer-
+Assisted Intervention.
+Springer, 2020, pp. 263–273.
+[27] F. Isensee, J. Petersen, A. Klein, D. Zimmerer, P. F. Jaeger, S. Kohl,
+J. Wasserthal, G. Koehler, T. Norajitra, S. Wirkert et al., “nnu-net: Self-
+adapting framework for u-net-based medical image segmentation,” arXiv
+preprint arXiv:1809.10486, 2018.
+[28] H. Zhou, J. Guo, Y. Zhang, L. Yu, L. Wang, and Y. Yu, “nnformer:
+Interleaved transformer for volumetric segmentation,” CoRR, vol.
+abs/2109.03201, 2021. [Online]. Available: https://arxiv.org/abs/2109.
+03201
+[29] J. M. J. Valanarasu, P. Oza, I. Hacihaliloglu, and V. M. Patel, “Medical
+transformer: Gated axial-attention for medical image segmentation,” in
+Medical Image Computing and Computer Assisted Intervention – MIC-
+CAI 2021, M. de Bruijne, P. C. Cattin, S. Cotin, N. Padoy, S. Speidel,
+Y. Zheng, and C. Essert, Eds. Cham: Springer International Publishing,
+2021, pp. 36–46.
+[30] D. Zhang, S. Liu, S. Chaganti, E. Gibson, Z. Xu, S. Grbic, W. Cai,
+and D. Comaniciu, “Graph attention network based pruning for recon-
+structing 3d liver vessel morphology from contrasted ct images,” arXiv
+preprint arXiv:2003.07999, 2020.
+[31] L. Yao, P. Jiang, Z. Xue, Y. Zhan, D. Wu, L. Zhang, Q. Wang, F. Shi, and
+D. Shen, “Graph convolutional network based point cloud for head and
+neck vessel labeling,” in International Workshop on Machine Learning
+in Medical Imaging.
+Springer, 2020, pp. 474–483.
+[32] X. Xu, T. Wang, Y. Shi, H. Yuan, Q. Jia, M. Huang, and J. Zhuang,
+“Whole heart and great vessel segmentation in congenital heart disease
+using deep neural networks and graph matching,” in International
+Conference on Medical Image Computing and Computer-Assisted In-
+tervention.
+Springer, 2019, pp. 477–485.
+[33] R. Achanta, A. Shaji, K. Smith, A. Lucchi, P. Fua, and S. S¨usstrunk,
+“Slic superpixels compared to state-of-the-art superpixel methods,” IEEE
+transactions on pattern analysis and machine intelligence, vol. 34,
+no. 11, pp. 2274–2282, 2012.
+[34] Y. Liu, F. Zhang, Q. Zhang, S. Wang, Y. Wang, and Y. Yu, “Cross-view
+correspondence reasoning based on bipartite graph convolutional net-
+work for mammogram mass detection,” in Proceedings of the IEEE/CVF
+Conference on Computer Vision and Pattern Recognition, 2020, pp.
+3812–3822.
+[35] Q. Yue, X. Luo, Q. Ye, L. Xu, and X. Zhuang, “Cardiac segmentation
+from lge mri using deep neural network incorporating shape and spatial
+priors,” in International Conference on Medical Image Computing and
+Computer-Assisted Intervention.
+Springer, 2019, pp. 559–567.
+[36] S. van der Walt, J. L. Sch¨onberger, J. Nunez-Iglesias, F. Boulogne,
+J. D. Warner, N. Yager, E. Gouillart, T. Yu, and the scikit-image
+contributors, “scikit-image: image processing in Python,” PeerJ, vol. 2,
+p. e453, 6 2014. [Online]. Available: https://doi.org/10.7717/peerj.453
+[37] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image
+recognition,” in Proceedings of the IEEE conference on computer vision
+and pattern recognition, 2016, pp. 770–778.
+[38] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan,
+T. Killeen, Z. Lin, N. Gimelshein, L. Antiga et al., “Pytorch: An
+imperative style, high-performance deep learning library,” in Advances
+in neural information processing systems, 2019, pp. 8026–8037.
+[39] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,”
+arXiv preprint arXiv:1412.6980, 2014.
+[40] Z. Zhang, Q. Liu, and Y. Wang, “Road extraction by deep residual u-
+net,” IEEE Geoscience and Remote Sensing Letters, vol. 15, no. 5, pp.
+749–753, 2018.
+[41] SLIC CUDA, https://github.com/fderue/SLIC CUDA. [Online]. Avail-
+able: https://github.com/fderue/SLIC CUDA
+

69AzT4oBgHgl3EQfEvpR/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:7962f394cfc25c59442e20668c3410a72f925f539d3346875b4d618ab639b934
+size 8388653

6NE0T4oBgHgl3EQfewDQ/content/tmp_files/2301.02396v1.pdf.txt

@@ -0,0 +1,1674 @@
+Don’t follow the leader: Independent thinkers create scientific
+innovation
+Sean Kelty,1 Raiyan Abdul Baten,2 Adiba Mahbub Proma,2
+Ehsan Hoque,2 Johan Bollen,3 and Gourab Ghoshal1, 2, ∗
+1Department of Physics & Astronomy,
+University of Rochester, Rochester, NY 14607, USA
+2Department of Computer Science, University of Rochester, Rochester, NY 14607, USA
+3Luddy School of Informatics, Computing and Engineering,
+919 E. 10th St., Bloomington, IN 47408, USA
+Abstract
+Academic success is distributed unequally; a few top scientists receive the bulk of attention,
+citations, and resources. However, do these “superstars” foster leadership in scientific innovation?
+We introduce three information-theoretic measures that quantify novelty, innovation, and impact
+from scholarly citation networks, and compare the scholarly output of scientists who are either not
+connected or strongly connected to superstar scientists. We find that while connected scientists do
+indeed publish more, garner more citations, and produce more diverse content, this comes at a cost
+of lower innovation and higher redundancy of ideas. Further, once one removes papers co-authored
+with superstars, the academic output of these connected scientists diminishes. In contrast, authors
+that produce innovative content without the benefit of collaborations with scientific superstars
+produce papers that connect a greater diversity of concepts, publish more, and have comparable
+citation rates, once one controls for transferred prestige of superstars. On balance, our results
+indicate that academia pays a price by focusing attention and resources on superstars.
+∗ Correspondence email address: gghoshal@pas.rochester.edu
+1
+arXiv:2301.02396v1  [cs.DL]  6 Jan 2023
+
+I.
+INTRODUCTION
+“To truly make an apple pie from scratch you must first invent the universe”—a quote
+attributed to Carl Sagan [1]—illustrates the idea that the process by which individuals cre-
+ate is contingent upon the elements on which that creation is based. Whether creating a
+new piece of music, going about daily routines, or engaging in scientific research, people’s
+actions are founded in the information, experiences, and relationships that they have estab-
+lish by themselves and through others [2–5]. Each person has their own basis of knowledge
+that stems from their own lived experiences while also existing in a network of relationships
+through which they share experiences and knowledge with each other, thereby informing a
+collective understanding among a network of connected individuals [6]. Within such net-
+works, hierarchies can emerge in which some actors exert greater social influence over the
+network and thus the creative process that it supports, while others may influence only those
+closest to them or no one at all [7]. This social hierarchy is common in the societal dynamics
+of government and politics, where some individuals and institutions exert a great degree of
+influence over the flow of information in the system and opinion formation [8–10].
+Academia is not immune from the emergence of social hierarchies; some academics can
+function as figures of authority due to the merit and influence of their work and their promi-
+nent position in a network of academic collaborations. Citations as an indicator of academic
+influence [11] have long been known to be distributed very unequally[12], with a minority
+of a few scientists receiving most citations. Such inequality may be increasing at a global
+level[13], at least with respect to citation numbers. In academic publishing, biasing effects
+like this have been studied under the lens of the Matthew Effect, where success begets more
+success and early success compounds into a cumulative advantage as the “rich get richer”
+[14]. There are arguments that this effect is beneficial for academia; the rewards of top
+researchers are proportional to their contributions, which ensures the “epistemic security”
+of the field [15]. This thinking is aligned with the notion that science should operate as a
+meritocracy; those who contribute the most are also valued the most, and will therefore be
+most influential. Indeed, there is a high degree of trust in our most successful academics and
+the value of their mentorship. For instance, junior researchers collaborating with top scien-
+tists at the early stages of their career are likely to become top-cited scientists themselves,
+2
+
+especially those at less prestigious universities [16]. Inexperienced academics can benefit
+from apprenticeships with top scientists; the “chaperoning” of early-career scientists leads to
+higher rates of publication in high-impact journals [17]. These relationships are frequently
+mutually beneficial. Less visible authors benefit from more opportunities to publish papers
+in high quality journals that attract larger audiences, whereas top scientists gain collabo-
+rators with unique skills to produce more high quality work [18]. Close collaboration of
+less visible academics with those in the upper echelons can furthermore create opportunities
+for a first-mover advantage, inducing a positive feedback loop and early bandwagoning of
+innovative ideas [19].
+While top academics (sometimes referred to as “superstars”) may make consistent and
+high impact contributions that benefit their field and collaborators, their status as super-
+stars may also have deleterious effects due to the subsequent concentration of resources and
+attention. For instance, it has been shown that the collaborators of academic superstars
+experience a 5 to 9% drop in publication rates after the sudden death of that superstar [20],
+highlighting their dependence on the superstar’s collaboration. In fact, it is unclear whether
+collaborating with superstars truly fosters independent career development [21, 22] Further-
+more, superstars can induce a high degree of inequality in the distribution of research funding
+due to a funding Matthew-effect. Those who receive funding accumulate twice as much re-
+search funding afterwards compared to those who submitted similarly valued proposals but
+found themselves, by chance, just below the funding threshold. There is no evidence that
+this accumulation of research funding is due to actual achievements enabled by previous
+funding [23, 24]. If successful collaborations with superstars lead to early funding success,
+this can induce a superstar-fueled funding cycle that increasingly widens the gap between
+scientific haves and have-nots.
+The topology, structure, and characteristics of scientific collaboration networks may play
+an important role in these effects since they shape both the production and dissemination
+of ideas, potentially with conflicting outcomes. Tightly connected networks could be more
+efficient in distributing and leveraging knowledge thereby yielding higher productivity, but
+may at the same time lead to a decline of diversity, reducing exploration and discovery [25–
+27]. Although some spillover effects may occur, i.e. collaborators of highly-acclaimed authors
+benefit by proxy [28], it is not clear whether the concentration of attention of resources
+3
+
+towards superstars yields more novel and innovative research. This is a particularly relevant
+issue with the rise of interdisciplinary research which relies on the ability of scientists to
+collaborate in equitable teams that foster creativity and innovation across various research
+fields [29].
+To investigate the effects of superstar influence on academic productivity, impact, and
+innovation, we perform a comprehensive analysis of the American Physical Society corpus.
+Following [20], we define superstars as academics who are among the top .1% in terms of
+their h-index [30, 31]. We extract the semantic content of over 250,000 abstracts, defining
+a number of information-theoretic measures to quantify the novelty and innovation of each
+paper. We augment this with analysis of publication and citation rates, and examine the
+difference in academic output between researchers who collaborate with or cite frequently
+papers by superstars against those with little-to-no connection to such superstars. We find
+that at the individual level, collaborators and frequent citers of superstars, publish more,
+garner higher citations and produce papers with more diverse content compared to other
+academics. However, their work is no more innovative than the rest of the corpus and its
+content is more redundant. Further, once one excludes papers co-authored with superstars,
+their publication and citation output are no different from the rest of the corpus and in some
+cases output is lower.
+Focusing on early career researchers, we find that those who frequently collaborate with
+superstars in the beginning of their careers, do eventually go on to produce impressive
+academic output, although once the collaboration is removed, their output in terms of publi-
+cation rates, citation impact, and innovation is significantly diminished. On the other hand,
+early career researchers that produce innovative content without the benefit of early super-
+star collaboration, continue to produce such content over the rest of their careers. They
+publish more then early collaborators of superstars and accrue similar citation numbers,
+once one controls for the collaboration itself.
+4
+
+II.
+RESULTS
+A.
+Data
+We use the American Physical Society (APS) corpus [32] that contains articles published
+in APS journals since 1893.
+The data set contains full citation data, i.e. the citations
+pointing from the references of one article to another, allowing a reconstruction of the full
+citation network among all articles, including article-specific fields such as DOI, journal,
+volume, issue, first page and last page OR article id and number of pages, title, authors,
+affiliations, publication history, PACS codes, table of contents heading, article type, and
+copyright information. Given that the data does not include article abstracts, we used a
+web-scraping algorithm [33] to collect abstracts for 250,628 articles corresponding to between
+35-40% of all published papers across the different APS journals (Fig. S1). We note that
+around 1% of these articles have references not contained in the APS citation network, and
+on average we scraped abstracts for 38% of paper references. The distribution of citations
+and h-index are both heavy-tailed (Fig. S2), with the average number of citations being 14.4
+and the average h-index 1.74. Author disambiguation was done using a rule-based scoring
+method [34] (Cf. Sec.S1.2) We consider authors who first publish on or after 1970, and define
+superstars as those with the top .1% of h-index in the corpus, corresponding to an h-index
+threshold of 21. This yields 303 superstars among 292,394 authors. The summary statistics
+can be found in Tab. S1.
+In order to extract topics from the collected abstracts, we use an unsupervised Latent
+Dirichlet Allocation (LDA) algorithm on phrases (P-LDA) [35] to establish vector embed-
+dings for phrases and documents within our corpus. Stop words in the corpus were removed,
+all words were lemmatized, and phrases were determined based on a significance score that
+determined whether or not phrases occurred due to random chance. These vector embed-
+dings have dimensionality k correspoding to the number of topics defined for our corpus.
+P-LDA utilizes Gibbs Sampling to generate distributions of topics over phrases as well as
+documents [36], from which novelty scores can be extracted based on topic-spread.
+We
+choose a number of topics k based on the UMass coherence measure ([37]), the value of
+which first stabilizes at k = 25 topics (Fig. S3). Tab. S2 shows the top 10 terms per topic.
+The resulting output for each document u is a k-dimensional vector vu whose elements vu
+i
+5
+
+correspond to the frequency of topic i extracted from its abstract (example in Tab. S3).
+B.
+Novelty, innovation and redundancy
+Novelty detection in the literature has been implemented in a variety of ways [38], such
+as contextualizing novelty in machine learning as information retrieval [39, 40], distant com-
+binations of ideas via citation relations [41], first-pass combinations of concepts never before
+connected [42], knowledge-graphs of concepts within social networks [26], and agent-based
+simulations of social and individual learning [27].
+Here we rely on document-level embeddings that represent a distribution of all topics
+contained within the abstract of given paper, using which one can define the topic diversity
+in terms of a paper, its references, and articles that cite the paper. Using this, we define a
+variety of metrics capturing different aspects of novelty and innovation.
+Coupling connections between authors and the content of their works can then elucidate
+the influence that superstars have on the success of and novelty produced by other academics.
+Entropy: For a given document u, we define the Shannon entropy as
+I(S)
+u
+= −
+k
+�
+i=1
+vu
+i ln vu
+i ,
+(1)
+The expression quantifies the average level of “surprise” or uncertainty over the outcomes
+of a random variable [43]. In this context, papers focusing on limited number of topics in
+abstracts will yield low values of I(S)
+u , whereas those with a wide diversity of topics will yield
+a larger value of the entropy.
+Reference and Citation Diversity: While I(S)
+u
+measures the “surprise” with respect to a
+paper’s content, in this case its abstract, references and citations refer to the degree that the
+ideas in a given paper were inspired by other papers (references) or of inspiration to other
+papers (citations). We can thus measure the novelty of a paper, or its Information Diversity
+[44], by evaluating the dispersion of the topics of its references or the citations its receives.
+The greater the variance of the topic distribution, the higher the information diversity. For
+a set Xu, that can represent either the references in paper u, or citations to paper u, we
+6
+
+define the quantity,
+I(X)
+u
+=
+1
+|Xu|
+�
+l∈Xu
+�
+1 − cos
+�
+vl, Xu��
+(2)
+where cos
+�
+vl, Xu�
+is the cosine similarity of the vector embedding of a particular refer-
+ence/citation vl with the average over the vector embeddings of all references/citations in
+the set Xu. We can as such define reference diversity and citation diversity as the information
+diversity over the references from a paper and citations to the paper respectively.
+Innovation: The metrics defined thus far are based on topic models expressed as topic dis-
+tributions per document derived from the words in their content (abstracts). These metrics
+capture topic diversity of the paper itself, or its influences, but does not express the degree
+to which the paper expanded the literature through innovation. In other words, they express
+what document themselves are about, but not whether this adds to the diversity of the
+literature. We therefore define Innovation as the degree to which the document adds topics
+in new combination to the literature [45, 46]. Specifically, innovation in this context, is a
+measurement of when terms were first introduced or combined in the corpus (cf. Sec. S1.4
+and Fig. S4). Coupled with the novelty measures, this allows us to track how the diversity of
+ideas correlates with new conceptual recombinations and co-occurrences of terms. Following
+this logic, we define the Innovativeness of paper u as
+I(I)
+u
+= 1
+2
+�
+w1̸=w2∈u
+I(w1, w2; u)
+(3)
+where w1 and w2 are distinct terms in paper u, I(w1, w2; u) is an indicator function that is
+1 if terms w1 and w2 are first seen within the corpus in paper u and 0 otherwise, and the 1
+2
+prefix accounts for double counting. To remove spurious conceptual links due to chance or
+extreme rarity, we calculate a point-wise mutual information for all links as the log ratio of
+co-occurrence probability over the individual probabilities of each concept [46]. In Fig. S5 we
+determine the Pearson’s r correlation coefficients between each measure and find only weak
+correlations, indicating that each measure captures a different aspect of academic output.
+Redundancy: Finally, in a related context, in the field of creative ideation, it has been
+reported that inspirees stimulated by highly creative alters, tend to generate more creative
+ideas [47–49]. However, as a group, the inspirees ideas was found to be similar to each other
+7
+
+leading to redundancy in generated ideas over time at the group level. To check whether
+a similar effect manifests in academic publishing, we compute the cosine similarity score
+between papers u, u′ in the set P(G, s, t) thus
+Sim(G, s, t) =
+2
+|P(G, s, t)| (|P(G, s, t)| − 1)
+�
+u,u′∈P(G,s,t)
+cos(vu, vu′).
+(4)
+C.
+Superstar statistics
+We next examine whether the novelty and innovation produced by superstars are sig-
+nificantly different from the rest of the academic corpus. In Fig. 1 we plot the Reference
+and Citation diversity (Eq. (2)), the Shannon entropy (Eq. (1)) and Innovation (Eq. (3))
+comparing the set of superstar academics against the rest of the authors in the corpus. In
+terms of reference diversity, citation diversity and Shannon entropy, superstars outperform
+the remaining academics by 20%, 15%, and 2% respectively. That is, superstars are inspired
+by a higher diversity of content, publish works that are more conceptually diverse, and in-
+spire a wider array of publications than non-superstars. The starkest contrast can be seen in
+terms of Innovation, where there is a factor of ten difference between superstars and other
+academics indicating that the former are more prolific in introducing new combinations of
+terms. We note that there is a monotonic dependence of the metrics with number of pub-
+lications for all academics, although the effect is more pronounced for superstars (Fig. S6).
+Furthermore, there is also a monotonic dependence of citations received by a paper u and
+the novelty/innovation metrics (once again more pronounced for superstars) indicating that
+an increase in conceptual diversity and the ability to connect concepts for the first time is
+rewarded in terms of more attention paid to that paper (Fig. S7).
+D.
+Superstar influence
+Having established that superstars outperform other academics in terms of our metrics,
+we next determine to what degree superstars affect the academic output of their collabo-
+rators and their “inspirees” (those inspired by their work). Inspirees are authors that cite
+a superstar’s papers, for whom we determine the degree of inspiration by the frequency of
+8
+
+Non-Superstar
+Superstar
+0.09
+0.10
+0.11
+0.12
+0.13
+I(R )
+(A)
+Non-Superstar
+Superstar
+0.08
+0.10
+0.12
+I(C )
+(B)
+Non-Superstar
+Superstar
+3.0
+3.1
+3.2
+3.3
+I(S)
+(C)
+Non-Superstar
+Superstar
+0
+5
+10
+15
+20
+I(I)
+(D)
+FIG. 1. Average author-level statistics of novelty and innovation A Reference Diversity, B
+Citation Diversity, C Shannon Entropy, D Innovation. The orange bar is for superstars (h-index
+≥ 21) and the blue bars correspond to all other authors in the corpus.
+citations. We examine inspirees both at the group- and individual-levels. At the group-
+level, we center the superstar in a network of inspirees where the degree of inspiration is
+the number of times a researcher cites the superstar. We then partition the inspirees into
+groups based on their degree of inspiration, where the upper bounds for each bin are the top
+10% of inspirees, 20%, 30%, 50%, and 100%. These groups represent increasingly weakening
+ties to a given superstar; those in the top 10 percent are the most actively inspired, while
+the bottom 50 percent typically cite the superstar only once. Note that some inspirees in
+the bottom 50 group of one superstar may be in the top group of another superstar. The
+increasing bin sizes are chosen to account for the decreasing frequency of inspired citations
+among the least-inspired inspirees, such that there are sufficient number of papers compared
+between groups.
+Given that we are interested in the temporal evolution of superstar influence on the novelty
+and innovation of the inspirees, we denote the year of the first superstar publication as t0 = 0
+and for every susbsequent year t > t0, we consider the set of publications by the inspirees
+who cite the superstar. For each partitioned group, we calculate the average novelty of all
+of the publications in year t per partition. Denoting the set of papers inspired by superstar
+9
+
+0
+10
+20
+30
+40
+50
+Years After First Superstar Publication
+3.1
+3.2
+3.3
+3.4
+I(S)
+(A)
+0.00-0.10
+0.10-0.20
+0.20-0.30
+0.30-0.50
+0.50-1.00
+0
+10
+20
+30
+40
+50
+0.0
+0.4
+0.8
+1.2
+1.6
+I(I)
+(B)
+0
+5
+10
+15
+20
+Years after Inspired-Paper Pub.
+0.0
+0.4
+0.8
+1.2
+1.6
+Citations per Paper
+(C)
+0.00-0.10
+0.10-0.20
+0.20-0.30
+0.30-0.50
+0.50-1.00
+3.254
+3.258
+3.262
+3.266
+I(S)
+(D)
+0.00-0.10
+0.10-0.20
+0.20-0.30
+0.30-0.50
+0.50-1.00
+Inspiration Groups
+0.20
+0.24
+0.28
+I(I)
+(E)
+0.00-0.10
+0.10-0.20
+0.20-0.30
+0.30-0.50
+0.50-1.00
+0.35
+0.40
+0.45
+0.50
+0.55
+0.60
+Citations per Paper
+(F)
+FIG. 2. Novelty and Innovation statistics at the group-level Temporal trajectory of average
+paper-level statistics. A: Shannon Entropy, B: Innovation, C: Citations per-paper. Aggregated
+group-level statistics D: Shannon Entropy, E: Innovation, F: Citations per-paper. Curves indicate
+averages, shaded area 95% confidence interval.
+s for partition G at year t as P(G, s, t), the average novelty scores are computed as
+⟨I(l)
+u ⟩G,s,t =
+1
+|P(G, s, t)|
+�
+u∈P(G,s,t)
+I(l)
+u
+(5)
+where l = S, X, I is the novelty or innovation score of paper u.
+We plot the results of our analysis in Fig. 2. In terms of the temporal evolution of the
+Shannon entropy, while there is a monotonic increase—reflecting an increase in the body
+of knowledge with time (Fig. S8)—we find little-to-no differences across the groups as seen
+in Fig. 2A. Averaging over the entire temporal range also indicates a flat trend (Fig. 2D).
+Similar trends are seen for the reference diversity both in terms of its temporal evolution
+(upper panel of Fig. S9A,B) as well as their temporally averaged values (lower panel). Unlike
+the entropy or reference diversity, there is a decreasing trend in time for the citation diversity.
+We observe a 5% decrease in the measure between those in the top 10% as compared to the
+bottom 50%. Figure 2B,E indicates the same trend for Innovation which also decreases in
+time across all groups, reflecting a saturation in the number of combinations of new terms
+10
+
+0.2
+0.4
+0.6
+0.8
+0.07
+0.097
+0.123
+0.15
+0.177
+I(R )
+(A)
+All Papers
+Excluding Superstar Papers
+0.2
+0.4
+0.6
+0.8
+0.07
+0.082
+0.093
+0.105
+I(C )
+(B)
+0.2
+0.4
+0.6
+0.8
+3.05
+3.167
+3.283
+3.4
+I(S)
+(C)
+0.2
+0.4
+0.6
+0.8
+0.0
+1.333
+2.667
+4.0
+I(I)
+(D)
+0.2
+0.4
+0.6
+0.8
+P ercent of Author P apers that Cite a Superstar
+12.0
+16.0
+20.0
+24.0
+Avg. Author Citation Count
+(E)
+0.2
+0.4
+0.6
+0.8
+8.0
+16.0
+24.0
+32.0
+Avg. Author Publication Count
+(F)
+FIG. 3. Novelty and Innovation statistics at the individual author-level. A Reference
+Diversity, B Citation Diversity, C Shannon Entropy, D Innovation, E Average citation count, F
+Average publication count.
+that are combined by authors as their career progresses. The difference between the top and
+bottom groups is now around 15%. Finally, citations to papers experience an initial boost
+and then decreases in time as seen in Fig. 2C, with now much clearer differences between
+the groups. Indeed, there is a 40% difference in citations per-paper between the most and
+least inspired groups as seen in Fig. 2F.
+In terms of redundancy, in Fig. S9C we plot the cosine similarity (Eq. (4). As the figure
+indicates, across all groups there is a decreasing trend in the temporal evolution of the
+similarity, yet a clear difference exists, whereby papers published by the top 10% are on
+average 8% more similar to each other in terms of content when compared to the bottom
+50%. Taken together, the results indicate that groups of authors who cite superstar papers
+often do get a citation boost as compared to other sets of authors. However, their output is
+modestly more innovative and equally novel as compared to the rest of the corpus. Rather
+their content is more redundnant than the remaining sets of authors.
+Next, we dis-aggregate the group-level results and examine the degree of superstar in-
+fluence at the individual author level. In Fig. 3 we plot the averages of the novelty and
+innovation metrics as well as citations and publication counts across authors as a function
+of the fraction of their papers that cite superstars. Given that many authors co-publish
+11
+
+with superstars, the blue curve indicates the results when including such papers, while the
+orange curve shows the results excluding these papers. Figure 3A-C indicate that as au-
+thors cite more superstars they experience an increase in reference and citation diversity as
+well as the Shannon entropy irrespective of whether one includes their collaboration with
+superstars. While we see no indications of novelty of content being driven by superstar-
+influence at the group-level, at the individual level the benefits are clear. On the other hand,
+when looking at Innovation (Fig. 3D), the trend is either flat when including all papers,
+and decreasing when co-authored publications are excluded. Indeed, it appears that the
+more authors cite superstars, the less innovative their own publications become (i.e those
+not co-authored with a superstar). The benefit of collaborating with a superstar becomes
+even more apparent when looking at citations (Fig. 3E) and number of publications (Fig. 3
+F). For the former when including collaborations there is a dramatic benefit in terms of
+garnered citations (approximately 67% more citations on average) that drops considerably
+when excluding collaborations. Indeed, the citation-benefit appears to be driven primarily
+by being collaborators of superstars who by definition have the largest number of citations to
+their papers. The same appears to be the case for the latter, with the number of publications
+increasing when including collaborations, and decreasing when excluded.
+E.
+Early Collaborators and Early Innovators
+The results thus far provide evidence for academics inspired by superstars producing out-
+put with diverse content and that receives visibility via citations, while not necessarily being
+innovative in the sense of tying together new concepts. On the other hand, there is also ev-
+idence that these features are significantly boosted by direct collaboration with superstars,
+and when left to their own devices their publication output, novelty and innovation is lower
+than the rest of the corpus. Indeed, it begs the question whether superstars foster indepen-
+dent individual success, or rather inhibits it? For instance, as shown, at the aggregate level,
+the group of authors that cite superstars the most often tend to publish on mostly the same
+topics.
+To further probe this we restrict our analysis to early-career scientists. Given that findings
+from prior studies have shown that collaboration with successful scientists provides a boost
+12
+
+0
+5
+10
+15
+20
+25
+0
+2
+4
+6
+8
+10
+Citations Per Publication
+(A)
+Including Superstar Papers
+Collaborator
+Early Innovators
+0
+5
+10
+15
+20
+25
+0
+1
+2
+3
+4
+5
+6
+(B)
+Excluding Superstar Papers
+0
+5
+10
+15
+20
+25
+t − t0 (yr)
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+Innovation
+(C)
+0
+5
+10
+15
+20
+25
+0.00
+0.25
+0.50
+0.75
+1.00
+1.25
+1.50
+(D)
+Citation and Novelty Statistics per Academic Group
+FIG. 4. Citations and Innovation for frequent collaborators and early innovators A
+Citations per paper when including superstar papers, B The same when excluding superstar papers.
+C Temporal evolution of Innovation. D The same when excluding superstar papers. The horizontal
+axis t − t0 indicates the time elapsed from the t0 the time of first publication for authors in either
+group.
+for early career researchers [16], and that early success generates a cumulative advantage of
+long-term career success [14], we define early collaborators as those authors who collaborate
+with superstars in at least half of their papers in the first five years of their career. As a
+point of comparison, we define another set of authors who do not collaborate with, or cite
+superstar papers, but are in the top 10% of the corpus in terms of Innovation as measured
+by their first five years of publications. We term these authors early innovators. We use
+innovation as a metric, given that this is the measure by which superstars outperform other
+academics the most (Fig. 1D) and therefore might serve as a robust indicator of academic
+potential.
+13
+
+For academics in each group we track the temporal evolution of the citations per-paper,
+the number of publications, as well as the Innovation, measured from the date of first pub-
+lication t0 for authors in either group. Early collaborators get more citations per paper
+(Fig. 4A) and publish more than early innovators (Fig. S10A) particularly within the first
+ten years of their career. However, when one removes superstar publications, the trend re-
+verses where now early innovators publish more (Fig. S10B) and garner a comparable rate of
+citations as the other group (Fig. 4B ). Additionally the early innovators maintain a higher
+degree of Innovation throughout their careers as compared to early collaborators (Fig. 4C,
+D) with or without including collaborations to superstars. Thus the evidence suggests that
+while early career scientists indeed get a boost from collaborating with superstars, their own
+academic output is less innovative and equally visible in terms of citations, as compared
+to other early career scientists who produce innovative output without the benefit of such
+collaborations.
+III.
+CONCLUSION AND DISCUSSION
+In the exponentially growing knowledge-base of academia in which visibility and funding
+are increasingly being biased towards top academics and institutions, we examine the influ-
+ence that superstar academics have on the community as a whole and in terms of novelty
+and career success. Superstars provide an irreplaceable source of novel ideas and contribu-
+tions at rates that exceed those of other academics in the corpus; our metrics support that
+their accolades are well deserved and should be rewarded as such. We find superstars are
+highly novel and inspire a higher diversity of concepts among their followers and collabo-
+rators. However they do inhibit innovation potential. Those academics most inspired by a
+superstar are individually themselves more diverse in their papers, but at the group level
+add little intrinsic novelty than groups more weakly inspired by the superstar, even though
+they achieve higher citations.
+Additionally, we find indications of a strong Matthew Effect whereby academics who cite a
+superstar highly receive higher citations when collaborating with the superstar than without,
+despite higher gains in concept diversity than academic counterparts. Though collaboration
+with successful academics can stimulate a successful career path, we find these collaborations
+14
+
+can stifle innovation and may not provide the best indicator of long-term independent career
+success.
+Collaboration is a requirement to tackle increasingly difficult interdisciplinary problems.
+Superstars are well-positioned to foster interdisciplinary research efforts by supporting early-
+career researchers. Although the latter receive a citation boost when collaborate with a
+superstar, this does not imply that they are developing more novel work than their colleagues
+who are less connected to top academics. In fact, our results indicate that those closest to
+a superstar show the lowest innovation potential. This is slightly surprising given that the
+literature have shown junior researchers that collaborate with superstars are more likely
+to publish in high quality journals and have increased chances of engaging in high quality
+research with other top scientists. On balance, however, we find that this does not stimulate
+long term independent career success. This could be an indication of individuals getting
+lost in the wake of a superstar, meaning these researchers “bandwagon” off the ideas and
+visibility of their respective superstars and iterate on the superstar’s work. Although there is
+value in iterating upon already developed research questions, this may not foster innovative
+work and stimulate individual careers. Indeed, very recently it has been shown that there
+is a decline in disruptive ideas in both scientific publications and patents [50]. The authors
+attribute this to an ever increasing reliance on a narrower set of extant scientific knowledge
+on which to build ideas, a finding very much in line with our observation that followers of
+superstars produce redundant and less innovative content as a group.
+The observed effects could be a consequence of superstars’ strong hold over their respec-
+tive fields. It’s been shown that paradigm shifts in thinking occur after the sudden deaths of
+superstars. Collaborators of superstars suffer a drop in publication rate after their superstar
+death, and the field may experience a surge of contributions by outsiders who are dispro-
+portionately likely to be highly-cited [51]. One can infer that collaborators of superstars
+are successful because they are collaborating with superstars. Care should be taken when
+considering these proteges themselves for matters of funding and academic hiring. If the
+goal is to foster highly novel work, elements outside of prestige and social connection, such
+as efficacy, equity, and innovation, should be considered.
+Our findings are not limited solely to early innovators, collaborators, and inspirees.
+Though we provide early innovators as an example, many other groups [52] can be iso-
+15
+
+lated and studied in the way we have done here to identify promising academics based on
+early signatures of novelty or a range of social parameters. We outlined multiple differ-
+ent definitions of novelty in the introduction which we have not further developed in this
+study. Implementing the different definitions and distinguishing different types of novelty
+can elucidate what types of novelty are stifled or enhanced by different social configurations.
+A subject that we have not probed but is directly relevant to our discussion is the matter
+of funding. In recent times, funding has increasingly become more biased towards top insti-
+tutions [53], with 90% of NSF funding in 2018 going to 22% of funded institutions, serving
+43% of all institutions and 34% of underrepresented minorities [54]. This is coupled with a
+history of funding disparities with respect to race and underrepresented communities [55–57].
+Additionally, underrepresented groups produce novel works at higher rates yet are taken up
+by other scholars at lower rates than novel contributions by gender and racial majorities [46].
+Equitable funding programs have been shown to enhance research infrastructure, investiga-
+tor capabilities, and intra- and inter-university collaborations at less prominent institutions
+[58]. As we have shown, those that are least influenced by superstars innovate the most
+and consequently have higher citation rates. Coupling these results with added attention to
+equitable funding practices [59] we believe will reduce the growing inequality in academia
+and stimulate novel and innovative research.
+Finally, we note that our investigation necessarily comes with limitations.
+Given our
+sole focus on the APS body of literature, one should be careful to extrapolate this to other
+academic disciplines. This is also an incomplete subset of the entire journal, so a full corpus
+with an entire citation network would give a more accurate picture.
+[1] Cliff, H. How to make an Apple Pie From Scratch In Search of the Recipe for our Universe
+(Picador, London, 2021).
+[2] McAndrew, S. & Everett, M.
+Music as collective invention:
+A social network analy-
+sis of composers.
+Cultural Sociology 9, 56–80 (2014).
+URL https://doi.org/10.1177/
+1749975514542486.
+[3] Muller, E. & Peres, R.
+The effect of social networks structure on innovation perfor-
+mance: A review and directions for research.
+International Journal of Research in Mar-
+16
+
+keting 36, 3–19 (2019).
+URL https://www.sciencedirect.com/science/article/pii/
+S0167811618300284.
+[4] Hazarie, S., Barbosa, H., Frank, A., Menezes, R. & Ghoshal, G. Uncovering the differences
+and similarities between physical and virtual mobility. Journal of The Royal Society Interface
+17, 20200250 (2020). URL https://doi.org/10.1098/rsif.2020.0250.
+[5] Chen, Z. et al.
+Contrasting social and non-social sources of predictability in human
+mobility.
+Nature Communications 13, 1922 (2022).
+URL https://doi.org/10.1038/
+s41467-022-29592-y.
+[6] Nathaniel Rodriguez, Y.-Y. A., Johan Bollen. Collective dynamics of belief evolution under
+cognitive coherence and social conformity. PLoS ONE 11, e0165910 (2016).
+[7] Holme, P. & Ghoshal, G. Dynamics of networking agents competing for high centrality and
+low degree. Physical Review Letters 96, 098701– (2006). URL https://link.aps.org/doi/
+10.1103/PhysRevLett.96.098701.
+[8] Ghoshal, G. & Newman, M. E. J. Growing distributed networks with arbitrary degree distri-
+butions. The European Physical Journal B 58, 175–184 (2007). URL https://doi.org/10.
+1140/epjb/e2007-00208-2.
+[9] Recuero, R., Zago, G. & Soares, F. Using social network analysis and social capital to iden-
+tify user roles on polarized political conversations on twitter.
+Social Media + Society 5,
+2056305119848745 (2019). URL https://doi.org/10.1177/2056305119848745.
+[10] Dubois, E. & Gaffney, D. The multiple facets of influence: Identifying political influentials
+and opinion leaders on twitter. American Behavioral Scientist 58, 1260–1277 (2014). URL
+https://doi.org/10.1177/0002764214527088.
+[11] Radicchi, F., Weissman, A. & Bollen, J. Quantifying perceived impact of scientific publica-
+tions. Journal of Informetrics 11, 704–712 (2017). URL https://www.sciencedirect.com/
+science/article/pii/S1751157717300846.
+[12] Hirsch, J. E. An index to quantify an individual’s scientific research output. Proceedings of
+the National Academy of Sciences 102, 16569–16572 (2005).
+[13] Nielsen, M. W. & Andersen, J. P. Global citation inequality is on the rise. Proceedings of the
+National Academy of Sciences 118, e2012208118 (2021).
+[14] Merton, R. K. The matthew effect in science. Science 159, 56–63 (1968).
+17
+
+[15] Runco, M. & Pritzker, S. Encyclopedia of Creativity. Encyclopedia of Creativity (Elsevier
+Science, 2011).
+[16] Li, W., Aste, T., Caccioli, F. & Livan, G. Early coauthorship with top scientists predicts
+success in academic careers. Nature Communications 10, 5170 (2019).
+[17] Sekara, V. et al. The chaperone effect in scientific publishing. Proceedings of the National
+Academy of Sciences 115, 12603–12607 (2018).
+[18] Xie, Q., Zhang, X., Kim, G. & Song, M. Exploring the influence of coauthorship with top
+scientists on researchers’ affiliation, research topic, productivity, and impact. Journal of Infor-
+metrics 16, 101314 (2022). URL https://www.sciencedirect.com/science/article/pii/
+S1751157722000669.
+[19] Abrahamson, E. & Rosenkopf, L. Social network effects on the extent of innovation diffusion:
+A computer simulation. Organization Science 8, 289–309 (1997). URL http://www.jstor.
+org/stable/2635149.
+[20] Azoulay, P., Graff Zivin, J. S. & Wang, J. Superstar Extinction. The Quarterly Journal of
+Economics 125, 549–589 (2010). URL https://doi.org/10.1162/qjec.2010.125.2.549.
+https://academic.oup.com/qje/article-pdf/125/2/549/5319678/125-2-549.pdf.
+[21] Clauset, A., Arbesman, S. & Larremore, D. B. Systematic inequality and hierarchy in faculty
+hiring networks. Science Advances 1, e1400005 (2015). URL https://doi.org/10.1126/
+sciadv.1400005.
+[22] Janosov, M., Battiston, F. & Sinatra, R. Success and luck in creative careers. EPJ Data
+Science 9, 9 (2020). URL https://doi.org/10.1140/epjds/s13688-020-00227-w.
+[23] Bol, T., de Vaan, M. & van de Rijt, A. The matthew effect in science funding. Proceedings of
+the National Academy of Sciences 115, 4887–4890 (2018). URL https://www.pnas.org/doi/
+abs/10.1073/pnas.1719557115. https://www.pnas.org/doi/pdf/10.1073/pnas.1719557115.
+[24] Petersen, A. M., Jung, W.-S., Yang, J.-S. & Stanley, H. E. Quantitative and empirical demon-
+stration of the matthew effect in a study of career longevity.
+Proceedings of the National
+Academy of Sciences 108, 18–23 (2011). URL https://www.pnas.org/doi/abs/10.1073/
+pnas.1016733108. https://www.pnas.org/doi/pdf/10.1073/pnas.1016733108.
+[25] Lazer, D. & Friedman, A. The network structure of exploration and exploitation. Adminis-
+trative Science Quarterly 52, 667 – 694 (2007).
+18
+
+[26] Rodan, S. & Galunic, C. More than network structure: How knowledge heterogeneity influ-
+ences managerial performance and innovativeness. Strategic Management Journal 25, 541–562
+(2004). URL http://www.jstor.org/stable/20142143.
+[27] Chang, M. & Joseph E. Harrington, J. Discovery and diffusion of knowledge in an endogenous
+social network. American Journal of Sociology 110, 937–976 (2005). URL http://www.jstor.
+org/stable/10.1086/426555.
+[28] Trapido, D. How novelty in knowledge earns recognition: The role of consistent identities.
+Research Policy 44, 1488–1500 (2015).
+URL https://www.sciencedirect.com/science/
+article/pii/S0048733315000839.
+[29] Xu, F. & Evans, J. Flat teams drive scientific innovation. Proceedings of the National Academy
+of Sciences 119 (2022).
+[30] Hirsch, J. E. Does the h-index have predictive power? Proceedings of the National Academy
+of Sciences 104, 19193–19198 (2007).
+[31] Hirsch, J. E. An index to quantify an individual’s scientific research output. Proceedings of the
+National Academy of Sciences 102, 16569–16572 (2005). URL https://www.pnas.org/doi/
+abs/10.1073/pnas.0507655102. https://www.pnas.org/doi/pdf/10.1073/pnas.0507655102.
+[32] American Physical Society. https://journals.aps.org/datasets.
+[33] Richardson,
+L.
+https://sethc23.github.io/wiki/Python/Beautiful_Soup_
+Documentation.pdf.
+[34] Caron, E. & van Eck, N.-J. Large scale author name disambiguation using rule-based scor-
+ing and clustering. In Noyons, E. (ed.) Proceedings of the Science and Technology Indicators
+Conference 2014, 79–86 (Universiteit Leiden, 2014). URL http://sti2014.cwts.nl. Interna-
+tional conference on science and technology indicators, STI 2014 ; Conference date: 03-09-2014
+Through 05-09-2014.
+[35] El-Kishky, A., Song, Y., Wang, C., Voss, C. R. & Han, J. Scalable topical phrase mining from
+text corpora. Proc. VLDB Endow. 8, 305–316 (2014). URL https://doi.org/10.14778/
+2735508.2735519.
+[36] Lee, S. Y. Gibbs sampler and coordinate ascent variational inference: A set-theoretical review.
+Communications in Statistics - Theory and Methods 51, 1549–1568 (2022).
+URL https:
+//doi.org/10.1080/03610926.2021.1921214.
+19
+
+[37] Mimno, D., Wallach, H., Talley, E., Leenders, M. & McCallum, A.
+Optimizing semantic
+coherence in topic models. In Proceedings of the 2011 Conference on Empirical Methods in
+Natural Language Processing, 262–272 (Association for Computational Linguistics, Edinburgh,
+Scotland, UK., 2011). URL https://aclanthology.org/D11-1024.
+[38] Ouafae, B., Oumaima, L., Mariam, R. & Abdelouahid, L. Novelty detection review state of
+art and discussion of new innovations in the main application domains. In 2020 1st Inter-
+national Conference on Innovative Research in Applied Science, Engineering and Technology
+(IRASET), 1–7 (2020).
+[39] Soboroff, I. & Harman, D. Overview of the TREC 2003 novelty track. In Voorhees, E. M.
+& Buckland, L. P. (eds.) Proceedings of The Twelfth Text REtrieval Conference, TREC
+2003, Gaithersburg, Maryland, USA, November 18-21, 2003, vol. 500-255 of NIST Special
+Publication, 38–53 (National Institute of Standards and Technology (NIST), 2003).
+URL
+http://trec.nist.gov/pubs/trec12/papers/NOVELTY.OVERVIEW.pdf.
+[40] Ghosal, T., Saikh, T., Biswas, T., Ekbal, A. & Bhattacharyya, P.
+Novelty Detection:
+A Perspective from Natural Language Processing.
+Computational Linguistics 48, 77–117
+(2022). URL https://doi.org/10.1162/coli_a_00429. https://direct.mit.edu/coli/article-
+pdf/48/1/77/2006641/coli a 00429.pdf.
+[41] Uzzi, B., Mukherjee, S., Stringer, M. & Jones, B. Atypical combinations and scientific impact.
+Science 342, 468–472 (2013). URL https://www.science.org/doi/abs/10.1126/science.
+1240474. https://www.science.org/doi/pdf/10.1126/science.1240474.
+[42] Schumpeter, J. A. The theory of economic development: An inquiry into profits, capital, credit,
+interest, and the business cycle (Theorie der wirtschaftlichen Entwicklung) (Transaction, Edi-
+son, NJ, 1934). Translated by Redvers Opie.
+[43] Cover, T. & Thomas, J. A. Elements of Information Theory. Wiley Series in Telecommunica-
+tions and Signal Processing (Wiley-Interscience, New York, New York, USA, 2006).
+[44] Aral, S. & Dhillon, P.
+What (exactly) is novelty in networks?
+unpacking the vision ad-
+vantages of brokers, bridges, and weak ties. Institute for Operations Research and the Man-
+agement Sciences (INFORMS) (2021). URL http://dx.doi.org/10.2139/ssrn.2388254.
+https://ssrn.com/abstract=2388254.
+[45] Kuhn, T. S. The Structure of Scientific Revolutions (University of Chicago Press, Chicago,
+20
+
+1962).
+[46] Hofstra, B. et al. The diversityx2013;innovation paradox in science. Proceedings of the National
+Academy of Sciences 117, 9284–9291 (2020).
+URL https://www.pnas.org/doi/abs/10.
+1073/pnas.1915378117. https://www.pnas.org/doi/pdf/10.1073/pnas.1915378117.
+[47] Baten, R. A. et al. Creativity in temporal social networks: how divergent thinking is impacted
+by one’s choice of peers. Journal of The Royal Society Interface 17, 20200667 (2020).
+[48] Baten, R. A., Aslin, R. N., Ghoshal, G. & Hoque, E. Cues to gender and racial identity
+reduce creativity in diverse social networks. Scientific Reports 11, 10261 (2021). URL https:
+//doi.org/10.1038/s41598-021-89498-5.
+[49] Baten, R. A., Aslin, R. N., Ghoshal, G. & Hoque, M. E.
+Novel idea generation in social
+networks is optimized by exposure to a “goldilocks” level of idea-variability. PNAS Nexus 1,
+pgac255 (2022).
+[50] Park, M., Leahey, E. & Funk, R. J. Papers and patents are becoming less disruptive over time.
+Nature 613, 138–144 (2023). URL https://doi.org/10.1038/s41586-022-05543-x.
+[51] Azoulay, P., Fons-Rosen, C. & Graff Zivin, J. S.
+Does science advance one funeral at a
+time? American Economic Review 109, 2889–2920 (2019). URL https://www.aeaweb.org/
+articles?id=10.1257/aer.20161574.
+[52] He, B., Ding, Y., Tang, J., Reguramalingam, V. & Bollen, J. Mining diversity subgraph in mul-
+tidisciplinary scientific collaboration networks: A meso perspective. Journal of Informetrics
+7, 117–128 (2013).
+[53] Murray, D. L. et al. Bias in research grant evaluation has dire consequences for small universi-
+ties. PLOS ONE 11, 1–19 (2016). URL https://doi.org/10.1371/journal.pone.0155876.
+[54] of Government Affairs, O. Building america’s stem workforce: Eliminating barriers and un-
+locking advantages. Tech. Rep., American Physical Society, 1 Physics Ellipse, College Park,
+MD 20740-3844 (2021).
+[55] Woodson,
+T. & Boutilier,
+S.
+Impacts for whom?
+Assessing inequalities in NSF-
+funded broader impacts using the Inclusion-Immediacy Criterion.
+Science and Pub-
+lic
+Policy
+49,
+168–178
+(2021).
+URL
+https://doi.org/10.1093/scipol/scab072.
+https://academic.oup.com/spp/article-pdf/49/2/168/43395599/scab072.pdf.
+[56] Chen, C. Y. et al. Decades of systemic racial disparities in funding rates at the national science
+21
+
+foundation (2022). URL osf.io/xb57u.
+[57] Ginther, D. et al. Race, ethnicity, and nih research awards. Science (New York, N.Y.) 333,
+1015–9 (2011).
+[58] Harris, L. A. Established program to stimulate competitive research (epscor): Background
+and selected issues. Tech. Rep. R44689, Congressional Research Service, 1 Physics Ellipse,
+College Park, MD 20740-3844 (2017).
+[59] Bollen, J., Crandall, D., Junk, D., Ding, Y. & B¨orner, K.
+From funding agencies to sci-
+entific agency. EMBO reports 15, 131–133 (2014). URL https://doi.org/10.1002/embr.
+201338068.
+22
+
+Supplementary Information
+Creativity and Production in Academic Social Networks
+Sean Kelty, Raiyan Abdul Baten, Adiba Proma, Ehsan Hoque, Johann Bollen, Gourab
+Ghoshal
+CONTENTS
+S1. Data
+S-2
+S1.1. Summary statistics
+S-2
+S1.2. Author Disambiguation
+S-4
+S1.3. Coherence of Topic Model
+S-5
+S1.4. Example of Topic Representation
+S-6
+S1.5. Innovation Example
+S-8
+S2. Trends of Novelty and Innovation
+S-10
+S3. Author and Paper-Level Novelty and Output
+S-11
+S-1
+arXiv:2301.02396v1  [cs.DL]  6 Jan 2023
+
+S1.
+DATA
+S1.1.
+Summary statistics
+TABLE S1: Summary of citation statistics of corpus with
+and without abstracts.
+No. Dois
+678,916
+No. Dois (w/ abstracts)
+250,628
+No. Authors (after disambiguation)
+307,894
+No. Superstars
+303
+h-index cutoff for superstars
+21
+Avg. h-index
+1.74
+Avg. No. References per paper
+13.5
+Avg. No. References per paper (w/ abstracts)
+5.57
+Avg. No. Citations per paper
+14.4
+Avg. No. Citations per paper (w/ abstracts)
+6.88
+S-2
+
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+Ratio of Extract Abstracts
+Ratio of Extracted Abstracts to Total Papers per Journal
+PRA
+PRAB
+PRAPPLIED
+PRB
+PRC
+PRD
+PRE
+PRFLUIDS
+PRL
+PRMATERIALS
+PRPER
+PRRESEARCH
+PRSTAB
+PRSTPER
+PRX
+PRXQUANTUM
+Physical Review
+RMP
+Journal
+101
+102
+103
+104
+105
+Count of Total Extracted Abstracts
+Paper Statistics per Journal in APS
+FIG. S1. Upper panel: Proportion of analyzed papers in across APS journals. Lower panel: Count
+of all papers in each journal.
+100
+101
+102
+Paper-Level Citations
+10−5
+10−4
+10−3
+10−2
+10−1
+Density
+Citation Distributions
+Papers with Abstracts Only
+Full Corpus
+100
+101
+h-index
+10−6
+10−5
+10−4
+10−3
+10−2
+10−1
+100
+h-index Distribution
+Citation and h-index Distributions
+FIG. S2. Left: Distribution of citations for all papers (orange curve), for papers with extracted
+abstracts (blue curve). Right: Distribution of the h-index across all authors in the corpus.
+S-3
+
+S1.2.
+Author Disambiguation
+We use a scoring-based method to disambiguate authors in our dataset.
+1. Initials
+• Two Initials : 5
+• More than 2 initials : 10
+• Conflicting Initials : -10
+2. First Name
+• General Name : 3
+• Non-General Name : 6
+(A name is considered general if it has been seen more than 1000 times)
+3. Address/Affiliation
+• Country,City : 4
+• Country,City,Organization : 7
+• Country,City,Organization,Department : 10
+4. Shared Co-Authors
+• one : 4
+• two : 7
+• more than 2 : 10
+5. Source
+• Journal : 6
+6. Self-Citation : 10
+7. Bibliographic Coupling (two works referencing a common third work)
+• one : 2
+• two : 4
+• three : 6
+• four : 8
+• More than four : 10
+8. Co-citation (Number of times a third work has cited two works)
+S-4
+
+• one : 2
+• two : 3
+• three : 4
+• four : 5
+• More than 4 : 6
+S1.3.
+Coherence of Topic Model
+We apply the UMass coherence measure to determine a stable number of topics for
+our topic model. This coherence score measures how similar the top words in a topic
+are to each other. We aim for the highest possible coherence value that stabilizes in a
+neighborhood of the number of topics k. Fig. S3 shows the coherence stablizing at roughly
+k = 25 topics.
+0
+20
+40
+60
+80
+100
+k
+4.0
+3.8
+3.6
+3.4
+3.2
+3.0
+2.8
+2.6
+Coherence (umass)
+Coherence of Topic Model
+FIG. S3. Coherence Scores of P-LDA Topic Model
+S-5
+
+S1.4.
+Example of Topic Representation
+Words and phrases in the corpus, which will generally be referred to as "terms", are
+represented by a distribution over latent topics that is the frequency of topic assignments
+of the term over the entire corpus. Topics are characterized by the frequency of terms
+associated with the topic. For each topic, all terms are ranked based on their relative topic
+frequency of their own distribution of the given topic. For example, if a phrase had a
+topic distribution for k = 3 topics of [.1,.2,.7], the phrase is representative of topic 3. Terms
+are pre-processed by removing stop words and stemming words such that conjugated
+versions of the same word can be represented as the same word.
+TABLE S2: Topic Model Summary of most representative
+terms per topic.
+Topic
+Number
+Representative Terms
+Topic 1
+crystal film, ultrathin, mtj, stm tip, stack, freestand, high resolution angle, franz, stm, force
+micrscop
+Topic 2
+center cubic fcc, temperature addit, measur x, tc cuprat, temperature down k, temperature k k, tc
+k, tc superconduct, tc superconductor, temperature tc k
+Topic 3
+spectral line, ωp, raman line, absorpt part, absorpt line, nd3, electroreflect. eliashberg, b1g, endor
+Topic 4
+axial magnet, spin angular, moment inertia, moment magnet, parallel magnet field, magnet
+revers, torqu, interlay exchange, spin texture, moriya
+Topic 5
+collim, electron eject, ion yield, ion trap, n4, ion produc, ion plasma, damag, wall carbon, electron
+drift
+Topic 6
+cauchi, broken time, takahashi, hamilton jacobi, symmetri spontan, tachyon, ward ident, polyakov,
+loop quantum cosmolog, coulomb guage
+Topic 7
+excitatori, hub, infect, epidem, volatil, exactli solvabl model, network model, synaps, synapt,
+integr fire
+Topic 8
+nonequilibrium phase transit, first order phase transit, j’, glass order, thouless transit, glass like,
+glass former, triangluar lattic, nearest neighbor coupl, nearest neighbor distanc
+S-6
+
+Topic 9
+magnitude higher, larg part, fourth gener, even though, order qcd, select rule, third, mach zehnder
+interferomet, even larger, order raman
+Topic 10
+quasilinear, langevin equat, gilbert equat, equate state eo, sand, attractor, classic chaotic, eulerian,
+chimera state, euler equat
+Topic 11
+advanc ligo, mit bag, catalog, model background, dark sector, dark matter, sight, model dark, sky,
+sno
+Topic 12
+nest, der waal force, nodal line, helic edg, non fermi, state degeneraci, hove, majorana zero,
+majorana bound, sdh
+Topic 13
+three dimension 3d, basin attract, fuld ferrel, dimension squar, lz, trap bose, bodi effect, bodi forc,
+hard core boson, fermion atom
+Topic 14
+highest occupi molecular, muffin tin orbit, gaas1, clathrat, cl2, cl, hexagon boron, interstiti, gell, ci
+Topic 15
+puls width, optic parametr, sapphir laser, exciton biexciton, optic pump, harmon gener shg, optic
+puls, inxga1 xa, optic nonlinear, ultrastrong
+Topic 16
+clauser, horn shimoni holt, simpl analyt express, us deriv, part paper, analyt formula, cb, exact
+forumla, exact expression, pauli exclus
+Topic 17
+agre reason, foudn good agreement, recent experiment data, find excel agreement, find good
+agreement, theoret data, theoret cross, reason agreement experiment, found excel agreement,
+good agreement experimental result
+Topic 18
+qutrit, regist, processor, studi entagle, protocol, markovian dynam, purif, decoy state, qkd, error
+correct
+Topic 19
+nucleon nucleon scatter, deep inelast scatter, total angular momentum, inclus cross, transfer cross
+section, multifragment, multiperipher, depend cross section, forward angle, πn
+Topic 20
+full potenti linear augment plane wave, wave born, wannier function, impuls, infield, use path,
+use mont, within densiti function, jastrow, use harte
+Topic 21
+avoid walk, nonergod, time τ, time tail, time t2, time t2, dimension diffus, time random, nonex-
+ponenti, msd
+Topic 22
+even even nuclei, xe136, rich nuclei, gt, v’, p0, cf252, α p, α reaction, p1
+Topic 23
+director field, shear modulu, homeotrop, tλ, antivortex, humid, u0, hydrophil, shear band, shear
+strain
+S-7
+
+Topic 24
+signific role, key role, kibbl zurek, amino acid, play essenti, play domin, play crucial, play critical,
+play central, remain elus
+Topic 25
+paramet η, ev2, rev c, rev lett, eq, right hand, right left, e e collide, e e annhil, f0
+TABLE S3: Example of Phrase-Topic Distributions.
+Term
+Topic-Embedding
+...
+Quantiz
+[1, 0, 0, 0, 0 2259, 0, 0, 560, 0, 0, 882, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 677, 0, 0]
+Quantum
+[29, 0, 0, 21, 0, 4304, 1069, 4276, 0, 308, 0, 6008, 454, 46, 14920, 0, 0, 35931, 0, 1828, 0, 0,
+1384, 7, 1]
+Quark
+[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 239, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 14542]
+Quarkonia
+[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 125]
+Quarkonium
+[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 299]
+Quarter
+[0, 0, 0, 0, 30, 0, 0, 0, 0, 0, 0, 321, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
+...
+Quantum Wire
+[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 342, 0, 0, 292, 0, 0, 23, 0, 0, 0, 0, 91, 0, 0]
+Quantum Zeno
+[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 31, 0, 0, 0, 0, 0, 0, 0]
+Quark Antiquark
+[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 433]
+Quark Condens
+[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 107]
+Quark Decay
+[[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 25]
+...
+S1.5.
+Innovation Example
+The innovation metric counts the first time a term or a new combination of terms have
+been seen in an article over the entire corpus. Fig. S4 shows the introduction of the terms
+"quantum" and "cosmolog" in the corpus. Note that "cosmolog" is the root of words such
+as "cosmology" and "cosmological" that were lemmatized in pre-processing. We plot the
+frequency of the terms in time as well as vertical lines representing the first year the term
+S-8
+
+has been seen. We also plot the counts of the phrase "quantum cosmolog" which is an
+additionally considered term in our topic model.
+1925
+1950
+1975
+2000
+Year
+0
+500
+1000
+1500
+2000
+Yearly Counts
+Quantum
+Cosmolog
+Quantum Cosmolog
+1925
+1950
+1975
+2000
+Year
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+Density
+FIG. S4. Example of innovation measure with terms "quantum" and "cosmolog"
+S-9
+
+S2.
+TRENDS OF NOVELTY AND INNOVATION
+FIG. S5. Correlations between all novelty and innovation measures based on Pearson’s r.
+S-10
+
+Correlation of Novelty Measures
+R^2:0.00137
+R^2:0.000218
+R^2:0.00737
+0.40
+0.40
+0.35
+0.35 
+4.0
+3.5
+OE0
+Citation Diversity
+3.0
+0.25
+0.25
+Reference
+0.20
+0.20 
+2.0
+ 0.15 
+0.15
+0.10
+0.10
+1.0
+0.05
+0.05
+0.5 
+0.00
+0.00
+0.0
+10
+12
+12
+14
+Innovativeness
+Innovativeness
+Innovativeness
+R^2:0.062
+R^2:0.0947
+R^2: 0.0529
+0.40
+0.35
+0't
+0't
+3.5
+3.5 
+OE0
+Diversity
+0.25 
+2.5
+2.5 
+0.20 
+Citation
+2.0
+2.0 
+0.15
+15
+0.10
+1.0
+0.05 
+0.5 
+0.5
+0.00
+0.0 
+0.40
+0.10
+0.0 -
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.00
+0.05
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+0.40
+Reference Diversity
+Reference Diversity
+Citation DiversityS3.
+AUTHOR AND PAPER-LEVEL NOVELTY AND OUTPUT
+0
+20
+40
+0.095
+0.102
+0.108
+0.115
+Author-Level Novelty
+I(R)(A)
+All Papers
+Without Superstar Papers
+0
+20
+40
+Number of Author Publications
+0.075
+0.083
+0.092
+0.1
+I(C)(B)
+0
+20
+40
+3.16
+3.183
+3.207
+3.23
+3.253
+I(S)(C)
+0
+20
+40
+0.0
+3.333
+6.667
+10.0
+I(I) (D)
+Author-Level Novelty Scores vs Author Success
+FIG. S6. Novelty and innovation metrics of an author’s publication record as a function of
+their number of publications. Authors with between 1-50 publications in the corpus have been
+considered.
+100
+101
+102
+0.09
+0.103
+0.117
+0.13
+Paper-Level Novelty
+I(R)(A)
+All Papers
+Without SS Papers
+101
+102
+Number of Citations
+0.02
+0.07
+0.12
+0.17
+I(C)(B)
+100
+101
+102
+3.1
+3.15
+3.2
+3.25
+I(S)(C)
+100
+101
+102
+0.0
+0.333
+0.667
+1.0
+I(I) (D)
+Paper-Level Novelty Scores vs Paper Success
+FIG. S7. Novelty and innovation metrics of an author’s publication record as a function of the
+number of citations their papers garner.
+S-11
+
+1970
+1980
+1990
+2000
+2010
+2020
+3.050
+3.075
+3.100
+3.125
+3.150
+3.175
+3.200
+3.225
+3.250
+Entropy
+1970
+1980
+1990
+2000
+2010
+2020
+Year
+0.080
+0.085
+0.090
+0.095
+0.100
+0.105
+0.110
+0.115
+0.120
+Reference Diversity
+1970
+1980
+1990
+2000
+2010
+2020
+0.04
+0.06
+0.08
+0.10
+0.12
+Citaiton Diversity
+1970
+1980
+1990
+2000
+2010
+2020
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Innovation
+Average Novelties per Year
+FIG. S8. All Novelty Measures per Year.
+0
+10
+20
+30
+40
+50
+0.09
+0.10
+0.11
+0.12
+0.13
+0.14
+0.15
+Temporal
+Reference Diversity (A)
+group
+0.00-0.10
+0.10-0.20
+0.20-0.30
+0.30-0.50
+0.50-1.00
+0
+10
+20
+30
+40
+50
+Years After First SS Pub.
+0.08
+0.09
+0.10
+0.11
+0.12
+0.13
+0.14
+0.15
+Citation Diversity (B)
+group
+0.00-0.10
+0.10-0.20
+0.20-0.30
+0.30-0.50
+0.50-1.00
+0
+10
+20
+30
+40
+50
+0.500
+0.525
+0.550
+0.575
+0.600
+0.625
+0.650
+0.675
+0.700
+Similarities (C)
+group
+0.00-0.10
+0.10-0.20
+0.20-0.30
+0.30-0.50
+0.50-1.00
+0.00-0.10
+0.10-0.20
+0.20-0.30
+0.30-0.50
+0.50-1.00
+0.1290
+0.1295
+0.1300
+0.1305
+0.1310
+Aggregate
+0.00-0.10
+0.10-0.20
+0.20-0.30
+0.30-0.50
+0.50-1.00
+Inspiration Groups
+0.099
+0.100
+0.101
+0.102
+0.103
+0.104
+0.00-0.10
+0.10-0.20
+0.20-0.30
+0.30-0.50
+0.50-1.00
+0.54
+0.55
+0.56
+0.57
+0.58
+Concept Diversity and Similarities of Inspired Groups
+FIG. S9. (A) Reference Diversity, (B) Citation Diversity, (C) Within-group paper similarities for
+the followers of a superstar partitioned by level of inspiration. Upper panel: temporal evolution.
+Lower panel: averaged in time.
+S-12
+
+0
+5
+10
+15
+20
+25
+t − t0 (yr)
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+Yearly Publications
+Including Superstar Papers
+Early Collaborators
+Early Innovators
+0
+5
+10
+15
+20
+25
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Excluding Superstar Papers
+Publication Rate per Academic Group
+FIG. S10. Publication rates of academic groups, LEFT including superstar collabortions and RIGHT
+excluding superstar collaborations
+S-13
+

6NE4T4oBgHgl3EQf1g37/content/2301.05292v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:e1cee2983f7fd812c2867320eb5e7c069305ce96a456d0d9f0e2951486e1b431
+size 2373256

6NE4T4oBgHgl3EQf1g37/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:9fa7bf2455072f437c03d335783c83b931e57ce97f93c301ebddd0f6926f2e13
+size 1048621

6NE4T4oBgHgl3EQf1g37/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:76d84708d419c91c8af96023665df90600e1fc8c21c871e08825cbaa81b2e3db
+size 48592

6NFKT4oBgHgl3EQfTS23/content/tmp_files/2301.11779v1.pdf.txt

@@ -0,0 +1,568 @@
+Invariant Meta Learning for Out-of-Distribution Generalization
+Penghao Jiang, Ke Xin, Zifeng Wang, Chunxi Li
+The Australian National University, Canberra, Australia *
+Abstract
+Modern deep learning techniques have illustrated their
+excellent capabilities in many areas, but relies on large
+training data.
+Optimization-based meta-learning train a
+model on a variety tasks, such that it can solve new learn-
+ing tasks using only a small number of training samples.
+However, these methods assumes that training and test data
+are identically and independently distributed. To overcome
+such limitation, in this paper, we propose invariant meta
+learning for out-of-distribution tasks. Specifically, invari-
+ant meta learning find invariant optimal meta-initialization,
+and fast adapt to out-of-distribution tasks with regulariza-
+tion penalty. Extensive experiments demonstrate the effec-
+tiveness of our proposed invariant meta learning on out-
+ofdistribution few-shot tasks.
+1. Introduction
+Modern deep learning techniques have illustrated their
+excellent capabilities in many areas like computer vision,
+natural language processing and recommendation, etc [11].
+However, these methods relies on large training data. To
+overcome this limitation, few-shot learning methods such
+as meta learning has been proposed [6]. Most popular meta
+learning approaches is the optimization-based metalearning
+[4, 16], which is model-agnostic and can be applied to var-
+ious downstream tasks. However, many recent researches
+have revealed the vulnerability of machine learning model
+when exposed to data with different distributions.
+Such massive gap is induced by the violation of a funda-
+mental assumption that training and test data are identically
+and independently distributed (a.k.a.
+i.i.d.
+assumption),
+upon which most of the existing meta learning models are
+developed [4, 16]. In many real cases where i.i.d. assump-
+tion can hardly be satisfied, especially those high-stake ap-
+plications such as healthcare, military and autonomous driv-
+ing, instead of generalization within the training distribu-
+tion, the ability to generalize under distribution shift is of
+more critical significance. As shown in Figure 1, given tran-
+* The first two authors contributed equally as joint first authorship.
+The last two authors contributed equally as joint second authorship.
+Figure 1. Illustration example of how the distribution shifts be-
+tween training data and testing data hamper the performance of
+model predictions.
+Figure 2. Causal framework of dog perdiction task. Due to the
+spurious correlation, the model tends to focus on both grass and
+dog, which lead to failed prediction in other distributions.
+ing data where dogs are on the grass, model could not make
+accurate predictions in testing data where dogs are in water,
+cage or street. The reason is that the supurious correlation
+between grass and dog in traning data hamper the perfor-
+mance of model. Due to the spurious correlation, the model
+tends to focus on both grass and dog, which lead to failed
+prediction in other distribution such as dogs are in water,
+cage or street as shown in Figure 2. However, recent meta
+learning methods could not overcome the distribution shifts
+between training and testing data. In this paper, we con-
+sider a realistic scenario where tasks come from different
+distributions (out-of-distribution, OOD).
+In this paper, to overcome the problem mentioned above,
+we propose Invariant Meta Learning (IML) for out-of- dis-
+arXiv:2301.11779v1  [cs.LG]  26 Jan 2023
+
+Athome
+onbeach
+eating
+incage
+inwater
+lying
+ongrass
+instreet
+running
+Training data
+Model
+Testing dataGrass--Label:Strongcorrelation
+CausalFramework
+Weakcausation
+Dog noseLabel:Strong correlation
+X
+Strong causation
+T:
+grass
+X: dog nose
+Y:labeltribution tasks, a general learning framework that jointly ad-
+justs gradient magnitudes and directions. Specifically, in-
+variant meta learning find invariant optimal metainitializa-
+tion, and fast adapt to out-of-distribution tasks with regular-
+ization penalty. To summarize, our main contributions are:
+• We consider the challenge of out-of-distribution tasks
+faced by few-shot learning, we show a natural idea to
+jointly adjust gradient magnitudes and directions of all
+tasks in the meta optimization process;
+• We propose Invariant Meta Learning (IML) for out-
+ofdistribution tasks, a general learning framework that
+jointly adjusts gradient magnitudes and directions;
+• We conduct extensive experiments and analysis to
+demonstrate that our approach effectively improves the
+performance and generalization ability under both in-
+distribution and out-of-distribution few-shot settings,
+and thus it can be regarded as a better baseline.
+2. Method
+In this section, we introduce our proposed Invariant Meta
+Learning (IML) to address the out-of-distribution problem
+in few-shot tasks.
+IML learns invariant optimal predic-
+tors based on optimization based meta learning framework.
+To learn invariant optimal meta-initialization in optimiza-
+tion based meta learning, the main challenge is that OOD
+problem exacerbates the inconsistency in both task-gradient
+magnitudes and directions.
+To overcome such problem,
+IML finds invariant optimal initialization, and adapt to
+outof- distribution tasks with regularization penalty.
+Model-agnostic meta-learning (MAML) [4] is an ap-
+proach to optimization-based meta-learning that is related
+to our work. For some parametric model fθ, MAML aims
+to find a single set of parameters θ which, using a few op-
+timization steps, can be successfully adapted to any novel
+task sampled from the same distribution. For a particular
+task instance Ti =
+�
+Dtr, Dval�
+, the parameters are adapted
+to task-specific model parameters θ′
+i by applying some dif-
+ferentiable function, typically an update rule of the form:
+θ′
+i = G
+�
+θ, Dtr�
+,
+(1)
+where G is typically implemented as a step of gradi-
+ent descent on the few-shot training set Dtr , θ′
+i = θ−
+α∇θLtr
+Ti (fθ).
+Generally, multiple sequential adaptation
+steps can be applied. The learning rate α can also be met-
+alearned concurrently, in which case we refer to this algo-
+rithm as Meta-SGD [13]. During meta-training, the param-
+eters θ are updated by back-propagating through the adap-
+tation procedure, in order to reduce errors on the validation
+set Dval :
+θ ← θ − η∇θ
+�
+Ti∼p(T )
+Lval
+Ti
+�
+fθ′
+i
+�
+.
+(2)
+The
+approach
+includes
+the
+main
+ingredients
+of
+optimization-based meta-learning with neural networks:
+initialization is done by maintaining an explicit set of
+model parameters θ; the adaptation procedure, or “inner
+loop”, takes θ as input and returns θ′
+i adapted specifically
+for task instance Ti, by iteratively using gradient descent
+(Eq.
+1); and termination, which is handled simply by
+choosing a fixed number of optimization steps in the “inner
+loop”.
+MAML updates θ by differentiating through the
+“inner loop” in order to minimize errors of instance-specific
+adapted models fθ′
+i on the corresponding validation set
+(Eq. 2). We refer to this process as the “outer loop” of
+meta-learning. We use the same stages to describe IML.
+Invariant Meta Learning (IML) finds invariant opti-
+mal meta-initialization, and fast adapt to out-of-distribution
+tasks with regularization penalty. MAML fast adapt net-
+work to new task during the inner loop and learns univer-
+sal meta-initialization in outer loop. Similarly, in IML, we
+update network with the bi-level update, optimizing clas-
+sifier in the inner loop and learning feature representation
+in the outer loop. For the inner-level optimization, the pa-
+rameters θ of the predictor become θi while adapting to the
+task ti ∈ Ttr. This correspond to the inner optimization of
+MAML, except that each task ti has a corresponding net-
+work θi. The optimization in the inner loop can be defined
+as follows:
+θ′
+i = θ − α∇θLtr
+Ti (fθ)
+(3)
+where α is a learning rate of the inner optimization.
+With inner optimized network fθ′
+i, we have outer loop
+objective function with variance penalty regularizer:
+Lval =
+�
+Ti∼p(T tr)
+�
+Tj∼p(T val)
+Lval
+Tj
+�
+fθ′
+i
+�
+(4)
+θ ← θ − η∇θLval − βλ trace
+�
+VarT val
+�
+∇θLval��
+(5)
+where η, β are the learning rate of the outer loop optimiza-
+tion, tj is task j for outer loop optimization for the net-
+work θ′
+i, L is the loss function for outer loop optimization.
+Note that the inner optimized network fθ′
+i is used to up-
+date meta-initialization in outer loop with tj whereas it is
+updated from meta-initialization with ti in ther inner loop.
+IML learn invariant meta-initialization obtained from the
+discrepancy among different training tasks with variance
+penalty regularizer.
+3. Experiments
+Datasets.
+In this paper, we address the few-shot clas-
+sification problem under both in-distribution and out-
+ofdistribution FSL settings. These settings are conducted
+on three benchmark datasets: miniImageNet [23], Caltech-
+UCSD-Birds 200-2011 (CUB) [25], and SUN Attribute
+Database (SUN) [15].
+
+Method
+miniImageNet
+CUB
+SUN
+5-way 1-shot
+5-way 5-shot
+5-way 1-shot
+5-way 5-shot
+5-way 1-shot
+5-way 5-shot
+Meta-Learner LSTM
+24.99
+29.79
+36.23
+44.39
+30.99
+44.86
+MAML
+45.69
+60.90
+48.87
+63.99
+57.75
+71.45
+Reptile
+26.59
+39.87
+27.21
+42.35
+28.30
+51.62
+Matching Network
+47.63
+56.28
+53.06
+62.19
+55.02
+62.57
+Prototypical Network
+46.15
+65.56
+48.21
+57.80
+55.70
+67.32
+Relation Network
+47.64
+63.65
+52.76
+64.71
+58.29
+72.15
+Baseline
+23.84
+32.09
+25.14
+35.35
+27.44
+34.54
+Baseline++
+30.15
+41.19
+32.48
+42.43
+35.56
+44.42
+IML
+48.35
+67.21
+54.18
+65.85
+59.24
+74.18
+Table 1. Average accuracy (%) comparison to state-of-the-arts with 95% confidence intervals on 5-way classification tasks under the
+in-distribution FSL setting. Best results are displayed in boldface.
+Method
+miniImageNet→ CUB
+miniImageNet→ SUN
+CUB→miniImageNet
+5-way 1-shot
+5-way 5-shot
+5-way 1-shot
+5-way 5-shot
+5-way 1-shot
+5-way 5-shot
+Meta-Learner LSTM
+23.77
+30.58
+25.52
+32.14
+22.58
+28.18
+MAML
+40.29
+53.01
+46.07
+59.08
+33.36
+41.58
+Reptile
+24.66
+40.86
+32.15
+50.38
+24.56
+40.60
+Matching Network
+38.34
+47.64
+39.58
+53.20
+26.23
+32.90
+Prototypical Network
+36.60
+54.36
+46.31
+66.21
+29.22
+38.73
+Relation Network
+39.33
+50.64
+44.55
+61.45
+28.64
+38.01
+Baseline
+24.16
+32.73
+25.49
+37.15
+22.98
+28.41
+Baseline++
+29.40
+40.48
+30.44
+41.71
+23.41
+25.82
+IML
+41.27
+57.34
+50.42
+69.15
+34.26
+44.17
+Table 2. Average accuracy (%) comparison to state-of-the-arts with 95% confidence intervals on 5-way classification tasks under the
+in-distribution FSL setting. Best results are displayed in boldface.
+Baselines.
+To evaluate the effectiveness of the proposed
+framework, we consider the following representative meta
+learning methods on the few-shot image classification task:
+MAML [5], Reptile [14], Matching Network [23], Proto-
+typical Network [20], Relation Network [21], Baseline and
+Baseline++ [3].
+Experimental Settings.
+We conduct experiments on 5-
+way 1-shot and 5-way 5 -shot settings, there are 15 query
+samples per class in each task. We report the average ac-
+curacy (%) and the corresponding 95% confidence interval
+over the 2000 tasks randomly sampled from novel classes.
+To fairly evaluate the original performance of each method,
+we use the same 4-layer ConvNet [23] as the backbone for
+all methods and do not adopt any data augmentation during
+training. All methods are trained via SGD with Adam [10],
+and the initial learning rate is set to e−3. For each method,
+models are trained for 40,000 tasks at most, and the best
+model on the validation classes is used to evaluate the final
+reporting performance in the meta-test phase.
+Evaluation Using the In-Distribution Setting.
+Table 1
+shows the comparative results under the in-distribution FSL
+setting on three benchmark datasets.
+It is observed that
+IML outperforms the original MAML in all in-distribution
+FSL scenarios.
+For 1-shot and 5-shot on miniImageNet
+→ miniImageNet, IML achieves about 1% higher perfor-
+mance than Prototypical Network. However, IML achieves
+5% and 10% higher performance for 1-shot and 5-shot
+on CUB → CUB, and 3% and 6% higher performance
+on SUN → SUN. As the latter two scenarios are con-
+ducted on finegrained classification datasets, we attribute
+the promising improvement to that the categories in these
+fine-grained datasets share more local concepts than those
+in coarsegrained datasets, and thus a more discriminative
+space can be rapidly learned with a few steps of adaptation.
+Moreover, IML achieves the best performance among all
+baselines in all in-distribution FSL scenarios, which shows
+that our approach can be considered as a better baseline op-
+tion under the in-distribution FSL setting.
+Evaluation Using the Out-of-Distribution Setting.
+We
+also conduct out-of-distribution FSL experiments and re-
+port the comparative results in Table 2. Compared to the re-
+sults under the in-distribution setting, it can be observed that
+all approaches suffer from a larger discrepancy between the
+
+distributions of training and testing tasks, which results in
+a performance decline in all scenarios. However, IML still
+outperforms the original MAML in all out-of-distribution
+FSL scenarios, demonstrating that the bilevel optimization
+strategy for adaptation and the learning of transferable la-
+tent factors can be utilized to improve simple meta learning
+approaches. Also, IML achieves all the best results, indicat-
+ing that our approach can be regarded as a promising base-
+line under the out-of-distribution setting.
+4. Conclusion
+In this paper,
+we consider the challenge of out-
+ofdistribution tasks faced by few-shot learning. We propose
+Invariant Meta Learning (IML) for out-of-distribution tasks,
+a general learning framework that jointly adjusts gradient
+magnitudes and directions. Extensive experiments demon-
+strate that our approach effectively improves the perfor-
+mance and generalization ability under both in-distribution
+and out-of-distribution few-shot settings, and thus it can be
+regarded as a better baseline.
+References
+[1] Yoshua Bengio, Samy Bengio, and Jocelyn Cloutier. Learn-
+ing a synaptic learning rule. Citeseer, 1990.
+[2] Fei Chen, Mi Luo, Zhenhua Dong, Zhenguo Li, and
+Xiuqiang He.
+Federated meta-learning with fast con-
+vergence and efficient communication.
+arXiv preprint
+arXiv:1802.07876, 2018.
+[3] Wei-Yu Chen, Yen-Cheng Liu, Zsolt Kira, Yu-Chiang Frank
+Wang, and Jia-Bin Huang. A closer look at few-shot classi-
+fication. arXiv preprint arXiv:1904.04232, 2019. 3
+[4] Chelsea Finn, Pieter Abbeel, and Sergey Levine.
+Model-
+agnostic meta-learning for fast adaptation of deep networks.
+In International conference on machine learning, pages
+1126–1135. PMLR, 2017. 1, 2
+[5] Chelsea Finn, Pieter Abbeel, and Sergey Levine.
+Model-
+agnostic meta-learning for fast adaptation of deep networks.
+In Proceedings of the 34th International Conference on Ma-
+chine Learning, pages 1126–1135. PMLR, 2017. 3
+[6] Chelsea Finn and Sergey Levine.
+Meta-learning and
+universality:
+Deep representations and gradient descent
+can approximate any learning algorithm.
+arXiv preprint
+arXiv:1710.11622, 2017. 1
+[7] Timothy Hospedales, Antreas Antoniou, Paul Micaelli, and
+Amos Storkey. Meta-learning in neural networks: A survey.
+arXiv preprint arXiv:2004.05439, 2020.
+[8] Simon Jenni and Paolo Favaro. Deep bilevel learning. In
+Proceedings of the European conference on computer vision
+(ECCV), pages 618–633, 2018.
+[9] Taewon Jeong and Heeyoung Kim.
+Ood-maml:
+Meta-
+learning for few-shot out-of-distribution detection and clas-
+sification. Advances in Neural Information Processing Sys-
+tems, 33:3907–3916, 2020.
+[10] Diederik P Kingma and Jimmy Ba. Adam: A method for
+stochastic optimization.
+arXiv preprint arXiv:1412.6980,
+2014. 3
+[11] Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple
+layers of features from tiny images. 2009. 1
+[12] Hae Beom Lee, Hayeon Lee, Donghyun Na, Saehoon Kim,
+Minseop Park, Eunho Yang, and Sung Ju Hwang. Learn-
+ing to balance: Bayesian meta-learning for imbalanced and
+out-of-distribution tasks. arXiv preprint arXiv:1905.12917,
+2019.
+[13] Zhenguo Li, Fengwei Zhou, Fei Chen, and Hang Li. Meta-
+sgd: Learning to learn quickly for few-shot learning. arXiv
+preprint arXiv:1707.09835, 2017. 2
+[14] Alex Nichol, Joshua Achiam, and John Schulman.
+On
+first-order
+meta-learning
+algorithms.
+arXiv
+preprint
+arXiv:1803.02999, 2018. 3
+[15] Genevieve Patterson, Chen Xu, Hang Su, and James Hays.
+The sun attribute database: Beyond categories for deeper
+scene understanding. International Journal of Computer Vi-
+sion, 108(1):59–81, 2014. 2
+[16] Aravind Rajeswaran, Chelsea Finn, Sham M Kakade, and
+Sergey Levine. Meta-learning with implicit gradients. Ad-
+vances in neural information processing systems, 32, 2019.
+1
+[17] Sachin Ravi and Hugo Larochelle. Optimization as a model
+for few-shot learning. 2016.
+[18] Andrei A Rusu, Dushyant Rao, Jakub Sygnowski, Oriol
+Vinyals, Razvan Pascanu, Simon Osindero, and Raia Had-
+sell.
+Meta-learning with latent embedding optimization.
+arXiv preprint arXiv:1807.05960, 2018.
+[19] Amrith Setlur, Oscar Li, and Virginia Smith.
+Is support
+set diversity necessary for meta-learning?
+arXiv preprint
+arXiv:2011.14048, 2020.
+[20] Jake Snell, Kevin Swersky, and Richard Zemel. Prototypical
+networks for few-shot learning. Advances in neural informa-
+tion processing systems, 30, 2017. 3
+[21] Flood Sung, Yongxin Yang, Li Zhang, Tao Xiang, Philip HS
+Torr, and Timothy M Hospedales. Learning to compare: Re-
+lation network for few-shot learning. In Proceedings of the
+IEEE conference on computer vision and pattern recogni-
+tion, pages 1199–1208, 2018. 3
+[22] Sebastian Thrun and Lorien Pratt.
+Learning to learn.
+Springer Science & Business Media, 2012.
+[23] Oriol Vinyals, Charles Blundell, Timothy Lillicrap, Daan
+Wierstra, et al. Matching networks for one shot learning. Ad-
+vances in neural information processing systems, 29, 2016.
+2, 3
+[24] Risto Vuorio, Shao-Hua Sun, Hexiang Hu, and Joseph J
+Lim.
+Multimodal model-agnostic meta-learning via task-
+aware modulation. Advances in Neural Information Process-
+ing Systems, 32, 2019.
+[25] Catherine Wah, Steve Branson, Peter Welinder, Pietro Per-
+ona, and Serge Belongie. The caltech-ucsd birds-200-2011
+dataset. 2011. 2
+

6NFKT4oBgHgl3EQfTS23/content/tmp_files/load_file.txt

@@ -0,0 +1,301 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf,len=300
+page_content='Invariant Meta Learning for Out-of-Distribution Generalization  Penghao Jiang, Ke Xin, Zifeng Wang, Chunxi Li  The Australian National University, Canberra, Australia *  Abstract  Modern deep learning techniques have illustrated their  excellent capabilities in many areas, but relies on large  training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Optimization-based meta-learning train a  model on a variety tasks, such that it can solve new learn-  ing tasks using only a small number of training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  However, these methods assumes that training and test data  are identically and independently distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' To overcome  such limitation, in this paper, we propose invariant meta  learning for out-of-distribution tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Specifically, invari-  ant meta learning find invariant optimal meta-initialization,  and fast adapt to out-of-distribution tasks with regulariza-  tion penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Extensive experiments demonstrate the effec-  tiveness of our proposed invariant meta learning on out-  ofdistribution few-shot tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Introduction  Modern deep learning techniques have illustrated their  excellent capabilities in many areas like computer vision,  natural language processing and recommendation, etc [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  However, these methods relies on large training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' To  overcome this limitation, few-shot learning methods such  as meta learning has been proposed [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Most popular meta  learning approaches is the optimization-based metalearning  [4, 16], which is model-agnostic and can be applied to var-  ious downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' However, many recent researches  have revealed the vulnerability of machine learning model  when exposed to data with different distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Such massive gap is induced by the violation of a funda-  mental assumption that training and test data are identically  and independently distributed (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  assumption),  upon which most of the existing meta learning models are  developed [4, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' In many real cases where i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' assump-  tion can hardly be satisfied, especially those high-stake ap-  plications such as healthcare, military and autonomous driv-  ing, instead of generalization within the training distribu-  tion, the ability to generalize under distribution shift is of  more critical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' As shown in Figure 1, given tran-  The first two authors contributed equally as joint first authorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  The last two authors contributed equally as joint second authorship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Illustration example of how the distribution shifts be-  tween training data and testing data hamper the performance of  model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Causal framework of dog perdiction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Due to the  spurious correlation, the model tends to focus on both grass and  dog, which lead to failed prediction in other distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  ing data where dogs are on the grass, model could not make  accurate predictions in testing data where dogs are in water,  cage or street.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' The reason is that the supurious correlation  between grass and dog in traning data hamper the perfor-  mance of model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Due to the spurious correlation, the model  tends to focus on both grass and dog, which lead to failed  prediction in other distribution such as dogs are in water,  cage or street as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' However, recent meta  learning methods could not overcome the distribution shifts  between training and testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' In this paper, we con-  sider a realistic scenario where tasks come from different  distributions (out-of-distribution, OOD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  In this paper, to overcome the problem mentioned above,  we propose Invariant Meta Learning (IML) for out-of- dis-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='11779v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='LG]  26 Jan 2023  Athome  onbeach  eating  incage  inwater  lying  ongrass  instreet  running  Training data  Model  Testing dataGrass--Label:Strongcorrelation  CausalFramework  Weakcausation  Dog noseLabel:Strong correlation  X  Strong causation  T:  grass  X: dog nose  Y:labeltribution tasks, a general learning framework that jointly ad-  justs gradient magnitudes and directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Specifically, in-  variant meta learning find invariant optimal metainitializa-  tion, and fast adapt to out-of-distribution tasks with regular-  ization penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' To summarize, our main contributions are:  We consider the challenge of out-of-distribution tasks  faced by few-shot learning, we show a natural idea to  jointly adjust gradient magnitudes and directions of all  tasks in the meta optimization process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  We propose Invariant Meta Learning (IML) for out-  ofdistribution tasks, a general learning framework that  jointly adjusts gradient magnitudes and directions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  We conduct extensive experiments and analysis to  demonstrate that our approach effectively improves the  performance and generalization ability under both in-  distribution and out-of-distribution few-shot settings,  and thus it can be regarded as a better baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Method  In this section, we introduce our proposed Invariant Meta  Learning (IML) to address the out-of-distribution problem  in few-shot tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  IML learns invariant optimal predic-  tors based on optimization based meta learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  To learn invariant optimal meta-initialization in optimiza-  tion based meta learning, the main challenge is that OOD  problem exacerbates the inconsistency in both task-gradient  magnitudes and directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  To overcome such problem,  IML finds invariant optimal initialization, and adapt to  outof- distribution tasks with regularization penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Model-agnostic meta-learning (MAML) [4] is an ap-  proach to optimization-based meta-learning that is related  to our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' For some parametric model fθ, MAML aims  to find a single set of parameters θ which, using a few op-  timization steps, can be successfully adapted to any novel  task sampled from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' For a particular  task instance Ti =  �  Dtr, Dval�  , the parameters are adapted  to task-specific model parameters θ′  i by applying some dif-  ferentiable function, typically an update rule of the form:  θ′  i = G  �  θ, Dtr�  ,  (1)  where G is typically implemented as a step of gradi-  ent descent on the few-shot training set Dtr , θ′  i = θ−  α∇θLtr  Ti (fθ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Generally, multiple sequential adaptation  steps can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' The learning rate α can also be met-  alearned concurrently, in which case we refer to this algo-  rithm as Meta-SGD [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' During meta-training, the param-  eters θ are updated by back-propagating through the adap-  tation procedure, in order to reduce errors on the validation  set Dval :  θ ← θ − η∇θ  �  Ti∼p(T )  Lval  Ti  �  fθ′  i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  (2)  The  approach  includes  the  main  ingredients  of  optimization-based meta-learning with neural networks:  initialization is done by maintaining an explicit set of  model parameters θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' the adaptation procedure, or “inner  loop”, takes θ as input and returns θ′  i adapted specifically  for task instance Ti, by iteratively using gradient descent  (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' and termination, which is handled simply by  choosing a fixed number of optimization steps in the “inner  loop”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  MAML updates θ by differentiating through the  “inner loop” in order to minimize errors of instance-specific  adapted models fθ′  i on the corresponding validation set  (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' We refer to this process as the “outer loop” of  meta-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' We use the same stages to describe IML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Invariant Meta Learning (IML) finds invariant opti-  mal meta-initialization, and fast adapt to out-of-distribution  tasks with regularization penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' MAML fast adapt net-  work to new task during the inner loop and learns univer-  sal meta-initialization in outer loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Similarly, in IML, we  update network with the bi-level update, optimizing clas-  sifier in the inner loop and learning feature representation  in the outer loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' For the inner-level optimization, the pa-  rameters θ of the predictor become θi while adapting to the  task ti ∈ Ttr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' This correspond to the inner optimization of  MAML, except that each task ti has a corresponding net-  work θi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' The optimization in the inner loop can be defined  as follows:  θ′  i = θ − α∇θLtr  Ti (fθ)  (3)  where α is a learning rate of the inner optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  With inner optimized network fθ′  i, we have outer loop  objective function with variance penalty regularizer:  Lval =  �  Ti∼p(T tr)  �  Tj∼p(T val)  Lval  Tj  �  fθ′  i  �  (4)  θ ← θ − η∇θLval − βλ trace  �  VarT val  �  ∇θLval��  (5)  where η, β are the learning rate of the outer loop optimiza-  tion, tj is task j for outer loop optimization for the net-  work θ′  i, L is the loss function for outer loop optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Note that the inner optimized network fθ′  i is used to up-  date meta-initialization in outer loop with tj whereas it is  updated from meta-initialization with ti in ther inner loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  IML learn invariant meta-initialization obtained from the  discrepancy among different training tasks with variance  penalty regularizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Experiments  Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  In this paper, we address the few-shot clas-  sification problem under both in-distribution and out-  ofdistribution FSL settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' These settings are conducted  on three benchmark datasets: miniImageNet [23], Caltech-  UCSD-Birds 200-2011 (CUB) [25], and SUN Attribute  Database (SUN) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Method  miniImageNet  CUB  SUN  5-way 1-shot  5-way 5-shot  5-way 1-shot  5-way 5-shot  5-way 1-shot  5-way 5-shot  Meta-Learner LSTM  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='99  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='79  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='23  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='39  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='99  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='86  MAML  45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='69  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='90  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='87  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='99  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='75  71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='45  Reptile  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='59  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='87  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='21  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='35  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='30  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='62  Matching Network  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='63  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='28  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='06  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='19  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='02  62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='57  Prototypical Network  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='15  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='56  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='21  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='80  55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='70  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='32  Relation Network  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='64  63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='65  52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='76  64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='71  58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='29  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='15  Baseline  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='84  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='09  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='14  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='35  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='44  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='54  Baseline++  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='15  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='19  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='48  42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='43  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='56  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='42  IML  48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='35  67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='21  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='18  65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='85  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='24  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='18  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Average accuracy (%) comparison to state-of-the-arts with 95% confidence intervals on 5-way classification tasks under the  in-distribution FSL setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Best results are displayed in boldface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Method  miniImageNet→ CUB  miniImageNet→ SUN  CUB→miniImageNet  5-way 1-shot  5-way 5-shot  5-way 1-shot  5-way 5-shot  5-way 1-shot  5-way 5-shot  Meta-Learner LSTM  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='77  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='58  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='52  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='14  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='58  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='18  MAML  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='29  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='01  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='07  59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='08  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='36  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='58  Reptile  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='66  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='86  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='15  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='38  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='56  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='60  Matching Network  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='34  47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='64  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='58  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='20  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='23  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='90  Prototypical Network  36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='60  54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='36  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='31  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='21  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='22  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='73  Relation Network  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='33  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='64  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='55  61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='45  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='64  38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='01  Baseline  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='16  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='73  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='49  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='15  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='98  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='41  Baseline++  29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='40  40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='48  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='44  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='71  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='41  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='82  IML  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='27  57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='34  50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='42  69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='15  34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='26  44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='17  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Average accuracy (%) comparison to state-of-the-arts with 95% confidence intervals on 5-way classification tasks under the  in-distribution FSL setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Best results are displayed in boldface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  To evaluate the effectiveness of the proposed  framework, we consider the following representative meta  learning methods on the few-shot image classification task:  MAML [5], Reptile [14], Matching Network [23], Proto-  typical Network [20], Relation Network [21], Baseline and  Baseline++ [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Experimental Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  We conduct experiments on 5-  way 1-shot and 5-way 5 -shot settings, there are 15 query  samples per class in each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' We report the average ac-  curacy (%) and the corresponding 95% confidence interval  over the 2000 tasks randomly sampled from novel classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  To fairly evaluate the original performance of each method,  we use the same 4-layer ConvNet [23] as the backbone for  all methods and do not adopt any data augmentation during  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' All methods are trained via SGD with Adam [10],  and the initial learning rate is set to e−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' For each method,  models are trained for 40,000 tasks at most, and the best  model on the validation classes is used to evaluate the final  reporting performance in the meta-test phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Evaluation Using the In-Distribution Setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Table 1  shows the comparative results under the in-distribution FSL  setting on three benchmark datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  It is observed that  IML outperforms the original MAML in all in-distribution  FSL scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  For 1-shot and 5-shot on miniImageNet  → miniImageNet, IML achieves about 1% higher perfor-  mance than Prototypical Network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' However, IML achieves  5% and 10% higher performance for 1-shot and 5-shot  on CUB → CUB, and 3% and 6% higher performance  on SUN → SUN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' As the latter two scenarios are con-  ducted on finegrained classification datasets, we attribute  the promising improvement to that the categories in these  fine-grained datasets share more local concepts than those  in coarsegrained datasets, and thus a more discriminative  space can be rapidly learned with a few steps of adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Moreover, IML achieves the best performance among all  baselines in all in-distribution FSL scenarios, which shows  that our approach can be considered as a better baseline op-  tion under the in-distribution FSL setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Evaluation Using the Out-of-Distribution Setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  We  also conduct out-of-distribution FSL experiments and re-  port the comparative results in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Compared to the re-  sults under the in-distribution setting, it can be observed that  all approaches suffer from a larger discrepancy between the  distributions of training and testing tasks, which results in  a performance decline in all scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' However, IML still  outperforms the original MAML in all out-of-distribution  FSL scenarios, demonstrating that the bilevel optimization  strategy for adaptation and the learning of transferable la-  tent factors can be utilized to improve simple meta learning  approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Also, IML achieves all the best results, indicat-  ing that our approach can be regarded as a promising base-  line under the out-of-distribution setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Conclusion  In this paper,  we consider the challenge of out-  ofdistribution tasks faced by few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' We propose  Invariant Meta Learning (IML) for out-of-distribution tasks,  a general learning framework that jointly adjusts gradient  magnitudes and directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Extensive experiments demon-  strate that our approach effectively improves the perfor-  mance and generalization ability under both in-distribution  and out-of-distribution few-shot settings, and thus it can be  regarded as a better baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  References  [1] Yoshua Bengio, Samy Bengio, and Jocelyn Cloutier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Learn-  ing a synaptic learning rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Citeseer, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  [2] Fei Chen, Mi Luo, Zhenhua Dong, Zhenguo Li, and  Xiuqiang He.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Federated meta-learning with fast con-  vergence and efficient communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  arXiv preprint  arXiv:1802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='07876, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  [3] Wei-Yu Chen, Yen-Cheng Liu, Zsolt Kira, Yu-Chiang Frank  Wang, and Jia-Bin Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' A closer look at few-shot classi-  fication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' arXiv preprint arXiv:1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='04232, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' 3  [4] Chelsea Finn, Pieter Abbeel, and Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Model-  agnostic meta-learning for fast adaptation of deep networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  In International conference on machine learning, pages  1126–1135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' PMLR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' 1, 2  [5] Chelsea Finn, Pieter Abbeel, and Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Model-  agnostic meta-learning for fast adaptation of deep networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  In Proceedings of the 34th International Conference on Ma-  chine Learning, pages 1126–1135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' PMLR, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' 3  [6] Chelsea Finn and Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Meta-learning and  universality:  Deep representations and gradient descent  can approximate any learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  arXiv preprint  arXiv:1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='11622, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' 1  [7] Timothy Hospedales, Antreas Antoniou, Paul Micaelli, and  Amos Storkey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Meta-learning in neural networks: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  arXiv preprint arXiv:2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='05439, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  [8] Simon Jenni and Paolo Favaro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Deep bilevel learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' In  Proceedings of the European conference on computer vision  (ECCV), pages 618–633, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  [9] Taewon Jeong and Heeyoung Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Ood-maml:  Meta-  learning for few-shot out-of-distribution detection and clas-  sification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Advances in Neural Information Processing Sys-  tems, 33:3907–3916, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  [10] Diederik P Kingma and Jimmy Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Adam: A method for  stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  arXiv preprint arXiv:1412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='6980,  2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' 3  [11] Alex Krizhevsky, Geoffrey Hinton, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Learning multiple  layers of features from tiny images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' 1  [12] Hae Beom Lee, Hayeon Lee, Donghyun Na, Saehoon Kim,  Minseop Park, Eunho Yang, and Sung Ju Hwang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Learn-  ing to balance: Bayesian meta-learning for imbalanced and  out-of-distribution tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' arXiv preprint arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='12917,  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  [13] Zhenguo Li, Fengwei Zhou, Fei Chen, and Hang Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Meta-  sgd: Learning to learn quickly for few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' arXiv  preprint arXiv:1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='09835, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' 2  [14] Alex Nichol, Joshua Achiam, and John Schulman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  On  first-order  meta-learning  algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  arXiv  preprint  arXiv:1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='02999, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' 3  [15] Genevieve Patterson, Chen Xu, Hang Su, and James Hays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  The sun attribute database: Beyond categories for deeper  scene understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' International Journal of Computer Vi-  sion, 108(1):59–81, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' 2  [16] Aravind Rajeswaran, Chelsea Finn, Sham M Kakade, and  Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Meta-learning with implicit gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Ad-  vances in neural information processing systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  1  [17] Sachin Ravi and Hugo Larochelle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Optimization as a model  for few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  [18] Andrei A Rusu, Dushyant Rao, Jakub Sygnowski, Oriol  Vinyals, Razvan Pascanu, Simon Osindero, and Raia Had-  sell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Meta-learning with latent embedding optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  arXiv preprint arXiv:1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='05960, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  [19] Amrith Setlur, Oscar Li, and Virginia Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Is support  set diversity necessary for meta-learning?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  arXiv preprint  arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='14048, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  [20] Jake Snell, Kevin Swersky, and Richard Zemel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Prototypical  networks for few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Advances in neural informa-  tion processing systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' 3  [21] Flood Sung, Yongxin Yang, Li Zhang, Tao Xiang, Philip HS  Torr, and Timothy M Hospedales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Learning to compare: Re-  lation network for few-shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' In Proceedings of the  IEEE conference on computer vision and pattern recogni-  tion, pages 1199–1208, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' 3  [22] Sebastian Thrun and Lorien Pratt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Learning to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Springer Science & Business Media, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  [23] Oriol Vinyals, Charles Blundell, Timothy Lillicrap, Daan  Wierstra, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Matching networks for one shot learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Ad-  vances in neural information processing systems, 29, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  2, 3  [24] Risto Vuorio, Shao-Hua Sun, Hexiang Hu, and Joseph J  Lim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  Multimodal model-agnostic meta-learning via task-  aware modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' Advances in Neural Information Process-  ing Systems, 32, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content='  [25] Catherine Wah, Steve Branson, Peter Welinder, Pietro Per-  ona, and Serge Belongie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' The caltech-ucsd birds-200-2011  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}
+page_content=' 2' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NFKT4oBgHgl3EQfTS23/content/2301.11779v1.pdf'}

7NE2T4oBgHgl3EQfPQZw/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:6686ae2509b51d0f7e491138107b3bf9830e28fe0329d08bf85e63195c4e4234
+size 232162

89E0T4oBgHgl3EQfwgGc/content/2301.02634v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:ec76062bdd94914f1b63f1a6c25edf2029673c24f6e641ee2943c96661e20f49
+size 238676

89E0T4oBgHgl3EQfwgGc/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:295dae4f778029a89ff3992a1d5ebc746d383d6e53e1abf3ce95c418083b79f2
+size 1966125

89E0T4oBgHgl3EQfwgGc/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:9ee97a0bb5cc35a3100c16769c7def0879699feb79cd3a9ad933353e34bfc4fb
+size 86167

8dAyT4oBgHgl3EQfp_jS/content/tmp_files/2301.00536v1.pdf.txt

@@ -0,0 +1,5015 @@
+arXiv:2301.00536v1  [math.PR]  2 Jan 2023
+Lp-SOLVABILITY AND H¨OLDER REGULARITY FOR STOCHASTIC
+TIME FRACTIONAL BURGERS’ EQUATIONS DRIVEN BY
+MULTIPLICATIVE SPACE-TIME WHITE NOISE
+BEOMSEOK HAN
+Abstract. We present the Lp-solvability for stochastic time fractional Burgers’ equations
+driven by multiplicative space-time white noise:
+∂α
+t u = aijuxixj + biuxi + cu + ¯biuuxi + ∂β
+t
+ˆ t
+0
+σ(u)dWt, t > 0; u(0, ·) = u0
+where α ∈ (0, 1), β < 3α/4 + 1/2, and d < 4 − 2(2β − 1)+/α. The operators ∂α
+t and
+∂β
+t are the Caputo fractional derivatives of order α and β, respectively. The process Wt
+is an L2(Rd)-valued cylindrical Wiener process, and the coefficients aij, bi, c and σ(u) are
+random.
+In addition to the existence and uniqueness of a solution, we also suggest the H¨older
+regularity of the solution. For example, for any constant T < ∞, small ε, δ > 0, and,
+almost sure ω ∈ Ω, we have
+sup
+x∈Rd |u(ω, ·, x)|
+C
+�
+α
+2 ((2−(2β−1)+/α−d/2)∧1)+ (2β−1)−
+2
+�
+∧1−ε
+([δ,T ])
+< ∞
+and
+sup
+t≤T
+|u(ω, t, ·)|
+C(2−(2β−1)+ /α−d/2)∧1−ε(Rd) < ∞.
+Moreover, δ can be 0 if the initial data u0 = 0. Additionally, the H¨older regularity of the
+solution in time changes behavior at β = 1/2. Furthermore, if β ≥ 1/2, then the H¨older
+regularity of the solution in time is α/2 times the one in space.
+1. Introduction
+This article investigates the existence, uniqueness, Lp-regularity, and maximal H¨older
+regularity of a solution to stochastic time fractional Burgers’ equations (STFBEs) driven
+by space-time white noise. We consider
+∂α
+t u = Lu + ¯biuuxi + ∂β
+t
+ˆ t
+0
+σ(u)dWt,
+(ω, t, x) ∈ Ω × (0, ∞) × Rd;
+u(0, ·) = u0,
+(1.1)
+where α ∈ (0, 1), β < 3
+4α + 1
+2, and d < 4 − 2(2β−1)+
+α
+. The operators ∂α
+t and ∂β
+t are the
+Caputo fractional derivatives of order α and β, and the operator L is the second order
+random differential operator defined as follows:
+(Lu)(ω, t, x) = aij(ω, t, x)uxixj + bi(ω, t, x)uxi + c(ω, t, x)u.
+2020 Mathematics Subject Classification. 35R11, 26A33, 60H15, 35R60.
+Key words and phrases. Stochastic partial differential equation, Time fractional derivative Stochastic
+Burgers’ equation, Time fractional Burgers’ equation, Space-time white noise, H¨older regularity.
+This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea
+government (MSIT) (No. NRF-2021R1C1C2007792) and the BK21 Fostering Outstanding Universities for
+Research (FOUR) funded by the Ministry of Education (MOE, Korea) and the National Research Foundation
+of Korea (NRF).
+1
+
+2
+BEOMSEOK HAN
+The random coefficients aij, bi, and c are predictable, differentiable (or continuous), and
+bounded functions. The diffusion coefficient σ(u) = σ(ω, t, x, u) is a predictable and mea-
+surable function satisfying growth conditions and Lipschitz continuity in u. The detailed
+conditions on aij, bi, c, and σ are described in Assumptions 3.1 and 3.3.
+The random
+measure dWt is induced from an L2(Rd)-valued cylindrical Wiener process Wt.
+When α = β = 1 in equation (1.1), the equation is said to be a stochastic Burgers’
+equation (SBE) of form
+∂tu = Lu + ¯buux + σ(u) ˙W,
+(ω, t, x) ∈ Ω × (0, ∞) × R;
+u(0, ·) = u0,
+(1.2)
+where
+˙W is the space-time white noise. Numerous studies have been conducted on the
+equation (1.2), but we only refer to the reader to [13, 14, 29]. In [13], the author proved
+the uniqueness, existence, and continuity of a solution to a semilinear equation, including
+an equation of type (1.2) on the unit interval (0, 1). Additionally, the same properties of a
+solution on R were obtained in [14] when the L2 bounded conditions on σ(u) were imposed.
+In [29], the authors investigated the H¨older regularity and moment estimates of the random
+field solution to (1.2) with L = ∆ and ¯b = −1.
+In contrast, (deterministic) partial differential equations with Caputo fractional deriva-
+tives have been used in many fields, such as electrochemical processes [5, 19], dielectric
+polarization [33], viscoelastic materials [32], biology [31], and physics [11, 18]. Especially,
+equation (1.1) with α ∈ (0, 1) and σ(u) = 0 is called a time fractional Burgers’ equa-
+tion (TFBE), which describes the propagation of waves through viscous media ([1, 2]).
+Indeed, various researches have been conducted on numerical analysis for the TFBE (see
+[3, 9, 10, 20, 30]). From a mathematical standpoint, it is reasonable to wonder whether it
+is possible to demonstrate the uniqueness and existence of a solution to STFBE (1.1), and
+also to obtain the H¨older regularity of the solution. To the best of our knowledge, [36] is
+the only study that answers this question. The authors of [36] demonstrate the existence,
+uniqueness, and regularity of the mild solution to SBEs with fractional derivatives in time
+and space on a bounded domain D ⊂ Rd.
+In this paper, we provide the Lp uniqueness, existence, and regularity of a strong solution
+to equation (1.1) with random second order differential operator L on the whole spatial
+domain Rd.
+Additionally, we achieve the H¨older regularity of the solution in time and
+space. In detail, if u(ω, t, x) denotes the solution to equation (1.1), then for any bounded
+stopping time τ ≤ T and small constant ε, δ > 0, almost surely,
+sup
+x∈Rd |u(ω, ·, x)|
+C
+�
+α
+2 ((2−(2β−1)+/α−d/2)∧1)+ (2β−1)−
+2
+�
+∧1−ε
+([δ,τ])
+< ∞,
+sup
+t≤τ
+|u(ω, t, ·)|C(2−(2β−1)+/α−d/2)∧1−ε(Rd) < ∞.
+(1.3)
+where a+ = (|a| + a)/2, a− = (|a| − a)/2, and Cγ(D) is the H¨older spaces. Observe that
+the behavior of the H¨older regularity of the solution in time changes from β = 1/2. For
+example, if β ≥ 1/2, then the H¨older regularity of the solution in time is α/2 times that
+of the regularity in space. Additionally, we can recover the the H¨older regularity results
+of SBEs by letting α, β ↑ 1. These results are consistent with the well-known results of
+stochastic heat equations driven by space-time white noise (e.g. [26, Remark 8.7] or [16,
+Corollary 3.1]).
+In contrast, if β < 1/2, the H¨older regularity in time gains additional
+regularity by as much as 1/2 − β. (Remark 3.12). Finally, δ = 0 is allowed if the initial
+data u0 is 0 (Remark 3.6).
+
+STFBES DRIVEN BY MULTIPLICATIVE SPACE-TIME WHITE NOISE
+3
+Several remarks about the proof are made. The proof strategy for the main theorem
+(Theorem 3.5) is based on [16]. However, some differences exist because since it is not
+certain that Itˆo’s formula and the maximum principle hold for STFBE (1.1).
+Thus, the proof proceeds as follows. As in [16], we focus on proving the uniqueness and
+existence of the Lp solution in each (large) p > 2, and the main difficulty is to demonstrating
+the existence of the solutions. Hence, we consider the cut-off form of equation (1.1) to obtain
+local solutions. Afterward, we construct a global solution candidate u by pasting the local
+solutions (Lemma 4.3 and Remark 4.4). A uniform Lp bound of u is required to show that
+our the candidate u is a global solution; thus, we divide the local solution into two parts:
+the noise-dominating and the nonlinear-dominating part. To estimate the noise-dominating
+parts, we employ the Lp bound of the diffusion coefficient σ(u) (Lemma 4.5). In contrast,
+to control the nonlinear-dominating part (Lemma 4.8), we employ an inequality similar to
+the chain rule (Lemma 4.6) and a version of the Gr¨onwall inequality including the Caputo
+fractional derivatives (Theorem 4.7).
+To obtain the maximal H¨older regularity of the solution to equation (1.1), we require two
+components: the H¨older embedding theorem for the solution space Hγ
+p(τ) (Theorem 2.16)
+and the uniqueness of the solution in p (Theorem 3.10). Indeed, when the Lp existence and
+uniqueness of a solution are given, we have the H¨older regularity of the solution in each
+(large) p > 2 by employing the H¨older embedding theorem for the solution space (Theorem
+2.16 and Theorem 3.5). The H¨older regularity of the solution becomes larger as a large p
+is chosen; thus, we have to select p that is as large as possible. Therefore, we require the
+uniqueness of solutions in p because p varies.
+This article is organized as follows. Section 2 introduces the definitions and properties
+of space-time white noise, fractional calculus, and stochastic Banach spaces. Additionally,
+we present the H¨older embedding theorem for the solution space Hγ
+p(τ). Section 3 states
+the main results of this article and suggests some remarks. The proof of the main results
+is presented in Section 4. Next, Section 5 proves the H¨older embedding theorem for the
+solution space Hγ
+p(τ).
+We finish this section with an introduction to the notation used in this paper. The sets
+N and R are sets of natural and real numbers, respectively.
+The set Rd denotes the d-
+dimensional Euclidean space of points x = (x1, . . . , xd) for xi ∈ R. Throughout this paper,
+we assume Einstein’s summation convention on i, j, k ∈ N. We use := to denote a definition.
+For a real-valued function f, we set the following:
+f+ := |f| + f
+2
+and
+f− := |f| − f
+2
+.
+For a normed space F, a measure space (X, M, µ), and p ∈ [1, ∞), a space Lp(X, M, µ; F)
+is a set of F-valued Mµ-measurable functions such that
+∥u∥Lp(X,M,µ;F ) :=
+�ˆ
+X
+∥u(x)∥p
+F µ(dx)
+�1/p
+< ∞.
+A set Mµ is the completion of M with respect to the measure µ.
+For γ ∈ (0, 1] and
+k = 0, 1, 2, . . . , a set Ck+γ(Rd) is the set of R-valued continuous functions u = u(x) such
+that
+|u|Cγ+k(Rd) :=
+sup
+x∈Rd,|β|=k
+���Dβu(x)
+��� +
+sup
+x,y∈Rd,x̸=y
+|β|=k
+��Dβu(x) − Dβu(y)
+��
+|x − y|γ
+< ∞,
+
+4
+BEOMSEOK HAN
+where β is a multi-index. Similarly, for γ ∈ (0, 1] and 0 ≤ δ < T < ∞, the set Cγ([δ, T]; F)
+is the set of F-valued continuous functions u such that
+|u|Cγ([δ,T];F ) := sup
+t∈[δ,T]
+|u(t)|F +
+sup
+t,s∈[δ,T],
+s̸=t
+|u(t) − u(s)|F
+|t − s|γ
+< ∞.
+For a, b ∈ R, we set a∧b := min{a, b} and a∨b := max{a, b}. Let S = S(Rd) denote the set
+of Schwartz functions on Rd. Let N = N(a1, a2, ..., ak) be a generic constant if N depends
+only on a1, a2, ..., ak. The constant N can vary line by line. For functions depending on ω, t,
+and x, the argument ω ∈ Ω is omitted. Finally, for x ∈ Rd, ¯xi := (x1, . . . , xi−1, xi+1, . . . , xd).
+2. Preliminaries
+In this section, we introduce the definitions and properties of space-time white noise,
+fractional calculus, and stochastic Banach spaces.
+Throughout this paper, (Ω, F, P) is
+a complete probability space equipped with a filtration {Ft}t≥0.
+Let {Ft}t≥0 denote a
+filtration satisfying the usual conditions. Let P be the predictable σ-field related to {Ft}t≥0.
+First, we present the space-time white noise ˙W to understand the stochastic part of (1.1).
+Definition 2.1 (Space-time white noise). A generalized random field ˙W is said to be the
+space-time white noise if it is a centered Gaussian random field such that its covariance is
+given by
+E ˙W(h) ˙W (g) =
+ˆ ∞
+0
+ˆ
+Rd h(t, x)g(t, x)dxdt,
+∀h, g ∈ L2((0, ∞) × Rd).
+Remark 2.2. We employ a series of Itˆo’s stochastic integral to interpret the stochastic part
+of equation (1.1). More precisely, let {ηk : k ∈ N} be an orthonormal basis on L2(Rd). If
+we define
+wk
+t :=
+ˆ t
+0
+ˆ
+Rd ηk(x) ˙W(ds, dx)
+using the Walsh integral (see [35]), then {wk
+t : k ∈ N} is a set of one dimensional indepen-
+dent Wiener processes. Then, if we set (see [26, Section 8.3], and [23, Section 7])
+Wt :=
+∞
+�
+k=1
+ηkwk
+t ,
+then Wt is an L2(Rd)-valued cylindrical Wiener process and dWt = �
+k ηkdwk
+t .
+Thus,
+equation (1.1) can be rewritten as
+∂α
+t u = Lu + ¯biuuxi + ∂β
+t
+ˆ t
+0
+σ(u)ηkdwk
+t ,
+(ω, t, x) ∈ Ω × (0, ∞) × Rd;
+u(0, ·) = u0.
+Next, we review the facts of fractional calculus. For more information, we refer to the
+reader to [6, 17, 21, 32].
+Definition 2.3. Let α > 0, and for ϕ ∈ L1((0, T)), the Riemann-Liouville fractional
+integral of the order α is defined as follows:
+Iα
+t ϕ(t) := (Iα
+t ϕ)(t) :=
+1
+Γ(α)
+ˆ t
+0
+(t − s)α−1ϕ(s)ds
+for all
+t ∈ (0, T),
+where Γ(α) :=
+´ ∞
+0
+tα−1e−tdt.
+
+STFBES DRIVEN BY MULTIPLICATIVE SPACE-TIME WHITE NOISE
+5
+Remark 2.4. For any q ∈ [1, ∞], by Jensen’s inequality
+∥Iαϕ∥Lq((0,T)) ≤ N(α, p, T)∥ϕ∥Lq((0,T)).
+(2.1)
+Therefore, Iα
+t ϕ(t) is well-defined and finite for almost all t ≤ T. Additionally, Fubini’s
+theorem implies that, for α, β ≥ 0, we have
+Iα+βϕ(t) = IαIβϕ(t).
+Definition 2.5. For α > 0, let n ∈ N be a nonnegative integer such that n − 1 ≤ α < n.
+Suppose ϕ(t) is a real-valued function on [0, T] such that ϕ is (n − 1)-times differentiable
+and ( d
+dt)n−1ϕ is absolutely continuous on [0, T].
+(i) The Riemann-Liouville fractional derivative Dα
+t ϕ is defined as
+Dα
+t ϕ(t) :=
+1
+Γ(n − α)
+dn
+dtn
+ˆ t
+0
+(t − s)n−α−1ϕ(s)ds.
+(ii) The Caputo fractional derivative ∂α
+t ϕ is defined as
+∂α
+t ϕ :=
+1
+Γ(n − α)
+ˆ t
+0
+(t − s)n−α−1ϕ(n)(s)ds
+:=
+1
+Γ(n − α)
+d
+dt
+ˆ t
+0
+(t − s)n−α−1 �
+ϕ(n−1)(s) − ϕ(n−1)(0)
+�
+ds.
+Remark 2.6.
+(i) For any α, β ≥ 0, Dα
+t Dβ
+t ϕ = Dα+β
+t
+ϕ and
+Dα
+t Iβ
+t ϕ = Dα−β
+t
+ϕ1α>β + Iβ−α
+t
+ϕ1α≤β.
+Additionally, if α ∈ (0, 1), I1−α
+t
+ϕ is absolutely continuous, and I1−α
+t
+ϕ(0) = 0, then the
+following equality holds:
+Iα
+t Dα
+t ϕ(t) = ϕ(t).
+(ii) By the definition of fractional derivatives, if ϕ(0) = ϕ(1)(0) = · · · = ϕ(n−1)(0) = 0,
+then Dα
+t ϕ = ∂α
+t ϕ.
+Below we recall the definitions and properties of stochastic Banach spaces (for more
+detail, see [12, 25, 26, 27]). The solution space Hγ
+p(T) and embedding theorems for Hγ
+p(T)
+are suggested.
+Definition 2.7. Let p > 1 and γ ∈ R. The space Hγ
+p = Hγ
+p (Rd) is the set of all tempered
+distributions u on R such that
+∥u∥Hγ
+p :=
+���(1 − ∆)γ/2u
+���
+Lp =
+���F−1 �
+(1 + |ξ|2)γ/2F(u)(ξ)
+����
+Lp < ∞.
+Similarly, Hγ
+p (l2) = Hγ
+p (Rd; l2) is a space of l2-valued functions g = (g1, g2, · · · ) such that
+∥g∥Hγ
+p (l2) :=
+����
+���(1 − ∆)γ/2g
+���
+l2
+����
+Lp
+=
+����
+���F−1 ��
+1 + |ξ|2�γ/2 F(g)(ξ)
+����
+l2
+����
+Lp
+< ∞.
+Remark 2.8. Let d ∈ N and γ ∈ (0, ∞). A nonnegative smooth function Rγ(x) exists on
+Rd such that, for u ∈ C∞
+c (Rd),
+�
+(1 − ∆)−γ/2 u
+�
+(x) =
+ˆ
+Rd Rγ(y)u(x − y)dy
+and
+|Rγ(x)| ≤ NAγ,d(x)1|x|≤2 + Ne−|x|/21|x|≥2,
+
+6
+BEOMSEOK HAN
+where N = N(γ, d) is a positive constant and
+Aγ,d(x) =
+
+
+
+
+
+|x|γ−d + 1 + O(|x|γ−d+2)
+for
+0 < γ < d,
+log(2/|x|) + 1 + O(|x|2)
+for
+γ = d,
+1 + O(|x|γ−d)
+for
+γ > d.
+For more detail, see [12, Proposition 1.2.5].
+We introduce the space of point-wise multipliers in Hγ
+p .
+Definition 2.9. Fix γ ∈ R and α ∈ [0, 1) such that α = 0 if γ ∈ Z and α > 0 if |γ| + α is
+not an integer. Define
+B|γ|+α =
+
+
+
+
+
+B(R)
+if γ = 0,
+C|γ|−1,1(R)
+if γ is a nonzero integer,
+C|γ|+α(R)
+otherwise,
+B|γ|+α(ℓ2) =
+
+
+
+
+
+B(R, ℓ2)
+if γ = 0,
+C|γ|−1,1(R, ℓ2)
+if γ is a nonzero integer,
+C|γ|+α(R, ℓ2)
+otherwise,
+where B(R) is the space of bounded Borel functions on R, C|γ|−1,1(R) represents the space
+of |γ| − 1 times continuous differentiable functions whose derivatives of the (|γ| − 1)th
+order derivative are Lipschitz continuous, and C|γ|+α is the real-valued H¨older spaces. The
+space B(ℓ2) denotes a function space with ℓ2-valued functions instead of real-valued function
+spaces.
+Below we collect the properties of Bessel potential spaces.
+Lemma 2.10. Let γ ∈ R and p > 1.
+(i) The space C∞
+c (Rd) is dense in Hγ
+p .
+(ii) Let γ − d/p = n + ν for some n = 0, 1, · · · and ν ∈ (0, 1].
+Then, for any k ∈
+{0, 1, · · · , n}, we have
+|Dku|C(Rd) + |Dnu|Cν(Rd) ≤ N∥u∥Hγ
+p ,
+(2.2)
+where Cν(Rd) is the Zygmund space.
+(iii) The operator Di : Hγ
+p → Hγ+1
+p
+is bounded. Moreover, for any u ∈ Hγ+1
+p
+,
+��Diu
+��
+Hγ
+p ≤ N∥u∥Hγ+1
+p
+,
+where N = N(γ, p).
+(iv) For γ1, γ2 ∈ R, and u ∈ Hγ1+γ2
+p
+, we have
+∥∆γ1/2u∥Hγ2
+p
+≤ N∥u∥Hγ1+γ2
+p
+,
+where N = N(γ1, γ2)
+(v) For γ ∈ (0, 1), and u ∈ Hγ
+p , we have
+∥(1 − ∆γ)u∥Lp ≤ N
+�
+∥u∥Lp + ∥(−∆)γu∥Lp
+�
+,
+where N = N(γ, p)
+(vi) For any µ, γ ∈ R, the operator (1 − ∆)µ/2 : Hγ
+p → Hγ−µ
+p
+is an isometry.
+
+STFBES DRIVEN BY MULTIPLICATIVE SPACE-TIME WHITE NOISE
+7
+(vii) Let
+ε ∈ [0, 1],
+pi ∈ (1, ∞),
+γi ∈ R,
+i = 0, 1,
+γ = εγ1 + (1 − ε)γ0,
+1/p = ε/p1 + (1 − ε)/p0.
+Then, we have
+∥u∥Hγ
+p ≤ ∥u∥ε
+Hγ1
+p1 ∥u∥1−ε
+Hγ0
+p0 .
+(viii) Let u ∈ Hγ
+p . Then, we have
+∥au∥Hγ
+p ≤ N∥a∥B|γ|+α∥u∥Hγ
+p
+and
+∥bu∥Hγ
+p (ℓ2) ≤ N∥b∥B|γ|+α(ℓ2)∥u∥Hγ
+p ,
+where N = N(γ, p) and B|γ|+α, B|γ|+α(ℓ2) are introduced in Definition 2.9.
+Proof. The above results are well known. For (i), (iii), (vi), and (vii), see Theorems 13.3.7
+(i), 13.8.1, 13.3.7 (ii), and Exercise 13.3.20 of [27], respectively. In the case of (ii) and (iv),
+see [34]. For (v), see Theorems 1.3.6 and 1.3.8 of [12]. For (viii), we refer the reader to [26,
+Lemma 5.2].
+□
+Definition 2.11 (Stochastic Banach spaces). Let τ ≤ T be a bounded stopping time, p ≥ 2,
+and γ ∈ R. Set |(0, τ]] := {(ω, t) : 0 < t ≤ τ(ω)} and define
+Hγ
+p(τ) := Lp
+�
+|(0, τ]], P, dP × dt; Hγ
+p
+�
+,
+Hγ
+p(τ, l2) := Lp
+�
+|(0, τ]], P, dP × dt; Hγ
+p (l2)
+�
+,
+U α,γ
+p
+:= Lp
+�
+Ω, F0, H
+γ− 2
+αp
+p
+�
+.
+We write u ∈ Hγ
+p if u ∈ Hγ
+p(τ) exists for any bounded stopping time τ. Additionally, if
+γ = 0, then we use L instead of H, ∥f∥Lp(τ) := ∥f∥H0p(τ). The norm of each space is defined
+naturally, for example,
+∥f∥Hγ
+p(τ) :=
+�
+E
+ˆ τ
+0
+∥f(t, ·)∥p
+Hγ
+p dt
+�1/p
+.
+Lemma 2.12 exhibits the relation between the stochastic and fractional integrals, which
+is employed when Iα
+t or Dα
+t is applied to the stochastic part of the SPDEs.
+Lemma 2.12. Let T < ∞ be a constant.
+(i) Let α ≥ 0 and h ∈ L2(Ω × [0, T], P; l2). Then, the equality
+Iα
+� ∞
+�
+k=1
+ˆ ·
+0
+hk(s)dwk
+s
+�
+(t) =
+∞
+�
+k=1
+�
+Iα
+ˆ ·
+0
+hk(s)dwk
+s
+�
+(t)
+holds for all t ≤ T almost surely and in L2(Ω × [0, T]), where the series on both sides
+converge in probability.
+(ii) If α ≥ 0 and hn → h in L2(Ω × [0, T], P; l2) as n → ∞, then
+∞
+�
+k=1
+�
+Iα
+ˆ ·
+0
+hk
+ndwk
+s
+�
+(t) →
+∞
+�
+k=1
+�
+Iα
+ˆ ·
+0
+hkdwk
+s
+�
+(t)
+in probability uniformly on [0, T].
+
+8
+BEOMSEOK HAN
+(iii) If α > 1/2 and h ∈ L2(Ω×[0, T], P; l2), then
+�
+Iα �∞
+k=1
+´ ·
+0 hk(s)dwk
+s
+�
+(t) is differentiable
+in t and
+∂
+∂t
+�
+Iα
+∞
+�
+k=1
+ˆ ·
+0
+hk(s)dwk
+s
+�
+(t) =
+1
+Γ(α)
+∞
+�
+k=1
+ˆ t
+0
+(t − s)α−1hk(s)dwk
+s
+(a.e.) on Ω × [0, T].
+Proof. See Lemmas 3.1 and 3.3 of [7].
+□
+Fix a small κ0 > 0. For α ∈ (0, 1) and β < α + 1/2, set
+c0 := (2β − 1)+
+α
++ κ01β=1/2.
+(2.3)
+Next, we introduce the solution spaces (for more detail, see Definitions 2.9 and 2.12 in
+[25]).
+Definition 2.13 (Solution spaces). Let τ ≤ T be a bounded stopping time, α ∈ (0, 1),
+β < α + 1/2, γ ∈ R, and p ≥ 2.
+(i) For u ∈ Hγ
+p(τ), we write u ∈ Hγ
+p(τ) if u0 ∈ U α,γ
+p
+, f ∈ Hγ−2
+p
+(τ), and g ∈ Hγ−2+c0
+p
+(τ, l2)
+such that
+∂α
+t u(t, x) = f(t, x) + ∂β
+t
+ˆ t
+0
+gk(s, x)dwk
+s ,
+0 < t ≤ τ;
+u(0, ·) = u0
+in the sense of distribution. In other words, for any φ ∈ S, the equality
+(u(t, ·), φ) = (u0, φ) + Iα
+t (f, φ) + Iα−β
+t
+∞
+�
+k=1
+ˆ t
+0
+(gk(s, ·), φ)dwk
+s
+(2.4)
+holds for a.e. (ω, t) ∈ Ω × [0, τ]. If α − β ∈ (−1/2, 0), we regard Iα−β
+t
+as
+∂
+∂tIα−β+1
+t
+.
+The norm in Hγ
+p(τ) is defined as follows:
+∥u∥Hγ
+p(τ) := ∥u∥Hγ
+p(τ) + ∥u0∥Uα,γ
+p
++ inf
+f,g
+�
+∥f∥Hγ−2
+p
+(τ) + ∥g∥Hγ−2+c0
+p
+(τ,l2)
+�
+.
+(2.5)
+(ii) We say u ∈ Hγ
+p,loc(τ) if there exists a sequence τn ↑ τ such that u ∈ Hγ
+p(τn) for
+each n. We write u = v in Hγ
+p,loc(τ) if a sequence of bounded stopping times τn ↑ τ
+exists such that u = v in Hγ
+p(τn) for each n. We omit τ if τ = ∞. In other words,
+Hγ
+p,loc = Hγ
+p,loc(∞).
+Remark 2.14. If α − β ≥ 0, the stochastic part of (2.4) is considered
+Iα−β
+t
+∞
+�
+k=1
+ˆ t
+0
+(gk(s, ·), φ)dwk
+s =
+∞
+�
+k=1
+Iα−β
+t
+ˆ t
+0
+(gk(s, ·), φ)dwk
+s .
+Otherwise, if α − β ∈ (−1/2, 0), we regard Iα−β
+t
+as
+∂
+∂tIα−β+1
+t
+. Then, by Lemma 2.12 (iii),
+the stochastic part of (2.4) is
+Iα−β
+t
+� ∞
+�
+k=1
+ˆ t
+0
+(gk(s, ·), φ)dwk
+s
+�
+= ∂
+∂t
+�
+Iα−β+1
+∞
+�
+k=1
+ˆ t
+0
+(gk(s, ·), φ)dwk
+s
+�
+=
+1
+Γ(α − β + 1)
+∞
+�
+k=1
+ˆ t
+0
+(t − s)α−β+1(gk(s, ·), φ)dwk
+s .
+
+STFBES DRIVEN BY MULTIPLICATIVE SPACE-TIME WHITE NOISE
+9
+Below, we provide the properties of the solution space Hγ
+p(τ).
+Theorem 2.15. Let τ ≤ T be a bounded stopping time.
+(i) For ν ∈ R, the map (1 − ∆)ν/2 : Hγ+2
+p
+(τ) → Hγ−ν+2
+p
+(τ) is an isometry.
+(ii) If γ ∈ R, α ∈ (0, 1), β < α + 1/2, and p ≥ 2, then Hγ
+p(τ) is a Banach space with the
+norm ∥ · ∥Hγ
+p(τ).
+Proof. The proof is a repeat of [25, Theorem 2.14] with τ instead of T.
+□
+Next, we suggest the H¨older embedding theorems for u ∈ Hγ
+p(τ). The proof of Theorem
+2.16 is contained in Section 5.
+Theorem 2.16. Let τ ≤ T be the bounded stopping time, γ ∈ R, α ∈ (0, 1), β < α + 1/2,
+and
+p > 2 ∨ 1
+α ∨
+1
+α − β + 1/2.
+(2.6)
+Suppose u ∈ Hγ
+p(τ).
+(i) Assume ν satisfies
+1
+αp < ν < 1 − c0
+2 ,
+(2.7)
+where c0 is the constant introduced in (2.3). Then, u ∈ C([0, τ]; Hγ−2ν
+p
+) almost surely
+and
+E sup
+t≤τ
+∥u(t, ·)∥p
+Hγ−2ν
+p
+≤ N∥u∥p
+Hγ
+p(τ),
+(2.8)
+where N = N(α, β, γ, d, p, T).
+(ii) Assume α, β, µ, and ν satisfy
+1
+αp < µ < (α(ν + c0/2) − β) ∧ 1/2 + 1/2
+α
+and
+1
+αp < ν < 1 − c0
+2 ,
+(2.9)
+where c0 is the constant introduced in (2.3). Then, for δ ∈ (0, T), u ∈ Cαµ−1/p([δ, τ]; Hγ−2ν
+p
+)
+almost surely and
+E∥u∥p
+Cαµ−1/p([δ,τ];Hγ−2ν
+p
+) ≤ N∥u∥p
+Hγ
+p(τ),
+(2.10)
+where N = N(α, β, γ, δ, d, p, T).
+Remark 2.17. Theorem 2.16 is consistent with the previous results ([26, Theorem 7.2]).
+In other words, if we let α, β ↑ 1 in Theorem 2.16, conditions (2.6), (2.7), and (2.9), and
+the results in (2.8) and (2.10) approach those of the case of α = β = 1.
+Remark 2.18. As stated in Theorem 2.16 (ii), the H¨older regularity of solution in time is
+given on [δ, T], where δ ∈ (0, T) (see Remark 5.6). Moreover, if u0 = 0, Theorem 2.16 (ii)
+holds for δ = 0 (see Remark 5.8).
+By combining Lemma 2.10 (ii) and Theorem 2.16, we have the H¨older embedding results
+of solution space H(2−c0−d/2)∧1
+p
+(τ) which is a preparation to obtain the maximum H¨older
+regularity of solutions.
+Corollary 2.19. Let τ ≤ T be a bounded stopping time, α ∈ (0, 1), β < α + 1/2, and
+0 < γ < (2 − c0 − d/2) ∧ 1, where c0 is introduced in (2.3). Suppose p satisfies (2.6) and
+u ∈ Hγ
+p(τ).
+
+10
+BEOMSEOK HAN
+(i) If α, β, γ, ν, d, and p satisfy (2.7) and
+ν < 1
+2
+�
+γ − d
+p
+�
+,
+(2.11)
+then u ∈ C([0, τ]; Cγ−2ν−d/p) almost surely and
+E sup
+t≤τ
+∥u(t, ·)∥p
+Cγ−2ν−d/p(Rd) ≤ N∥u∥p
+Hγ
+p(τ),
+where N = N(α, β, γ, d, p, T).
+(ii) If α, β, γ, µ, ν, d and p satisfy (2.9) and (2.11), then for a small δ > 0, we have
+u ∈ Cαµ−1/p([δ, τ]; Cγ−2ν−d/p)
+almost surely and
+E∥u∥p
+Cαµ−1/p([δ,τ];Cγ−2ν−d/p(Rd)) ≤ N∥u∥p
+Hγ
+p(τ),
+where N = N(α, β, γ, δ, d, p, T).
+Proof. To demonstrate (i), we employ Lemma 2.10 (ii) and Theorem 2.16 (i). Then, we
+have
+E sup
+t≤τ
+∥u(t, ·)∥p
+Cγ−2ν−d/p(Rd) ≤ NE sup
+t≤τ
+∥u(t, ·)∥p
+Hγ−2ν
+p
+(Rd) ≤ N∥u∥p
+Hγ
+p(τ).
+In the case of (ii), Lemma 2.10 (ii) and Theorem 2.16 (ii) imply
+E∥u∥p
+Cαµ−1/p([δ,τ];Cγ−2ν−d/p(Rd)) ≤ E∥u∥p
+Cαµ−1/p([δ,τ];Hγ−2ν
+p
+(Rd)) ≤ N∥u∥p
+Hγ
+p(τ).
+Thus, the corollary is proved.
+□
+3. Main Results
+This section presents the uniqueness, existence, Lp-regularity, and H¨older regularity of
+the solution to the following equation:
+∂α
+t u = Lu + ¯biuuxi + ∂β
+t
+�
+k
+ˆ t
+0
+σ(u)ηkdwk
+s,
+t > 0;
+u(0, ·) = u0,
+(3.1)
+where Lu = aijuxixj + biuxi + cu. The coefficients aij, bi, and c are P × B(Rd)-measurable,
+¯bi is P × B(Rd−1)-measurable, and aij, bi, c, and ¯bi (and their derivatives) are uniformly
+bounded (see Assumption 3.1). Additionally, we assume the coefficient ¯bi is independent
+of xi. Indeed, because ¯bi is independent of xi, we can employ the fundamental theorem
+of calculus to control the nonlinear term ¯biuuxi (see Remark 3.2). Moreover, the diffusion
+coefficient σ(u) is dominated by an Lp function h (see Assumption 3.3) and it is used to
+obtain a uniform Lp bound of the local solutions (see Remark 3.4).
+In Theorem 3.5, we obtain the existence and uniqueness of a solution in Hγ
+p, where
+γ ∈ (0, 2 − c0 − d/2) ∧ 1. The components of equation (3.1) affect the properties of the
+solution u. For example, if α, β, d, and p are given, the regularity γ is determined. Remarks
+3.7, 3.8, and 3.9 provide explanations for these relations.
+Additionally, in Corollary 3.11, we have the maximal H¨older regularity of the solution by
+employing the H¨older embedding theorem for solution spaces and the H¨older regularity of
+the solution is given in (1.3). Observe that (1.3) derives from Corollary 2.19, and we have
+the following. The constant δ can be taken as 0 if the initial data u0 = 0 (see Remarks 2.18
+and 3.6). Furthermore, depending on the range of β, the behavior of the H¨older regularity
+
+STFBES DRIVEN BY MULTIPLICATIVE SPACE-TIME WHITE NOISE
+11
+of the solution in time varies. In detail, when β ≥ 1/2, then the H¨older regularity of the
+solution in space is α/2 times of the H¨older regularity of the solution in time. Moreover,
+if we consider the case α, β ↑ 1, then the H¨older regularity in time and space approaches
+1/4 and 1/2, which are the results of the SPDEs driven by space-time white noise (e.g. [26,
+Remark 8.7] or [16, Corollary 3.1]). In the case of β < 1/2, 1/2− β of the H¨older regularity
+in time is obtained due to the regularity of the stochastic integral (Remark 3.12).
+The following are assumptions on coefficients.
+Assumption 3.1.
+(i) The coefficients aij = aij(t, x), bi = bi(t, x), and c = c(t, x) are
+P × B(Rd)-measurable.
+(ii) The coefficient ¯bi(t, ¯xi) = ¯bi(t, x1, . . . , xi−1, xi+1, . . . , xd) is P × B(Rd−1)-measurable.
+(iii) There exists K > 0 such that
+K−1|ξ|2 ≤ aij(t, x)ξiξj ≤ K|ξ|2
+for all
+(ω, t, x) ∈ Ω × [0, ∞) × Rd,
+ξ ∈ Rd,
+(3.2)
+and
+�
+i,j
+��aij(t, ·)
+��
+C2(Rd) +
+�
+i
+��bi(t, ·)
+��
+C2(Rd) + |c(t, ·)|C2(Rd) +
+�
+i
+��¯bi(t, ·)
+��
+C2(Rd−1) ≤ K
+(3.3)
+for all (ω, t) ∈ Ω × [0, ∞).
+Remark 3.2. To prove the existence of a global solution, we need to acquire a uniform Lp
+bound of the local solutions. Thus, we separate the local solutions into two parts: noise-
+dominating and nonlinear-dominating parts.
+In this remark, we consider the nonlinear-
+dominating parts related to ¯biuuxi.
+If coefficient ¯bi is independent of xi, coefficient ¯bi can be taken out of the integral for
+xi. Then, by the fundamental theorem of calculus to xi, the nonlinear term ¯biuuxi is elimi-
+nated in the Lp estimate of the nonlinear-dominating part of the local solutions. Thus, the
+nonlinear-dominating parts are controlled by the initial data and diffusion coefficient σ(u)
+(for more information, see Lemma 4.8).
+To introduce the assumptions on the diffusion coefficient, we may assume p ≥ 2.
+Assumption 3.3 (p).
+(i) The coefficient σ(t, x, u) is P × B(Rd) × B(R)-measurable.
+(ii) There exists a constant K such that
+|σ(t, x, u) − σ(t, x, v)| ≤ K|u − v|
+for all
+(ω, t, x) ∈ Ω × [0, ∞) × Rd,
+u, v ∈ R.
+(iii) There exists a P × B(Rd)-measurable function h ∈ Lp such that
+|σ(t, x, u)| ≤ |h(t, x)|
+for all
+(ω, t, x) ∈ Ω × [0, ∞) × Rd,
+u ∈ R.
+(3.4)
+Remark 3.4. As mentioned in Remark 3.2, we divide the local solutions into two parts, and
+the nonlinear-dominating parts are controlled by the initial data u0 and diffusion coefficients
+σ(u). Then, to deal with the noise-dominating term and the terms including σ(u), we employ
+the function h(t, x) introduced in Assumption 3.3 (p) (iii). Indeed, the terms related to the
+diffusion coefficient σ(u) are controlled by the initial data and h so that a uniform Lp bound
+of u is obtained (see Lemmas 4.5 and 4.8).
+Next, we introduce the main results.
+
+12
+BEOMSEOK HAN
+Theorem 3.5. Let
+α ∈ (0, 1),
+β < 3
+4α + 1
+2,
+d < 4 − 2c0,
+0 < γ < (2 − c0 − d/2) ∧ 1
+(3.5)
+and
+p = 2k
+for some
+k ∈ N
+and
+p > 2 ∨ 1
+α ∨
+1
+α − β + 1/2 ∨ 2 + αd
+αγ
+∨
+d
+1 − γ ,
+(3.6)
+where c0 are the constants introduced in (2.3). Suppose Assumptions 3.1 and 3.3 (p) hold.
+If u0 ∈ U α,γ
+p
+, then there exists a unique solution u ∈ Hγ
+p,loc satisfying (3.1). Furthermore,
+for ν satisfying (2.7) and (2.11), and for any T ∈ (0, ∞) and bounded stopping time τ ≤ T,
+we have
+u ∈ C([0, τ]; Cγ−2ν−d/p)
+and
+sup
+t≤τ
+∥u(t, ·)∥Cγ−2ν−d/p < ∞
+(3.7)
+almost surely. Additionally, for µ and ν satisfying (2.9) and (2.11), and for any T ∈ (0, ∞),
+bounded stopping time τ ≤ T, and small δ > 0, we have
+u ∈ Cαµ−1/p([δ, τ]; Cγ−2ν−d/p)
+and
+∥u∥
+Cαµ− 1
+p ([δ,τ];Cγ−2ν−d/p) < ∞
+(3.8)
+almost surely. If initial data u0 = 0, (3.8) holds with δ = 0.
+Proof. See Proof of Theorem 3.5 in Section 4.
+□
+Remark 3.6. If the initial data u0 = 0, we can consider the case δ = 0 because we employ
+Theorem 2.16 to obtain (3.8) (see Theorem 2.16 and Remark 2.18).
+Remark 3.7.
+(i) We assume
+α ∈ (0, 1)
+because an inequality acting like the chain rule is employed to deal with the nonlinear-
+dominating part of the local solution (see Lemma 4.6).
+(ii) The conditions
+β < 3α/4 + 1/2
+and
+d < 4 − 2c0
+are expected to obtain the uniqueness and existence of solutions to SPDEs with Caputo
+time fractional derivatives and space-time white noise even for the semilinear case. For
+example, see [23, Section 7]. Additionally, observe that the choice of α and β allows
+d = 1, 2, 3, where c0 is the constant introduced in (2.3).
+Remark 3.8.
+(i) For the existence and uniqueness of local solutions, we impose
+γ ∈ (0, 2 − c0 − d/2).
+(3.9)
+Heuristically, if u is a measurable, continuous, and bounded solution to equation (3.1),
+then for given T < ∞, we can define a bounded stopping time as follows:
+τm := inf
+�
+t ≥ 0 : sup
+x∈Rd |u(t, x)| ≥ m
+�
+∧ T.
+Then, the solution u satisfies the localized version of equation (3.1) on (0, τm). In
+other words,
+∂α
+t u = Lu + 1
+2
+¯bi �
+(|u| ∧ m)2�
+xi + ∂β
+t
+�
+k
+ˆ t
+0
+σ(u)ηkdwk
+s
+(3.10)
+
+STFBES DRIVEN BY MULTIPLICATIVE SPACE-TIME WHITE NOISE
+13
+holds on 0 < t < τm with u(0, ·) = u0. Then, as (3.10) is a semilinear equation, (3.9)
+has to be satisfied by [23, Theorem 7.1] (for more detail, see [23, Section 7] and [25,
+Section 5].
+(ii) The following condition
+γ ∈ (0, 1)
+(3.11)
+is assumed due to the nonlinear term ¯biuuxi lowering the regularity of the solution.
+Even for SBEs (α = β = 1), the condition in (3.11) is required (for more information,
+see [13, 14, 15, 16, 29]).
+Remark 3.9.
+(i) To obtain the local solution, we employ the Lp theory for the semilinear
+equation (see [26, Theorem 5.1]). When we control the nonlinear term ¯biuuxi in the
+Lp estimate, the kernel of (1 − ∆)− γ−1
+2
+has to be controlled. Hence,
+p >
+d
+1 − γ
+is imposed (see Lemma 4.3).
+(ii) We require R-valued continuous solutions to consider the cut-off version of equation
+(3.1). Therefore, we assume
+p > 2 ∧ 1
+α ∧
+1
+α − β + 1/2 ∧ 2 + αd
+αγ
+which is required to apply the H¨older embedding theorem for Hγ
+p (see Theorem 2.16
+and Corollary 2.19).
+(iii) As mentioned in Remark 3.7 (i), we employ an inequality similar to the chain rule.
+To apply (4.13) instead of chain rule for the Caputo fractional derivative, we assume
+p = 2k
+for some k ∈ N.
+To achieve the maximal H¨older regularity, we require the uniqueness of the solution in p.
+Theorem 3.10. Suppose all the conditions of Theorem 3.5 hold. Let u ∈ Hγ
+p,loc be the
+solution of equation (3.1) introduced in Theorem 3.5. If q > p, u0 ∈ U α,γ
+q
+, and Assumption
+3.3 (q) hold, then u ∈ Hγ
+q,loc.
+Proof. See Proof of Theorem 3.10 in Section 4.
+□
+Finally, we obtain the maximal H¨older regularity of the solution by combining Theorems
+3.5 and 3.10. Recall that c0 is introduced in (2.3).
+Corollary 3.11. Suppose α, β, d, and γ satisfy (3.5), and u0 ∈ ∩p>2U α,(2−c0−d/2)∧1
+p
+, and
+h ∈ ∩p>2Lp satisfies (3.4). Then, for T > 0, (1.3) holds almost surely.
+Proof. When α, β, d, and γ are given in (3.5), we choose p as in (3.6). For each p, there
+exists a unique solution up ∈ Hγ
+p,loc to equation (3.1). Due to Theorem 3.10, up ∈ Hγ
+q,loc for
+any q ≥ p so that we write u instead of up and u is independent of p. Thus, by letting p
+large in (3.7) and (3.8), we have (1.3). Thus, the corollary is proved.
+□
+Remark 3.12.
+(i) If 1/2 ≤ β < α + 1/2, the H¨older regularity in space is α/2 times that
+in time. Furthermore, we can recover the H¨older regularity results of SBEs (α = β = 1)
+by considering the case α, β ↑ 1. We cite [29, Proposition 5.1] or [16, Corollary 3.1]
+for reader’s convenience.
+
+14
+BEOMSEOK HAN
+(ii) If β < 1/2, then the H¨older regularity in time obtains additional regularity by as much
+as 1/2 − β. This phenomenon is caused by the stochastic integral of equation (3.1)
+adding the H¨older regularity of noise in time almost 1/2, and ∂β
+t reducing the regularity
+of the noise by β.
+4. Proof of Theorems 3.5 and 3.10
+We assume that all conditions in Theorem 3.5 hold for the remainder of this section.
+To establish the existence of a global solution, we need to obtain the uniqueness and
+existence of local solutions (Lemma 4.3). With these local solutions, we build a candidate
+for a global solution. More precisely, we paste the local solutions and demonstrate that the
+local existence time explodes almost surely (Lemma 4.9). To prove that the local existence
+time explodes almost surely, we demonstrate that a uniform Lp bound of local solutions
+exists. In detail, we separate the local solution into noise- and nonlinear-dominating parts.
+The noise-dominating part is affected by the stochastic part of the equation, and the other
+part is influenced by the nonlinear term biuuxi. When we deal with the noise-dominating
+part of the solution, the dominating function of the diffusion coefficient provides a uniform
+Lp bound for the noise-dominating part of the local solutions (see Assumption 3.3 (p) (iii)
+and Lemma 4.5). The other part is controlled by employing a version of the chain rule and
+Gr¨onwall inequality (see Lemmas 4.6 and 4.8 and Theorem 4.7).
+First, we introduce the uniqueness and existence theorem for semilinear SPDEs.
+Assumption 4.1 (τ).
+(i) The functions f(t, x, u) and gk(t, x, u) are P × B(Rd) × B(R)-
+measurable functions satisfying the following:
+f(t, x, 0) ∈ Hγ
+p(τ)
+and
+g(t, x, 0) = (g1(t, x, 0), g2(t, x, 0), . . . ) ∈ Hγ+1
+p
+(τ, l2).
+(ii) For any ε > 0, there exists a constant Nε such that for any u, v ∈ Hγ
+p(τ),
+∥f(u) − f(v)∥p
+Hγ−2
+p
+(τ) + ∥g(u) − g(v)∥p
+Hγ−2+c0
+p
+(τ,l2) ≤ ε∥u − v∥p
+Hγ
+p(τ) + Nε∥u − v∥p
+Hγ−2
+p
+(τ),
+where c0 is the constant introduced in (2.3).
+Lemma 4.2. Let τ ≤ T be a bounded stopping time. Suppose Assumption 4.1 (τ) hold.
+Then, for initial data u0 ∈ U α,γ
+p
+, the following equation:
+∂α
+t u = Lu + f(u) + ∂β
+t
+ˆ t
+0
+gk(u)dwk
+t ,
+0 < t ≤ τ;
+u(0, ·) = u0
+(4.1)
+has a unique solution u ∈ Hγ
+p(τ). Moreover,
+∥u∥p
+Hγ
+p(τ) ≤ N
+�
+∥u0∥p
+Uα,γ
+p
++ ∥f(0)∥p
+Hγ−2
+p
+(τ) + ∥g(0)∥p
+Hγ−2+c0
+p
+(τ,l2)
+�
+,
+(4.2)
+where N = N(α, β, γ, d, p, K, T) and c0 is the constant introduced in (2.3).
+Proof. Theorem 5.1 of [26] is the motivation of the proof. The case τ ≡ T is obtained by
+[25, Theorem 2.18]; thus, we only consider the case τ ≤ T.
+(Step 1). (Existence) Set
+¯f(t, u) := 1t≤τf(t, u)
+and
+¯g(t, u) := 1t≤τg(t, u).
+Additionally, ¯f(u) and ¯g(u) satisfy Assumption 4.1 (T). Then, by [25, Theorem 2.18], there
+exists a unique solution u ∈ Hγ
+p(T) such that u satisfies equation (4.1) with ¯f and ¯g, instead
+
+STFBES DRIVEN BY MULTIPLICATIVE SPACE-TIME WHITE NOISE
+15
+of f and g, respectively. As τ ≤ T, we have u ∈ Hγ
+p(τ) and u satisfies equation (4.1) and
+estimate (4.2) with f and g.
+(Step 2). (Uniqueness) Let u, v ∈ Hγ
+p(τ) be two solutions of equation (4.1). Then, [25,
+Theorem 2.18] yields there exists a unique solution ¯v ∈ Hγ
+p(T) satisfying
+∂α
+t ¯v = L¯v + ¯f(v) +
+∞
+�
+k=1
+∂β
+t
+ˆ t
+0
+¯gk(v)dwk
+t ,
+0 < t ≤ T ;
+¯v(0, ·) = u0.
+(4.3)
+Notice that in (4.3), ¯f(v) and ¯g(v) are used instead of ¯f(¯v) and ¯g(¯v), respectively. Set
+˜v := v − ¯v. Then, for fixed ω ∈ Ω, we have
+∂α
+t ˜v = L˜v,
+0 < t ≤ τ ;
+˜v(0, ·) = 0.
+By the deterministic version of [25, Theorem 2.18], we have ˜v = 0 in Lp((0, τ] × Rd) almost
+surely. Additionally, it implies v(t, ·) = ¯v(t, ·) in Lp((0, τ] × Rd) almost surely. Thus, in
+equation (4.3), we can replace ¯f(v) and ¯g(v) with ¯f(¯v) and ¯g(¯v). Therefore, ¯v ∈ Hγ
+p(T)
+satisfies equation (4.1) on (0, T] with ¯f, ¯g instead of f, g, respectively. Similarly, by following
+word for word, there exists ¯u ∈ Hγ
+p(T) such that ¯u satisfies equation (4.1) on (0, T] with
+¯f and ¯g instead of f and g. Thus, by the uniqueness result in Hγ
+p(T), we have ¯u = ¯v in
+Hγ
+p(T), which implies u = v in Hγ
+p(τ). Thus, the lemma is proved.
+□
+Next, we provide the uniqueness and existence of a local solution to equation (3.1). As an
+auxiliary function, we choose ρ(·) ∈ C∞
+c (R) such that ρ(z) ≥ 0 on z ∈ (−∞, ∞), ρ(z) = 1
+on |z| ≤ 1, ρ(z) = 0 on |z| ≥ 2, and
+d
+dzρ(z) ≤ 0 on z ≥ 0. We define the following:
+ρm(z) := ρ(z/m).
+(4.4)
+Lemma 4.3. Let τ ≤ T be a bounded stopping time. For m ∈ N, there exists um ∈ Hγ
+p(τ)
+such that
+∂α
+t um = Lum + ¯bi �
+u2
+mρm(um)
+�
+xi + ∂β
+t
+ˆ t
+0
+σ(t, x, um)ηk(x)dwk
+t , 0 < t ≤ τ; um(0, ·) = u0,
+where ρm is the function introduced in (4.4). Furthermore, um ∈ C([0, τ]; C(Rd)) almost
+surely and
+E sup
+t≤τ
+sup
+x∈Rd |um(t, x)|p ≤ N∥um∥p
+Hγ
+p(τ) < ∞
+(4.5)
+almost surely.
+Proof. Due to Lemma 4.2 and Corollary 2.19, it suffices to show that Assumption 4.1 (τ)
+holds. Because σ(t, x, 0) ≤ h(t, x) for all ω, t, x and h ∈ Lp, Assumption 4.1 (i) is satisfied.
+In the case of Assumption 4.1 (ii), notice that for u, v ∈ R, we have
+��u2ρm(u) − v2ρm(v)
+�� ≤ Nm|u − v|.
+Then, for u, v ∈ Hγ
+p(τ), by Remark 2.8 and Lemmas 2.10 (viii) and (iii), we have
+��¯bi �
+(u(t, ·))2ρm(u(t, ·)) − (v(t, ·))2ρm(v(t, ·))
+�
+xi
+��p
+Hγ−2
+p
+≤ N
+��(u(t, ·))2ρm(u(t, ·)) − (v(t, ·))2ρm(v(t, ·))
+��p
+Hγ−1
+p
+≤ N
+ˆ
+Rd
+�ˆ
+Rd |R1−γ(x − y)|
+�
+(u(·))2ρm(·, u(·)) − (v(·))2ρm(·, v(·))
+�
+(t, y)dy
+�p
+dx
+≤ Nm
+�ˆ
+Rd |R1−γ(x)|dx
+�p ˆ
+Rd |u(t, x) − v(t, x)|pdx
+(4.6)
+
+16
+BEOMSEOK HAN
+and
+∥σ(u)η − σ(v)η∥p
+Hγ−2+c0
+p
+(l2)
+≤
+ˆ
+Rd
+��
+k
+�ˆ
+Rd |R−γ+2−c0(x − y)| (σ(·, u(·)) − σ(·, v(·)))(t, y)ηk(y)dy
+�2�p/2
+dx
+≤
+ˆ
+Rd
+�ˆ
+Rd |R−γ+2−c0(x − y)|2 (σ(t, y, u(t, y)) − σ(t, y, v(t, y)))2 dy
+�p/2
+dx
+≤ Kp
+ˆ
+Rd
+�ˆ
+Rd |R−γ+2−c0(y)|2 (u(t, x − y) − v(t, x − y))2dy
+�p/2
+dx
+≤ Kp
+�ˆ
+Rd |R−γ+2−c0(y)|2 dy
+�p/2 ˆ
+Rd |u(t, x) − v(t, x)|pdx
+(4.7)
+on almost every (ω, t) ∈ |(0, τ]]. Due to Remark 2.8, we have
+ˆ
+Rd |R1−γ(y)| dy +
+ˆ
+Rd |R−γ+2−c0(y)|2 dy < ∞.
+By integrating with respect to (ω, t) to (4.6) and (4.7), employing Lemma 2.10 (vii), and
+Young’s inequality, we have
+��¯bi �
+u2ρm(u) − v2ρm(v)
+�
+xi
+��p
+Hγ−2
+p
+(τ) + ∥σ(u)η − σ(v)η∥p
+Hγ−2+c0
+p
+(τ,l2)
+≤ Nm∥u − v∥p
+Lp(τ)
+≤ ε∥u − v∥p
+Hγ
+p(τ) + Nm∥u − v∥p
+Hγ−2
+p
+(τ).
+(4.8)
+The lemma is proved.
+□
+Remark 4.4. We introduce a candidate for a global solution. Let T < ∞. For m ∈ N, let
+um ∈ Hγ
+p(T) be the solution introduced in Lemma 4.3. Then, for R ∈ {1, 2, . . . , m}, define
+a stopping time τ R
+m
+τ R
+m := inf
+�
+t ≥ 0 : sup
+x∈R
+|um(t, x)| ≥ R
+�
+∧ T.
+(4.9)
+Observe that
+τ R
+R ≤ τ m
+m
+(4.10)
+Indeed, if R = m, (4.10) is obvious. If R < m, we have um ∧ m = um ∧ m ∧ R = um ∧ R
+for t ≤ τ R
+m. Therefore, um and uR are solutions to equation
+∂α
+t u = Lu + ¯bi �
+u1+λρR(u)
+�
+xi + σ(u)ηkdwk
+t ,
+0 < t ≤ τ R
+m ;
+u(0, ·) = u0.
+In contrast, uR ∧ R = uR ∧ R ∧ m = uR ∧ m for t ≤ τ R
+R . Thus, um and uR are solutions to
+equation
+∂α
+t u = Lu + ¯b
+�
+u1+λρm(u)
+�
+xi + σ(u)ηkdwk
+t ,
+0 < t ≤ τ R
+R ;
+u(0, ·) = u0.
+Observe that the uniqueness and continuity results in Lemma 4.3 yields that um = uR for
+all t ≤ (τ R
+m ∨ τ R
+R ). Therefore, for t ≤ τ R
+m,
+sup
+s≤t
+sup
+x∈R
+|uR(s, x)| = sup
+s≤t
+sup
+x∈R
+|um(s, x)| ≤ R,
+
+STFBES DRIVEN BY MULTIPLICATIVE SPACE-TIME WHITE NOISE
+17
+and this implies τ R
+m ≤ τ R
+R . Similarly, τ R
+m ≥ τ R
+R ; thus,
+τ R
+R = τ R
+m
+almost surely. Moreover, we have τ R
+m ≤ τ m
+m because m > R. Therefore, we have (4.10).
+Further, we define
+u(t, x) := um(t, x)
+on
+t ≤ τ m
+m
+and set
+τ∞ := lim sup
+m→∞ lim sup
+T→∞
+τ m
+m .
+(4.11)
+It should be remarked that u(t, x) is well-defined on Ω × [0, ∞) × Rd and the nontrivial
+domain of u is Ω × [0, τ∞) × Rd.
+To obtain a uniform Lp bound of the local solution um, we separate um into noise- and
+nonlinear-dominating parts. Lemma 4.5 provides the existence, uniqueness, and estimate
+of noise-dominating parts of um.
+Lemma 4.5. Let T < ∞. Then, there exists v ∈ Hγ
+p(T) such that
+∂α
+t v = Lv + ∂β
+t
+ˆ t
+0
+σ(s, x, u)ηk(x)dwk
+s,
+0 < t ≤ T,
+u(0, ·) = u0
+Furthermore, v ∈ C([0, T]; C(Rd)) almost surely, and
+E sup
+t≤T
+sup
+x∈Rd |v(t, x)|p + E sup
+t≤T
+∥v(t, ·)∥p
+Lp ≤ N∥v∥p
+Hγ
+p(T) ≤ N∥u0∥p
+Uα,γ
+p
++ N∥h∥p
+Lp(τ),
+where N = N(α, β, γ, d, p, K, T).
+Proof. Similar to the proof of Lemma 4.3, it is enough to show that Assumption 4.1 (τ)
+holds. Set η = (η1, η2, . . . ). Then, by Remark 2.8, for t ≤ T
+∥σ(t, ·, u(t, ·))η∥p
+Hγ−2+c0
+p
+(l2)
+=
+ˆ
+Rd
+� ∞
+�
+k=1
+�ˆ
+Rd R−γ+2−c0(x − y)σ(t, y, u(t, y))ηk(y)dy
+�2�p/2
+dx
+=
+�ˆ
+Rd |R−γ+2−c0(x)|2dx
+�p/2 ˆ
+Rd |σ(t, y, u(t, y))|pdy
+≤
+�ˆ
+Rd |R−γ+2−c0(x)|2dx
+�p/2 ˆ
+Rd |h(t, y)|pdy
+≤ N∥h(t, ·)∥p
+Lp.
+(4.12)
+Therefore,
+∥σ(u)η∥p
+Hγ−2+c0
+p
+(T,l2) ≤ E
+ˆ T
+0
+∥σ(t, ·, u(t, ·))η∥p
+Hγ−2+c0
+p
+(l2)dt ≤ N∥h∥p
+Lp.
+Thus, the lemma is proved by Lemma 4.2.
+□
+Next, we control the nonlinear-dominating parts of the local solutions. The following two
+lemmas are crucial in obtaining uniform Lp bounds. Lemma 4.6 functions as a chain rule,
+and Theorem 4.7 is a version of the Gr¨onwall inequality.
+
+18
+BEOMSEOK HAN
+Lemma 4.6. Suppose α ∈ (0, 1) and k ∈ N. For any ψ ∈ C∞
+c ((0, ∞) × Rd), we have
+∂α
+t (ψ(·, x))2k(t) ≤ 2kψ(t, x)|ψ(t, x)|2k−2∂α
+t ψ(t, x),
+(4.13)
+for all (t, x) ∈ (0, ∞) × Rd.
+Proof. We employ the mathematical induction. The results and proof are motivated by
+(4.2) of [8].
+(Step 1). First, we consider the case k = 1. Although the proof is in the proof of [8,
+Proposition 4.1], we include the proof for the completeness of this paper.
+Let ψ ∈ C∞
+c ((0, ∞) × Rd) and t ∈ (0, ∞) and x ∈ Rd. For s ∈ (0, t], set
+F1(s) := 1
+2|ψ(s, x)|2,
+F2(s) := ψ(s, x)ψ(t, x),
+and
+F(s) := 1
+2
+�
+|ψ(s, x)|2 − |ψ(t, x)|2�
+− (ψ(s, x) − ψ(t, x))ψ(t, x).
+Further,
+F(s) = 1
+2|ψ(s, x) − ψ(t, x)|2 ≥ 0
+on s ≤ t, and the equality holds for s = t. Notice that the integration by parts implies that
+ˆ t
+0
+(t − s)−α(F ′
+1(s) − F ′
+2(s))ds =
+ˆ t
+0
+(t − s)−αF ′(s)ds ≤ 0.
+Then, by the definition of ∂α
+t (Definition 2.5), we have (4.13) with k = 1.
+(Step 2).
+Let n ∈ N and assume that the results hold for k = 1, 2, . . . , n − 1.
+Set
+˜ψ(t, x) := (ψ(t, x))2. Since ˜ψ(t, x) ∈ C∞
+c ((0, ∞) × Rd), we have
+∂α
+t (ψ(·, x))2n(t) = ∂α
+t ( ˜ψ(·, x))2n−1(t)
+≤ 2n−1 ˜ψ(t, x)
+��� ˜ψ(t, x)
+���
+2n−1−2
+∂α
+t ˜ψ(t, x)
+= 2n−1 |ψ(t, x)|2n−2 ∂α
+t (ψ(t, x))2
+≤ 2nψ(t, x) |ψ(t, x)|2n−2 ∂α
+t ψ(t, x).
+The lemma is proved.
+□
+Theorem 4.7 (Theorem 8 of [4]). Let ψ(t) be a nonnegative integrable function on [0, T].
+For a constant N1, if the function ψ satisfies
+ψ(t) ≤ ψ0 + N1Iα
+t ψ
+on t ∈ [0, T], then
+ψ(t) ≤
+�
+1 +
+∞
+�
+k=0
+N k
+1
+Γ(kα)
+(Γ(α)tα)k
+kα
+�
+ψ0
+on t ∈ [0, T].
+We consider following lemma to control the remainder of the local solution um.
+
+STFBES DRIVEN BY MULTIPLICATIVE SPACE-TIME WHITE NOISE
+19
+Lemma 4.8. Let um ∈ Hγ
+p(T) and v ∈ Hγ
+p(T) be functions introduced in Lemmas 4.3 and
+4.5, and τ m
+m be the stopping time introduced in (4.9). Then,
+∥um(t, ·) − v(t, ·)∥p
+Lp
+≤ N sup
+t≤T
+sup
+x∈Rd |v(t, x)|p sup
+t≤T
+∥v(s, ·)∥p
+Lp
+�
+1 +
+∞
+�
+k=0
+�
+1 + sups≤t,x∈Rd |v(s, x)|2�k
+Γ(kα)
+(Γ(α)T α)k
+kα
+�
+for all t ≤ τ m
+m almost surely, where N = N(p, K).
+Proof. Set
+wm := um − v
+and
+fm := Lwm + ¯bi((um)2ρm(um))xi.
+Then, we have fm ∈ Hγ−2
+p
+(T) since wm, um ∈ Hγ
+p(T) and estimates similar to (4.8). Ad-
+ditionally, ∂α
+t wm = fm.
+Let (ω, t) ∈
+|(0, τ m
+m ]].
+Due to [23, Remark 2.9], there exists
+wn
+m ∈ C∞
+c ((0, ∞) × Rd) such that wn
+m → wm in Lp((0, t); Hγ
+p ), and ∂α
+t wn
+m is a Cauchy
+in Lp((0, t); Hγ−2
+p
+). Define
+f n
+m := ∂α
+t wn
+m.
+Moreover, fm is the limit of f n
+m as n → ∞ in Lp((0, t); Hγ−2
+p
+) (see [23, Remark 2.9]).
+Choose a nonnegative function ζ ∈ C∞
+c (Rd) with a unit integral and set ζε(x) := ε−dζ(x/ε)
+for ε > 0. For h ∈ L1,loc(Rd), set h(ε)(x) :=
+´
+Rd h(y)ζε(x − y)dy.
+Next, let ε > 0 and x ∈ Rd. Since wn(ε)
+m
+∈ C∞
+c ((0, ∞) × Rd) and p = 2k, Lemma 4.6
+yields
+1
+p∂α
+t
+�
+wn(ε)
+m
+(·, x)
+�p
+(t) ≤ f n(ε)
+m
+(t, x)wn(ε)
+m
+(t, x)
+���wn(ε)
+m
+(t, x)
+���
+p−2
+(4.14)
+on t ∈ (0, ∞). Additionally, as wn
+m(0, x) = 0, we have
+wn(ε)
+m
+(0, x) = 0.
+(4.15)
+Thus, if we take stochastic integral Iα
+t on both sides of (4.14), we have
+1
+p
+���wn(ε)
+m
+(t, x)
+���
+p
+≤ Iα
+t
+�
+f n(ε)
+m
+(·, x)wn(ε)
+m
+(·, x)
+���wn(ε)
+m
+(·, x)
+���
+p−2�
+(4.16)
+due to
+�
+wn(ε)
+m
+�p
+∈ C∞
+c ((0, ∞)×Rd), (4.15), and Remark 2.6. Observe that (2.1) with q = ∞
+and the H¨older inequality imply that
+����Iα
+·
+�
+f n(ε)
+m
+(·, x)wn(ε)
+m
+(·, x)
+���wn(ε)
+m
+(·, x)
+���
+p−2
+− f (ε)
+m (·, x)w(ε)
+m (·, x)
+���w(ε)
+m (·, x)
+���
+p−2�����
+L1((0,t))
+≤
+ˆ t
+0
+����f n(ε)
+m
+(s, x)wn(ε)
+m
+(s, x)
+���wn(ε)
+m
+(s, x)
+���
+p−2
+− f (ε)
+m (s, x)w(ε)
+m (s, x)
+���w(ε)
+m (s, x)
+���
+p−2���� ds
+≤ N
+ˆ t
+0
+���f n(ε)
+m
+(s, x) − f (ε)
+m (s, x)
+���
+���wn(ε)
+m
+(s, x)
+���
+p−1
+ds
++ N
+ˆ t
+0
+����f (ε)
+m (s, x)
+�
+wn(ε)
+m
+(s, x)
+���wn(ε)
+m
+(s, x)
+���
+p−2
+− w(ε)
+m (s, x)
+���w(ε)
+m (s, x)
+���
+p−2����� ds
+≤ N
+�
+An
+���wn(ε)
+m
+(·, x)
+���
+2
+Lp(0,t) + BnCn
+���f (ε)
+m (·, x)
+���
+Lp(0,t)
+� ���wn(ε)
+m
+(·, x)
+���
+p−3
+Lp(0,t) ,
+(4.17)
+
+20
+BEOMSEOK HAN
+where
+An =
+���f n(ε)
+m
+(·, x) − f (ε)
+m (·, x)
+���
+Lp(0,t)
+Bn =
+���wn(ε)
+m
+(·, x) − w(ε)
+m (·, x)
+���
+Lp(0,t) ,
+and
+Cn =
+���wn(ε)
+m
+(·, x)
+���
+Lp(0,t) +
+���w(ε)
+m (·, x)
+���
+Lp(0,t) .
+Moreover,
+An, Bn → 0
+and
+Cn → 2
+���w(ε)
+m (·, x)
+���
+Lp(0,t)
+as
+n → ∞
+(4.18)
+since wn
+m → wm and f n
+m → fm in Lp((0, t); Hγ
+p ). Then, by applying (4.18) to (4.17), we
+have
+����Iα
+·
+�
+f n(ε)
+m
+(·, x)wn(ε)
+m
+(·, x)
+���wn(ε)
+m
+(·, x)
+���
+p−2
+− f (ε)
+m (·, x)w(ε)
+m (·, x)
+���w(ε)
+m (·, x)
+���
+p−2�����
+L1((0,t))
+→ 0
+as n → ∞. Therefore, there exists a sequence nl such that wnl(ε)
+m
+(·, x) → w(ε)
+m (·, x) and
+Iα
+·
+�
+f nl(ε)
+m
+wnl(ε)
+m
+���wnl(ε)
+m
+���
+p−2�
+→ Iα
+·
+�
+f (ε)
+m w(ε)
+m
+���w(ε)
+m
+���
+p−2�
+almost everywhere on [0, t]. Further-
+more, the convergence holds everywhere on [0, t] due to the continuity in t.
+Then, by
+considering sequence nl instead of n and letting l → ∞ for (4.16), we have
+1
+p
+���w(ε)
+m (t, x)
+���
+p
+≤ Iα
+t
+�
+f (ε)
+m (·, x)w(ε)
+m (·, x)
+���w(ε)
+m (·, x)
+���
+p−2�
+.
+Since t ≤ τ m
+m , ρm(um) = 1. By integrating with respect to x, we have
+Γ(α)
+p
+ˆ
+Rd
+���w(ε)
+m (t, x)
+���
+p
+dx
+≤
+ˆ t
+0
+(t − s)α−1
+ˆ
+Rd(Lwm(s, ·))(ε)(x)w(ε)
+m (s, x)
+���w(ε)
+m (s, x)
+���
+p−2
+dxds
++
+ˆ t
+0
+(t − s)α−1
+ˆ
+Rd
+�
+¯bi(s, ¯xi)
+�
+|wm(s, ·) + v(s, ·)|2�(ε)
+xi (x)
+�
+w(ε)
+m (s, x)
+���w(ε)
+m (s, x)
+���
+p−2
+dxds.
+(4.19)
+Furthermore, by integration by parts, we obtain
+ˆ
+Rd
+�
+(Lwm(s, ·))(ε)(x) + ¯bi �
+|wm(s, ·) + v(s, ·)|2�(ε)
+xi (x)
+�
+w(ε)
+m (s, x)
+���w(ε)
+m (s, x)
+���
+p−2
+dx
+≤ −(p − 1)
+ˆ
+Rd
+�
+aijwm
+�(ε)
+xj (s, x)
+���w(ε)
+m (s, x)
+���
+p−2
+w(ε)
+mxi(s, x)dx
++ (p − 1)
+ˆ
+Rd
+��
+2aij
+xj − bi�
+wm
+�(ε)
+(s, x)
+���w(ε)
+m (s, x)
+���
+p−2
+w(ε)
+mxi(s, x)dx
++
+ˆ
+Rd
+��
+aij
+xixj − bi
+xi + c
+�
+wm
+�(ε)
+(s, x)w(ε)
+m (s, x)
+���w(ε)
+m (s, x)
+���
+p−2
+dx
+− (p − 1)
+ˆ
+Rd
+¯bi(s, ¯xi)
+�
+(wm(s, ·) + v(s, ·))2�(ε) (x)
+���w(ε)
+m (s, x)
+���
+p−2
+w(ε)
+mxi(s, x)dx.
+(4.20)
+
+STFBES DRIVEN BY MULTIPLICATIVE SPACE-TIME WHITE NOISE
+21
+Additionally, observe that
+���
+�
+aijwm
+�(ε)
+xj (s, x) − aij(s, x)w(ε)
+mxj(s, x)
+���
+= ε−1
+����
+ˆ
+Rd
+�
+aij(s, x − εy) − aij(s, x)
+�
+wm(s, x − εy)ζyj(y)dy
+����
+≤ N(K)
+ˆ
+Rd |wm(s, x − εy)||y||ζy(y)|dy,
+(4.21)
+and by (3.2),
+−
+ˆ
+Rd aij(s, x)w(ε)
+mxi(s, x)w(ε)
+mxj(s, x)
+���w(ε)
+m (s, x)
+���
+p−2
+dx
+≤ −K−1
+ˆ
+Rd
+���w(ε)
+m (s, x)
+���
+p−2 ���w(ε)
+mx(s, x)
+���
+2
+dx.
+(4.22)
+Thus, by combining (4.21) and (4.22)
+−
+ˆ
+Rd
+�
+aijwm
+�(ε)
+xj (s, x)
+���w(ε)
+m (s, x)
+���
+p−2
+w(ε)
+mxi(s, x)dx
+= −
+ˆ
+Rd
+���w(ε)
+m (s, x)
+���
+p−2
+w(ε)
+mxi(s, x)
+�
+(awm)(ε)
+xj (s, x) − a(s, x)w(ε)
+mxj(s, x)
+�
+dx
+−
+ˆ
+Rd aij(s, x)w(ε)
+mxi(s, x)w(ε)
+mxj(s, x)
+���w(ε)
+m (s, x)
+���
+p−2
+dx
+≤ N
+ˆ
+Rd
+���w(ε)
+m (s, x)
+���
+p−2 ���w(ε)
+mxi(s, x)
+���
+ˆ
+Rd |wm(s, x − εy)||y||ζy(y)|dydx
+− K−1
+ˆ
+Rd
+���w(ε)
+m (s, x)
+���
+p−2 ���w(ε)
+mx(s, x)
+���
+2
+dx
+≤ N
+ˆ
+Rd
+���w(ε)
+m (s, x)
+���
+p−2 �ˆ
+Rd |wm(s, x − εy)||y||ζy(y)|dy
+�2
+dx
+− 1
+2K−1
+ˆ
+Rd
+���w(ε)
+m (s, x)
+���
+p−2 ���w(ε)
+mx(s, x)
+���
+2
+dx,
+(4.23)
+where N = N(K). Moreover,
+����
+��
+2aij
+xj − bi�
+wm
+�(ε)
+(s, x)
+���� =
+����
+ˆ
+Rd
+�
+2aij
+yj(s, y) − bi(s, y)
+�
+wm(s, y)ζε(x − y)dy
+����
+≤ K
+ˆ
+Rd |wm(s, y)|ζε(x − y)dy
+= K(|wm(s, ·)|)(ε)(x)
+(4.24)
+and
+����
+��
+aij
+xixj − bi
+xi + c
+�
+wm
+�(ε)
+(s, x)
+���� ≤ K(|wm(s, ·)|)(ε)(x).
+(4.25)
+
+22
+BEOMSEOK HAN
+Thus, by applying H¨older’s inequality, (4.23), (4.24), and (4.25) to (4.20), we have
+ˆ
+Rd
+�
+(Lwm(s, ·))(ε)(x) + ¯bi �
+|wm(s, ·) + v(s, ·)|2�(ε)
+xi (x)
+�
+w(ε)
+m (s, x)
+���w(ε)
+m (s, x)
+���
+p−2
+dx
+≤ N
+ˆ
+R
+���w(ε)
+m (s, x)
+���
+p−2 �ˆ
+R
+|wm(s, x − εy)||y||ζy(y)|dy
+�2
+dx
+− p − 1
+4K
+�
+i
+ˆ
+R
+���w(ε)
+m (s, x)
+���
+p−2 ���w(ε)
+mxi(s, x)
+���
+2
+dx
++ N
+�
+i
+ˆ
+Rd
+�
+(|wm(s, ·)|)(ε)(x)
+�2 ���w(ε)
+m (s, x)
+���
+p−2
+dx
++ N
+ˆ
+Rd(|wm(s, ·)|)(ε)(x)
+���w(ε)
+m (s, x)
+���
+p−1
+dx.
+− (p − 1)
+�
+i
+ˆ
+Rd
+¯bi(s, ¯xi)
+�
+(wm(s, ·) + v(s, ·))2�(ε) (x)
+���w(ε)
+m (s, x)
+���
+p−2
+w(ε)
+mxi(s, x)dx,
+(4.26)
+where N = N(K). Furthermore, note that
+ˆ
+Rd
+���w(ε)
+m (s, x)
+���
+p
+w(ε)
+mxi(s, x)dx = 0
+for
+s ≤ t.
+(4.27)
+Indeed, take a nonnegative smooth function φ ∈ C∞
+c (Rd) such that φ(x) = 1 on |x| < 1,
+φ(x) = 0 on |x| > 2, and supx∈Rd |φ′(x)| ≤ 2. Then, integration by parts yields
+ˆ
+Rd
+���w(ε)
+m (s, x)
+���
+p
+w(ε)
+mxi(s, x)φ(x/n)dx
+= −p
+ˆ
+Rd
+���w(ε)
+m (s, x)
+���
+p
+w(ε)
+mxi(s, x)φ(x/n)dx − 1
+n
+ˆ
+Rd
+���w(ε)
+m (s, x)
+���
+p
+w(ε)
+m (s, x)φ′(x/n)dx.
+Thus, we have
+lim sup
+n→∞
+����
+ˆ
+Rd
+���w(ε)
+m (s, x)
+���
+p
+w(ε)
+mxi(s, x)φ(x/n)dx
+���� ≤ lim sup
+n→∞
+2
+n(p + 1)
+ˆ
+Rd
+���w(ε)
+m (s, x)
+���
+p+1
+dx = 0
+(4.28)
+and (4.28) yields (4.27). Then, from the last term of (4.26), by applying (4.27) and the
+H¨older’s inequality, we have
+�����
+�
+i
+ˆ
+Rd−1
+¯bi(s, ¯xi)
+ˆ
+R
+�
+|wm(t, ·) + v(t, ·)|2�(ε)
+(x)
+���w(ε)
+m (t, x)
+���
+p−2
+w(ε)
+mxi(t, x)dxid¯xi
+�����
+≤ N
+�
+i
+ˆ
+Rd
+����
+�
+|wm(t, ·) + v(t, ·)|2�(ε)
+(x) −
+���w(ε)
+m (t, x)
+���
+2����
+���w(ε)
+m (t, x)
+���
+p−2 ���w(ε)
+mxi(t, x)
+��� dx
+≤ N
+ˆ
+Rd
+��
+|wm(t, ·) + v(t, ·)|2�(ε)
+(x) −
+���w(ε)
+m (t, x)
+���
+2�2 ���w(ε)
+m (t, x)
+���
+p−2
+dx
++
+1
+8KN
+�
+i
+ˆ
+Rd
+���w(ε)
+mxi(t, x)
+���
+2 ���w(ε)
+m (t, x)
+���
+p−2
+dx,
+(4.29)
+
+STFBES DRIVEN BY MULTIPLICATIVE SPACE-TIME WHITE NOISE
+23
+where N = N(K). Then, by applying (4.26) and (4.29) to (4.19), we have
+ˆ
+Rd
+���w(ε)
+m (t, x)
+���
+p
+dx
+≤ NIα
+t
+ˆ
+R
+���w(ε)
+m (s, x)
+���
+p−2 �ˆ
+R
+|wm(s, x − εy)||y||ζy(y)|dy
+�2
+dx
++ NIα
+t
+ˆ
+Rd
+���(|wm(s, ·)|)(ε)(x)
+���
+2 ���w(ε)
+m (s, x)
+���
+p−2
++ (|wm(s, ·)|)(ε)(x)
+���w(ε)
+m (s, x)
+���
+p−1
+dx
++ NIα
+t
+ˆ
+Rd
+��
+|wm(t, ·) + v(t, ·)|2�(ε)
+(x) −
+���w(ε)
+m (t, x)
+���
+2�2 ���w(ε)
+m (t, x)
+���
+p−2
+dx,
+where N = N(p, K). By letting ε ↓ 0, we have
+∥wm(t, ·)∥p
+Lp
+≤ NIα
+t
+ˆ
+R
+|wm(·, x)|p dx + NIα
+t
+ˆ
+Rd
+�
+|wm(·, x) + v(·, x)|2 − |wm(·, x)|2�2
+|wm(·, x)|p−2 dx
+≤ NIα
+t
+ˆ
+R
+|wm(·, x)|p dx + NIα
+t
+ˆ
+Rd |v(·, x)|2 |wm(·, x)|p + |v(·, x)|4 |wm(·, x)|p−2 dx
+≤ N
+�
+1 +
+sup
+s≤t,x∈Rd |v(s, x)|2
+�
+Iα
+t ∥wm(·, ·)∥p
+Lp + N sup
+s≤t
+∥v(s, ·)∥2p
+L2p.
+for all t ≤ τ m
+m . Then, by Theorem 4.7, we obtain
+∥wm(t, ·)∥p
+Lp ≤ N sup
+s≤t
+∥v(s, ·)∥2p
+L2p
+�
+1 +
+∞
+�
+k=0
+�
+1 + sups≤t,x∈Rd |v(s, x)|2�k
+Γ(kα)
+(Γ(α)T α)k
+kα
+�
+for all t ≤ τ m
+m . The lemma is proved.
+□
+Finally, we demonstrate that the global solution candidate does not explode in a finite
+time.
+Lemma 4.9. For any T < ∞, we have
+lim
+R→∞ P
+��
+ω ∈ Ω :
+sup
+t≤T,x∈Rd |u(t, x)| > R
+��
+= 0.
+Proof. Let v be the function introduced in Lemma 4.5. Define
+τ 1(S) := inf
+�
+t ≥ 0 : ∥v(t, ·)∥Lp ≥ S
+�
+∧ T,
+τ 2(S) := inf
+�
+t ≥ 0 : sup
+x∈Rd |v(t, x)| ≥ S
+�
+∧ T.
+and
+τ 0
+m(S) := τ m
+m ∧ τ 1(S) ∧ τ 2(S),
+
+24
+BEOMSEOK HAN
+where τ m
+m is the stopping time introduced in (4.9). Set r :=
+p
+p−1. Then, by Lemmas 4.2 and
+2.10, (viii), H¨older inequality, and Minkowski inequality, we have
+∥um∥p
+Hγ
+p(τ 0m(S)) − N∥u0∥p
+Uα,γ
+p
+≤ N
+�����
+�
+i
+¯bi(u2
+mρm(um))xi
+�����
+p
+Hγ−2
+p
+(τ 0m(S))
++ N∥σ(um)η∥p
+Hγ−2+c0
+p
+(τ 0m(S),l2)
+≤ N
+��u2
+m
+��p
+Hγ−1
+p
+(τ 0m(S)) + N∥σ(um)η∥p
+Hγ−2+c0
+p
+(τ 0m(S),l2)
+≤ NE
+ˆ τ 0
+m(S)
+0
+ˆ
+Rd
+����
+ˆ
+Rd R1−γ(y)|um(s, x − y)|2dy
+����
+p
+dxds
++ NE
+ˆ τ 0
+m(S)
+0
+ˆ
+Rd
+����
+ˆ
+Rd |R−γ+2−c0(y)|2 |um(s, x − y)|2dy
+����
+p/2
+dxds
+≤ NE
+ˆ τ 0
+m(S)
+0
+ˆ
+Rd
+����
+ˆ
+Rd |R1−γ(y)|r |um(s, x − y)|rdy
+����
+p/r
+dx
+ˆ
+Rd |um(s, x)|pdxds
++ NE
+ˆ τ 0
+m(S)
+0
+�ˆ
+Rd |R−γ+2−c0(x)|2 dx
+�p/2 ˆ
+Rd |um(s, x)|pdxds
+≤ N0E
+ˆ τ 0
+m(S)
+0
+�
+1 +
+ˆ
+Rd |um(s, x)|pdx
+� ˆ
+Rd |um(s, x)|pdxds,
+(4.30)
+where N0 = N(α, β, γ, d, p, K, T)
+��´
+Rd |R1−γ(x)|r dx
+�p/r +
+�´
+Rd |R−γ+2−c0(x)|2 dx
+�p/2�
+. Note
+that N0 < ∞ due to r <
+d
+d+γ−1 and Remark 2.8. Then, by Lemma 4.8 and the definitions
+of τ1(S) and τ2(S),
+ˆ
+Rd |um(t, x)|pdx
+≤ N
+ˆ
+Rd |um(t, x) − v(t, x)|p + |v(t, x)|pdx
+≤ N sup
+s≤t
+sup
+x∈Rd |v(t, x)| sup
+s≤t
+∥v(s, ·)∥p
+p
+�
+1 +
+∞
+�
+k=0
+�
+1 + sups≤t,x∈Rd |v(t, x)|2�k
+Γ(kα)
+(Γ(α)T α)k
+kα
+�
++
+ˆ
+Rd |v(t, x)|pdx
+< N(p, S, K).
+(4.31)
+Therefore, by combining (4.30) and (4.31), we have
+∥um∥p
+Hγ
+p(τ 0m(S)) ≤ N + N∥u0∥p
+Uα,γ
+p
+,
+(4.32)
+
+STFBES DRIVEN BY MULTIPLICATIVE SPACE-TIME WHITE NOISE
+25
+where N = N(α, β, γ, d, p, S, K, T). It should be noted that the right-hand side of (4.32) is
+independent of m. Therefore, by Chebyshev’s inequality and Lemma 4.3, we have
+P
+�
+sup
+t≤τ 0m(S)
+sup
+x∈Rd |u(t, x)| > R
+�
+≤ 1
+RpE
+sup
+t≤τ 0m,x∈Rd |u(t, x)|p
+≤ 1
+RpE
+sup
+t≤τ 0m,x∈Rd |um(t, x)|p
+≤ 1
+Rp∥um∥p
+Hγ
+p(τ 0m)
+≤ N
+Rp,
+where N = N(u0, α, β, γ, d, p, S, K, T). In contrast, by Lemma 4.5,
+P
+�
+τ 1(S) < T
+�
++ P
+�
+τ 2(S) < T
+�
+≤ P
+�
+sup
+t≤T
+∥v(t, ·)∥Lp > S
+�
++ P
+�
+sup
+t≤T
+sup
+x∈Rd |v(t, x)| > S
+�
+≤ 1
+Sp E sup
+t≤T
+∥v(t, ·)∥p
+Lp + 1
+Sp E
+sup
+t≤T,x∈Rd |v(t, x)|p
+≤ 1
+Sp N(u0, h, α, β, γ, d, p, K, T).
+Thus,
+P
+�
+sup
+t≤T,x∈Rd |u(t, x)| > R
+�
+≤ lim inf
+m→∞ P
+�
+sup
+t≤τ 0m(S),x∈Rd |u(t, x)| > R
+�
++ P
+�
+τ 1(S) < T
+�
++ P
+�
+τ 2(S) < T
+�
+≤ N1
+Rp + N2
+Sp ,
+where N1 = N1(u0, α, β, γ, d, p, S, K, T) and N2 = N2(u0, h, α, β, γ, d, p, K, T). The lemma
+is proved by letting R → ∞ and S → ∞ in order.
+□
+Proof of Theorem 3.5. Step 1. (Uniqueness).
+Suppose u, ¯u ∈ Hγ
+p,loc are nonnegative
+solutions of equation (3.1). By Definition 2.13, there are bounded stopping times τn (n =
+1, 2, · · · ) such that
+τn ↑ ∞
+and
+u, ¯u ∈ Hγ
+p(τn).
+Fix n ∈ N. Note that u, ¯u ∈ C([0, τn]; C(Rd)) almost surely and
+E sup
+t≤τn
+sup
+x∈Rd |u(t, x)|p + E sup
+t≤τn
+sup
+x∈Rd |¯u(t, x)|p < ∞.
+(4.33)
+
+26
+BEOMSEOK HAN
+Then, for m ∈ N, define
+τ 1
+m,n := inf
+�
+t ≥ 0 : sup
+x∈Rd |u(t, x)| > m
+�
+∧ τn,
+τ 2
+m,n := inf
+�
+t ≥ 0 : sup
+x∈Rd |¯u(t, x)| > m
+�
+∧ τn,
+and
+τm,n := τ 1
+m,n ∧ τ 2
+m,n.
+(4.34)
+Due to (4.33), τ 1
+m,n and τ 2
+m,n are well-defined stopping times; thus, τm,n is a stopping time.
+Observe that u, ¯u ∈ Hγ
+p(τm,n) and τm,n ↑ τn as m → ∞ almost surely. Fix m ∈ N. Notice
+that u, ¯u ∈ Hγ
+p(τm,n) are solutions to equation
+∂α
+t v = Lv + ¯bi �
+v2ρm(v)
+�
+xi + ∂β
+t
+ˆ t
+0
+σ(v)dWt,
+0 < t ≤ τm,n;
+v(0, ·) = u0,
+where Lv = aijvxixj + bivxi + cv. By the uniqueness result in Lemma 4.3, we conclude that
+u = ¯u in Hγ
+p(τm,n) for each m ∈ N. The monotone convergence theorem yields u = ¯u in
+Hγ
+p(τn), which implies u = ¯u in Hγ
+p,loc.
+Step 2 (Existence.). Let T < ∞. For m ∈ N, define τ m
+m and u as in Remark 4.4. Observe
+that
+P (τ m
+m < T) ≤ P
+�
+sup
+t≤T,x∈Rd |u(t, x)| ≥ m
+�
+.
+Indeed, if τ m
+m < T, then supt≤τ m
+m ,x∈Rd |u(t, x)| = supt≤τ m
+m ,x∈Rd |um(t, x)| = m almost surely.
+Then, by Lemma 4.9, we have
+lim sup
+m→∞ P (τ m
+m < T) ≤ lim sup
+m→∞ P
+�
+sup
+t≤T,x∈Rd |u(t, x)| ≥ m
+�
+= 0
+Since T < ∞ is arbitrary, τ m
+m → ∞ in probability. In addition, we conclude that τ m
+m ↑ ∞
+almost surely, because τ m
+m is increasing in m.
+Last, set τm := τ m
+m ∧ m. Note that (see Remark 4.4)
+u(t, x) = um(t, x)
+for
+t ∈ [0, τm].
+Observe that supx∈Rd |um(t, x)| ≤ m for t ∈ [0, τm]; thus, um satisfies (3.1) almost every-
+where t ∈ [0, τm] almost surely. Because u = um for t ∈ [0, τm] and um ∈ Hγ
+p(τm), it follows
+that u ∈ Hγ
+p(τm) and u satisfies (3.1) for all t ≤ τm almost surely. We have u ∈ Hγ
+p,loc
+because τm ↑ ∞ as m → ∞ almost surely. The theorem is proved.
+□
+Proof of Theorem 3.10. The proof of Theorem 3.10 is motivated by [26, Corollarly 5.11].
+Since q > p, by Theorem 3.5, there exists a unique solution ¯u ∈ Hγ
+q,loc satisfying equation
+(3.1). By Definition 2.13, there exists τn such that τn → ∞ almost surely as n → ∞,
+u ∈ Hγ
+p(τn) and ¯u ∈ Hγ
+q(τn). Fix n ∈ N. Because 2+αd
+αγ
+< p < q, we can define τm,n (m ∈ N)
+as in (4.34). Notice that for any p0 > p, we have
+u ∈ Lp0(τm,n)
+
+STFBES DRIVEN BY MULTIPLICATIVE SPACE-TIME WHITE NOISE
+27
+since
+E
+ˆ τm,n
+0
+ˆ
+R
+|u(t, x)|p0dxdt ≤ mp0−pE
+ˆ τm,n
+0
+ˆ
+R
+|u(t, x)|pdxdt < ∞.
+Observe that ¯biuuxi ∈ Hγ−2
+q
+(τm,n) ⊂ H−2
+q (τm,n). Indeed, similar to (4.6),
+E
+ˆ τm,n
+0
+����
+1
+2
+¯bi(s, ·)
+�
+(u(s, ·))2�
+xi
+����
+q
+Hγ−2
+q
+ds ≤ NE
+ˆ τm,n
+0
+ˆ
+R
+|u(s, x)|2qdxds < ∞.
+Additionally, we have
+auxx ∈ H−2
+q (τm,n),
+bux ∈ H−1
+q (τm,n),
+and
+cu ∈ Lq(τm,n).
+Therefore, because Lq(τm,n) ⊂ H−1
+q (τm,n) ⊂ H−2
+q (τm,n),
+aijuxixj + biuxi + cu + ¯biuuxi ∈ H−2
+q (τm,n).
+(4.35)
+Similar to (4.12), we have
+∥σ(u)η∥q
+Hγ−2+c0
+q
+(τm,n,l2) ≤ N
+ˆ τm,n
+0
+∥h(t, ·)∥q
+Lq dt < ∞.
+(4.36)
+Thus, we have
+σ(u)η ∈ Hγ−2+c0
+q
+(τm,n, l2) ⊂ H−2+c0
+q
+(τm,n, l2).
+(4.37)
+Due to (4.35), (4.37), and Lemma 4.2, u is in Lq(τm,n) and u satisfies (3.1) for almost
+everywhere t ≤ τm,n almost surely. On the other hand, since ¯biuuxi ∈ Hγ−2
+q
+(τm,n) and
+σ(u)η ∈ Hγ−2+c0
+q
+(τm,n, l2), Lemma 4.2 implies that there exists v ∈ Hγ
+q(τm,n) satisfying
+∂α
+t v = Lv + ¯biuuxi + ∂α
+t
+ˆ t
+0
+σk(u)dWt,
+0 < t ≤ τm,n ;
+v(0, ·) = u0,
+(4.38)
+where Lv = aijvxixj + bivxi + cv. In (4.38), note that ¯biuuxi and σk(u) are used instead of
+¯bivvxi and σk(v). Moreover, because u ∈ Lq(τm,n) satisfies equation (4.38), ¯v := u − v ∈
+Lq(τm,n) satisfies
+∂α
+t ¯v = aij¯vxixj + bi¯vxi + c¯v,
+0 < t ≤ τm,n ;
+¯v(0, ·) = 0.
+By the deterministic version of Lemma 4.2, we have ¯v = 0 in Lq(τm,n); thus, u = v in
+Lq(τm,n). Therefore, u is in Hγ
+q(τm,n). As ¯u ∈ Hγ
+q(τm,n) and ¯u satisfies equation (3.1), by
+Lemma 4.3, we have u = ¯u in Hγ
+q(τm,n). The theorem is proved.
+□
+5.
+Proof of Theorem 2.16
+This section provides a proof of the embedding theorem for solution spaces Hγ
+p(τ). Con-
+sider the following fractional diffusion equation
+∂α
+t u = ∆u
+t > 0 ;
+u(0, ·) = u0(·),
+(5.1)
+where α ∈ (0, 1) and u0(·) ∈ C∞
+c (Rd). It turns out that a fundamental solution p(t, x) exists
+such that
+p(t, ·) ∈ L1(Rd)
+and
+F(p(t, ·))(ξ) = Eα(−tα|ξ|2)
+(e.g., [24, Theorem 2.1]), and the solution of (5.1) is given by
+u(t, x) = (u0 ∗ p(t, ·))(x) =
+ˆ
+Rd u0(y)p(t, x − y)dy.
+
+28
+BEOMSEOK HAN
+For convenience, define
+qα,β(t, x) :=
+�
+Iα−βp(t, x)
+if
+α ≥ β
+Dβ−αp(t, x)
+if
+α < β.
+We gather some facts related to p(t, x) and qα,β(t, x) (for more information, see [22, 23, 24]).
+Lemma 5.1. Let d ∈ N, α ∈ (0, 1), β < α + 1/2, γ ∈ [0, 2), and σ ∈ R.
+(i) For all t ̸= 0 and x ̸= 0,
+∂α
+t p(t, x) = ∆p(t, x)
+and
+∂tp(t, x) = ∆qα,1(t, x).
+Additionally, for each x ̸= 0,
+∂
+∂tp(t, x) → 0 as t ↓ 0. Moreover,
+∂
+∂tp(t, x) is integrable
+in Rd uniformly on t ∈ [δ, T] for any δ > 0.
+(ii) There exist constants c = c(α, d) and N = N(α, d) such that if |x|2 ≥ tα,
+|p(t, x)| ≤ N|x|−d exp
+�
+−c|x|
+2
+2−α t−
+α
+2−α
+�
+.
+(iii) Let n ∈ N. Then, there exists N = N(α, γ, n) such that
+���Dσ
+t Dn
+x(−∆)γ/2qα,β(1, x)
+��� ≤ N
+�
+|x|−d+2−γ−n ∧ |x|−d−γ−n�
+.
+(iv) The scaling properties hold. In other words,
+qα,β(t, x) = t− αd
+2 +α−βqα,β(1, xt− α
+2 ),
+(−∆)γ/2qα,β(t, x) = t− α(d+γ)
+2
++α−β(−∆)γ/2qα,β(1, xt− α
+2 ).
+Proof. To see (i), (ii), and (iii) follow from Theorems 2.1 and 2.3 of [23]. For (iv), see (5.2)
+in [23].
+□
+Remark 5.2. To prove Theorem 2.16, we define the operators.
+Let φ ∈ C∞
+c (Rd) and
+f ∈ C∞
+c ((0, ∞) × Rd). Take a function g = (g1, g2, . . . ) satisfying the form
+gk(t, x) =
+��n
+i=1 1 |(τi−1,τi]](t)gik(x)
+for
+k = 1, 2, . . . , n,
+0
+for
+k = n + 1, . . .
+(5.2)
+for some n ∈ N, where τi is the bounded stopping time and gik ∈ C∞
+c (Rd). Further, we set
+T 1
+t φ(x) :=
+ˆ
+Rd p(t, x − y)φ(y)dy,
+(5.3)
+T 2
+t f(t, x) :=
+ˆ t
+0
+ˆ
+Rd qα,1(t − r, x − y)f(r, y)dyds,
+(5.4)
+T 3
+t g(t, x) :=
+�
+k
+ˆ t
+0
+ˆ
+Rd qα,β(t − r, x − y)gk(r, y)dydwk
+s .
+(5.5)
+It is well-known that T 1
+t φ, T 2
+t f, and T 3
+t g are solutions to
+∂α
+t u1 = ∆u1;
+u1(0, ·) = φ,
+∂α
+t u2 = ∆u2 + f;
+u2(0, ·) = 0,
+∂α
+t u3 = ∆u3 + ∂β
+t
+ˆ t
+0
+gkdwk
+s;
+u3(0, ·) = 0,
+
+STFBES DRIVEN BY MULTIPLICATIVE SPACE-TIME WHITE NOISE
+29
+respectively (for more information, see [22, 23, 24]).
+First, we provide a smoothing effect of T 1
+t , which implies the continuity of T 1
+t φ in t.
+Lemma 5.3. Let τ ≤ T be a bounded stopping time and α ∈ (0, 1). If p ∈ (1, ∞), θ ∈ [0, α),
+φ ∈ C∞
+c (Rd), and t ∈ (0, T), we have
+e−t ��T 1
+t φ
+��
+H
+2θ
+α
+p
+≤ Nt−θ∥φ∥Lp
+(5.6)
+where N = N(α, θ, d, p, T).
+Proof. In the case of θ = 0, by Mink¨owski’s inequality, we have
+∥T 1
+t φ∥Lp ≤ ∥p(t, ·) ∗ φ∥Lp ≤ ∥p(t, ·)∥L1∥φ∥Lp ≤ ∥φ∥Lp.
+(5.7)
+Thus, we have (5.6). For θ ∈ (0, α), observe that
+∥e−tT 1
+t φ∥H2θ/α
+p
+= ∥(1 − ∆)θ/α(e−tT 1
+t φ)∥Lp ≤ ∥e−tT 1
+t φ∥Lp + ∥(−∆)θ/α(e−tT 1
+t φ)∥Lp,
+where the last inequality follows from Lemma 2.10 (v). As e−t ≤ (Nt−θ ∧ 1), we have
+��e−tT 1
+t φ
+��
+H2θ/α
+p
+≤ N
+�
+t−θ ��T 1
+t φ
+��
+Lp +
+���(−∆)θ/αT 1
+t φ
+���
+Lp
+�
+.
+(5.8)
+By inequality (5.7), we have
+t−θ∥T 1
+t φ∥Lp ≤ t−θ∥φ∥Lp.
+(5.9)
+On the other hand, Minkowski’s inequality yields
+∥(−∆)θ/αT 1
+t h∥Lp = ∥(−∆)θ/α(p(t, ·) ∗ φ)∥Lp ≤ ∥((−∆)θ/αp)(t, ·)∥L1∥φ∥Lp.
+(5.10)
+Additionally Lemma 5.1 (iv), (ii), and (iii) imply
+∥((−∆)θ/αp)(t, ·)∥L1 =
+ˆ
+Rd |((−∆)θ/αp)(t, x)|dx
+≤ t−θ
+ˆ
+Rd |((−∆)θ/αp)(1, x)|dx
+≤ N(α, θ, d, p)t−θ.
+(5.11)
+Then, by applying (5.11) to (5.10), we have
+∥(−∆)θ/αT 1
+t φ∥Lp ≤ Nt−θ∥φ∥Lp.
+(5.12)
+Thus, by plugging in (5.9) and (5.12) into (5.8), we have (5.6). The lemma is proved.
+□
+To proceed further, we introduce one of the embedding theorems for Slobodetskii’s spaces.
+Lemma 5.4. If µp > 1 and p ≥ 1, for any continuous Lp-valued function φ and γ ≤ ρ, we
+have the following:
+∥φ(ρ) − φ(γ)∥p
+Lp ≤ N(ρ − γ)µp−1
+ˆ ρ
+γ
+ˆ ρ
+γ
+1t>s
+∥φ(t) − φ(s)∥p
+Lp
+|t − s|1+µp
+dsdt
+�0
+0 := 0
+�
+,
+(5.13)
+where N = N(µ, p). In particular,
+E
+sup
+0≤s<t≤T
+∥φ(t) − φ(s)∥p
+Lp
+|t − s|µp−1
+≤ N
+ˆ T
+0
+ˆ T
+0
+1t>s
+E ∥φ(t) − φ(s)∥p
+Lp
+|t − s|1+µp
+dsdt.
+(5.14)
+
+30
+BEOMSEOK HAN
+With the help of Lemma 5.4, we obtain the continuity of T 2
+t f and T 3
+t g on t ∈ [0, T], and
+the H¨older continuity of T 1
+t φ, T 2
+t f, and T 3
+t g on [δ, T].
+First, we suggest the H¨older continuity of T 1
+t φ in t.
+Lemma 5.5. Let T < ∞, δ > 0, and α ∈ (0, 1). If p ∈ (1, ∞) and µ ∈ (0, 1) satisfy
+1
+αp < µ < 1
+α,
+and φ ∈ C∞
+c (Rd), then
+sup
+δ≤s<t≤T
+��T 1
+t φ − T 1
+s ��
+��p
+Lp
+|t − s|αµp−1
+≤ N ∥φ∥p
+Lp
+(5.15)
+where N = N(α, δ, d, p, T).
+Proof. By (5.14), we have
+sup
+δ≤s<t≤T
+��T 1
+t φ − T 1
+s φ
+��p
+Lp
+|t − s|αµp−1
+=
+sup
+0≤s<t≤T−δ
+��T 1
+t+δφ − T 1
+s+δφ
+��p
+Lp
+|t − s|αµp−1
+≤ N
+ˆ T−δ
+0
+ˆ T−δ
+0
+1t>s
+��T 1
+t+δφ − T 1
+s+δφ
+��p
+Lp
+|t − s|1+αµp
+dsdt.
+By Minkowski’s inequality,
+��T 1
+t+δφ − T 1
+s+δφ
+��
+Lp ≤
+ˆ
+Rd |p(t + δ, y) − p(s + δ, y)| dy ∥φ∥Lp .
+Then, by the fundamental theorem of calculus, the change of variable, and Lemma 5.1 (i)
+- (iii),
+ˆ T−δ
+0
+ˆ T−δ
+0
+1t>s|t − s|−1−αµp ��T 1
+t+δφ − T 1
+s+δφ
+��p
+Lp dsdt
+≤
+ˆ T−δ
+0
+ˆ T−δ
+0
+1t>s|t − s|−1−αµp
+�ˆ
+Rd |p(t + δ, y) − p(s + δ, y)| dy ∥φ∥Lp
+�p
+dsdt
+=
+ˆ T−δ
+0
+t−1−αµp
+ˆ T−t
+δ
+�ˆ
+Rd |p(t + s, y) − p(s, y)| dy
+�p
+dsdt ∥φ∥p
+Lp
+≤
+ˆ T−δ
+0
+t−1−αµp+p
+ˆ T−t
+δ
+�ˆ
+Rd sup
+r∈[δ,T]
+|∂tp(r, y)| dy
+�p
+dsdt ∥φ∥p
+Lp
+≤ N ∥φ∥p
+Lp .
+(5.16)
+Thus, we have (5.15). The lemma is proved.
+□
+Remark 5.6. It should be remarked that we assume δ > 0 in Lemma 5.5, and it is a
+sufficient condition. Indeed, if we try δ = 0, the term
+ˆ T
+0
+t−1−αµp
+ˆ T−t
+0
+�ˆ
+Rd |p(t + s, y) − p(s, y)| dy
+�p
+dsdt
+in (5.16) blows up.
+Next we introduce the continuities of T 2
+t f and T 3
+t g.
+
+STFBES DRIVEN BY MULTIPLICATIVE SPACE-TIME WHITE NOISE
+31
+Lemma 5.7. Let α ∈ (0, 1).
+(i) If p ∈ (1, ∞), θ ∈ (0, α), and µ ∈ (0, 1) satisfy
+p >
+1
+α − θ,
+µ ∈
+� 1
+αp, α − θ
+α
+�
+and f ∈ Lp((0, T) × Rd), then for t, s ∈ (0, T),
+��T 2
+t f − T 2
+s f
+��p
+H2θ/α
+p
+≤ N|t − s|αµp−1
+ˆ T
+0
+∥f(r, ·)∥p
+Lp dr
+(5.17)
+where N = N(α, θ, d, p, T). Additionally,
+sup
+0≤s<t≤T
+��T 2
+t f − T 2
+s f
+��p
+H2θ/α
+p
+|t − s|αµp−1
+≤ N
+ˆ T
+0
+∥f(r, ·)∥p
+Lp dr,
+(5.18)
+where N = N(α, θ, d, p, T).
+(ii) Let τ ≤ T be a bounded stopping time. If β < α + 1
+2, p ∈ [2, ∞), θ ∈ (0, α − β + 1/2),
+and µ ∈ (0, 1) satisfy
+p >
+1
+(α − β − θ) ∧ 1/2 + 1/2,
+µ ∈
+� 1
+αp, (α − β − θ) ∧ 1/2 + 1/2
+α
+�
+,
+and g ∈ Lp(τ, l2), then, for t, s ∈ (0, T), we have
+E
+��T 3
+t∧τg − T 3
+s∧τg
+��p
+H2θ/α
+p
+(l2) ≤ N|t − s|αµp−1∥g∥p
+Lp(τ,l2),
+(5.19)
+where N = N(α, β, θ, d, p, T). Additionally,
+E
+sup
+0≤s<t≤τ
+��T 3
+t g − T 3
+s g
+��p
+H2θ/α
+p
+(l2)
+|t − s|αµp−1
+≤ N∥g∥p
+Lp(τ,l2),
+(5.20)
+where N = N(α, β, θ, d, p, T).
+Proof. Proof of (i) Let ρ > 0 and γ > 0. Notice that (5.13) yields
+��T 2
+ρ f − T 2
+γ f
+��p
+H2θ/α
+p
+≤ N|ρ − γ|αµp−1
+ˆ T
+0
+ˆ T
+0
+1t>s
+��T 2
+t f − T 2
+s f
+��p
+H2θ/α
+p
+|t − s|1+αµp
+dsdt.
+(5.21)
+Then, by definition of T 2
+t f (see (5.4)),
+��T 2
+t f − T 2
+s f
+��
+H2θ/α
+p
+≤
+����
+ˆ t
+s
+ˆ
+Rd qα,1(t − r, y)f(r, · − y)dydr
+����
+H2θ/α
+p
++
+����
+ˆ s
+0
+ˆ
+Rd (qα,1(t − r, y) − qα,1(s − r, y)) f(r, · − y)dydr
+����
+H2θ/α
+p
+.
+(5.22)
+Set
+I1 :=
+ˆ T
+0
+ˆ T
+0
+1t>s
+���
+´ t
+s
+´
+Rd qα,1(t − r, y)f(r, · − y)dydr
+���
+p
+H2θ/α
+p
+|t − s|1+αµp
+dsdt,
+I2 :=
+ˆ T
+0
+ˆ T
+0
+1t>s
+��´ s
+0
+´
+Rd (qα,1(t − r, y) − qα,1(s − r, y)) f(r, · − y)dydr
+��p
+H2θ/α
+p
+|t − s|1+αµp
+dsdt.
+(5.23)
+
+32
+BEOMSEOK HAN
+Then, apply (5.22) and (5.23) to (5.21),
+��T 2
+t f − T 2
+s f
+��p
+H2θ/α
+p
+≤ |t − s|αµp−1(I1 + I2).
+(5.24)
+To deal with I1, we employ Minkowski’s inequality, the change of variable, and Lemma 2.10
+(v). Then,
+I1 ≤
+ˆ T
+0
+t−1−αµp
+�ˆ t
+0
+ˆ
+Rd
+���
+�
+(1 − ∆)θ/αqα,1
+�
+(r, y)
+��� dydr
+�p
+dt
+ˆ T
+0
+∥f(s, ·)∥p
+Lp ds
+≤ I11 + I12,
+(5.25)
+where
+I11 :=
+ˆ T
+0
+t−1−αµp
+�ˆ t
+0
+ˆ
+Rd |qα,1(r, y)| dydr
+�p
+dt
+ˆ T
+0
+∥f(s, ·)∥p
+Lp ds,
+I12 :=
+ˆ T
+0
+t−1−αµp
+�ˆ t
+0
+ˆ
+Rd
+���
+�
+∆θ/αqα,1
+�
+(r, y)
+��� dydr
+�p
+dt
+ˆ T
+0
+∥f(s, ·)∥p
+Lp ds.
+Because µ < 1, Lemma 5.1 (iv) and (iii) yield
+I11 =
+ˆ T
+0
+t−1−αµp+αpdt
+�ˆ
+Rd |qα,1(1, y)| dy
+�p ˆ T
+0
+∥f(s, ·)∥p
+Lp ds
+≤ N
+ˆ T
+0
+∥f(s, ·)∥p
+Lp ds.
+(5.26)
+Similarly, since µ < 1 − θ/α,
+I12 =
+ˆ T
+0
+t−1−αµp+αp−θpdt
+�ˆ
+Rd
+���
+�
+∆θ/αqα,1
+�
+(1, y)
+��� dy
+�p ˆ T
+0
+∥f(s, ·)∥p
+Lp ds
+≤ N
+ˆ T
+0
+∥f(s, ·)∥p
+Lp ds.
+(5.27)
+Thus, by applying (5.26) and (5.27) to (5.25), we have
+I1 ≤ N
+ˆ T
+0
+∥f(s, ·)∥p
+Lpds.
+(5.28)
+Next, we address I2. Similar to the case for I1, we have
+I2 ≤
+ˆ T
+0
+´ T−t
+0
+�´ s
+0
+´
+Rd
+��(1 − ∆)θ/α (qα,1(t + r, y) − qα,1(r, y))
+�� dy∥f(s − r, ·)∥Lpdr
+�p ds
+t1+αµp
+dt
+≤ I21 + I22,
+(5.29)
+where
+I21 :=
+ˆ T
+0
+´ T−t
+0
+�´ s
+0
+´
+Rd |qα,1(t + r, y) − qα,1(r, y)| dy∥f(s − r, ·)∥Lpdr
+�p ds
+t1+αµp
+dt,
+I22 :=
+ˆ T
+0
+´ T−t
+0
+�´ s
+0
+´
+Rd
+��∆θ/α (qα,1(t + r, y) − qα,1(r, y))
+�� dy∥f(s − r, ·)∥Lpdr
+�p ds
+t1+αµp
+dt.
+
+STFBES DRIVEN BY MULTIPLICATIVE SPACE-TIME WHITE NOISE
+33
+Since µ < 1, by Minkowski’s inequality and the fundamental theorem of calculus, we have
+I21 ≤
+ˆ T
+0
+t−1−αµp
+�ˆ T−t
+0
+ˆ
+Rd |qα,1(t + r, y) − qα,1(r, y)| dydr
+�p
+dt
+ˆ T
+0
+∥f(s, ·)∥p
+Lp(Rd)ds
+≤
+ˆ T
+0
+t−1−αµp
+�ˆ T−t
+0
+ˆ t+r
+r
+ˆ
+Rd |qα,2(s, y)| dydsdr
+�p
+dt
+ˆ T
+0
+∥f(s, ·)∥p
+Lp(Rd)ds
+≤ N
+ˆ T
+0
+t−1−αµp
+�ˆ T−t
+0
+rα−1 − (t + r)α−1dr
+�p
+dt
+ˆ T
+0
+∥f(s, ·)∥p
+Lp(Rd)ds
+≤ N
+ˆ T
+0
+t−1−αµp+αpdt
+ˆ T
+0
+∥f(s, ·)∥p
+Lp(Rd)ds
+≤ N
+ˆ T
+0
+∥f(s, ·)∥p
+Lp(Rd)ds.
+(5.30)
+Additionally, since µ < 1 − θ/α,
+I22 ≤
+ˆ T
+0
+t−1−αµp
+�ˆ T−t
+0
+ˆ
+Rd
+ˆ t+r
+r
+���(∆θ/αqα,2(s, y)
+��� dsdydr
+�p
+dt
+ˆ T
+0
+∥f(s, ·)∥p
+Lpds
+≤ N
+ˆ T
+0
+t−1−αµp+αp−θpdt
+ˆ T
+0
+∥f(s, ·)∥p
+Lpds
+≤ N
+ˆ T
+0
+∥f(s, ·)∥p
+Lpds.
+(5.31)
+Therefore, by employing (5.30) and (5.31) to (5.29), we have
+I2 ≤ N
+ˆ T
+0
+∥f(s, ·)∥p
+Lpds,
+(5.32)
+and thus by combining (5.28) and (5.32) to (5.24), we have (5.17).
+To obtain (5.18), employ (5.14) instead of (5.13) and repeat the proof word for word.
+Proof of (ii) By (5.13), we have
+E
+��T 3
+ρ g − T 3
+γ g
+��p
+H2θ/α
+p
+(l2) ≤ N|ρ − γ|αµp−1
+ˆ T
+0
+ˆ T
+0
+1t>s
+E
+��T 3
+t g − T 3
+s g
+��p
+H2θ/α
+p
+(l2)
+|t − s|1+αµp
+dsdt.
+
+34
+BEOMSEOK HAN
+Notice that the Burkholder-Davis-Gundy and Minkowski’s inequalities imply that
+E
+��T 3
+t g − T 3
+s g
+��p
+H2θ/α
+p
+(l2)
+≤ N
+ˆ
+Rd E
+��
+k
+����
+ˆ t
+s
+ˆ
+Rd((1 − ∆)θ/α qα,β)(t − r, y)gk(r, x − y)dydwk
+r
+����
+2�p/2
++
+��
+k
+����
+ˆ s
+0
+ˆ
+Rd
+�
+(1 − ∆)θ/α (qα,β(t − r, y) − qα,β(s − r, y))
+�
+gk(r, x − y)dydwk
+r
+����
+2�p/2
+dx
+≤ N
+ˆ
+Rd E
+�ˆ t−s
+0
+�ˆ
+Rd
+���((1 − ∆)θ/α qα,β)(t − s − r, y)
+��� |g(s + r, x − y)|l2dy
+�2
+dr
+�p/2
++
+�ˆ s
+0
+�ˆ
+Rd
+���(1 − ∆)θ/α (qα,β(t − r, y) − qα,β(s − r, y))
+��� |g(r, x − y)|l2dy
+�2
+dr
+�p/2
+dx
+≤ NE
+�ˆ t−s
+0
+�ˆ
+Rd
+���((1 − ∆)θ/α qα,β)(t − s − r, y)
+��� dy
+�2
+∥g(s + r, ·)∥2
+Lp(l2)dr
+�p/2
++ NE
+�ˆ s
+0
+�ˆ
+Rd
+���(1 − ∆)θ/α (qα,β(t − r, y) − qα,β(s − r, y))
+��� dy
+�2
+∥g(r, ·)∥2
+Lp(l2)dr
+�p/2
+.
+Then, set
+I3 :=
+ˆ T
+0
+ˆ T
+0
+1t>s|t − s|−1−αµpE
+�ˆ t−s
+0
+A(t, s, r)∥g(s + r, ·)∥2
+Lp(l2)dr
+�p/2
+dsdt,
+I4 :=
+ˆ T
+0
+ˆ T
+0
+1t>s|t − s|−1−αµpE
+�ˆ s
+0
+B(t, s, r)∥g(r, ·)∥2
+Lp(l2)dr
+�p/2
+dsdt,
+where
+A(t, s, r) =
+�ˆ
+Rd | (1 − ∆)θ/α qα,β(t − s − r, y)|dy
+�2
+,
+B(t, s, r) =
+�ˆ
+Rd
+���(1 − ∆)θ/α (qα,β(t − r, y) − qα,β(s − r, y))
+��� dy
+�2
+.
+Note that Minkowski’s inequality and Lemma 2.10 (v) imply that
+I3 ≤
+ˆ T
+0
+t−1−αµp
+�ˆ t
+0
+�ˆ
+Rd
+���((1 − ∆)θ/α qα,β)(t − r, y)
+��� dy
+�2
+dr
+�p/2
+dt∥g∥p
+Lp(T,l2)
+≤ I31 + I32,
+(5.33)
+where
+I31 :=
+ˆ T
+0
+t−1−αµp
+�ˆ t
+0
+�ˆ
+Rd |qα,β(r, y)|dy
+�2
+dr
+�p/2
+dt∥g∥p
+Lp(T,l2),
+I32 :=
+ˆ T
+0
+t−1−αµp
+�ˆ t
+0
+�ˆ
+Rd
+���
+�
+∆θ/αqα,β
+�
+(r, y)
+��� dy
+�2
+dr
+�p/2
+dt∥g∥p
+Lp(T,l2).
+
+STFBES DRIVEN BY MULTIPLICATIVE SPACE-TIME WHITE NOISE
+35
+Since 1
+α
+�
+α − β + 1
+2
+�
+> µ, by Lemma 5.1 (iii), we have
+I31 =
+ˆ T
+0
+t−1−αµp
+�ˆ t
+0
+�ˆ
+Rd |qα,β(r, y)|dy
+�2
+dr
+�p/2
+dt∥g∥p
+Lp(T,l2)
+=
+ˆ T
+0
+t−1−αµp
+�ˆ t
+0
+r2(α−β)dr
+�p/2
+dt
+�ˆ
+Rd |qα,β(1, y)|dy
+�p
+∥g∥p
+Lp(T,l2)
+=
+ˆ T
+0
+t−1+(α−β+1/2−αµ)pdt
+�ˆ
+Rd |qα,β(1, y)|dy
+�p
+∥g∥p
+Lp(T,l2)
+≤ N∥g∥p
+Lp(T,l2).
+(5.34)
+Similarly, as 1
+α
+�
+α − β − θ + 1
+2
+�
+> µ,
+I32 ≤
+ˆ T
+0
+t−1+(−αµ+α−β−θ+1/2)pdt
+�ˆ
+Rd |(∆θ/αqα,β)(1, y)|dy
+�p
+∥g∥p
+Lp(T,l2)
+≤ N∥g∥p
+Lp(T,l2).
+(5.35)
+Therefore, by employing (5.34) and (5.35) to (5.33), we have
+I3 ≤ N∥g∥p
+Lp(T,l2).
+(5.36)
+In the case of I4, by Minkowski’s inequality and Lemma 2.10 (v), we have
+I4 ≤ I41 + I42.
+Further,
+I41 :=
+ˆ T
+0
+t−1−αµpE
+ˆ T−t
+0
+�ˆ s
+0
+�ˆ
+Rd |C(t, r, y)|dy
+�2
+∥g(s − r, ·)∥2
+Lp(l2)dr
+�p/2
+dsdt,
+I42 :=
+ˆ T
+0
+t−1−αµpE
+ˆ T−t
+0
+�ˆ s
+0
+�ˆ
+Rd |∆θ/αC(t, r, y)|dy
+�2
+∥g(s − r, ·)∥2
+Lp(l2)dr
+�p/2
+dsdt,
+where
+C(t, r, y) = qα,β(t + r, y) − qα,β(r, y).
+We address I41.
+By Minkowski’s inequality, the fundamental theorem of calculus, and
+Lemma 5.1, we have
+I41 ≤ N
+ˆ T
+0
+t−1−αµp
+�ˆ T−t
+0
+�ˆ
+Rd |qα,β(t + r, y) − qα,β(r, y)|dy
+�2
+dr
+�p/2
+dt∥g∥p
+Lp(T,l2)
+≤ N
+ˆ T
+0
+t−1−αµp
+�ˆ T−t
+0
+�ˆ t+r
+r
+ˆ
+Rd |qα,β+1(s, y)|dyds
+�2
+dr
+�p/2
+dt∥g∥p
+Lp(T,l2)
+≤ NH1∥g∥p
+Lp(T,l2),
+(5.37)
+where
+H1 :=
+ˆ T
+0
+t−1−αµp
+�ˆ T−t
+0
+�ˆ t+r
+r
+sα−β−1ds
+�2
+dr
+�p/2
+dt
+(5.38)
+
+36
+BEOMSEOK HAN
+and N = N(α, β, d, p, T). Next, we claim that
+H1 < ∞
+(5.39)
+To demonstrate (5.39), set
+χ(t) :=
+ˆ T−t
+0
+�ˆ t+r
+r
+sα−β−1ds
+�2
+dr.
+Furthermore,
+H1 =
+ˆ T
+0
+t−1−αµp(χ(t))p/2dt
+(5.40)
+Depending on the range of α − β, we consider the following five cases.
+(Case 1.) −1/2 < α − β < 0
+For t ∈ (0, T), we have
+0 ≤ χ(t) ≤ N
+ˆ T−t
+0
+r2(α−β) − (t + r)2(α−β)dr ≤ Nt2(α−β)+1,
+(5.41)
+where N = N(α, β). Then, since 1
+α(α − β + 1/2) > µ, by combining (5.40) and (5.41),
+H1 ≤ N(α, β, p)
+ˆ T
+0
+t−1+p(−αµ+α−β+1/2)dt < ∞.
+(Case 2.) α − β = 0
+Notice that
+χ(t) =
+ˆ T−t
+0
+�ˆ t+r
+r
+s−1ds
+�2
+dr =
+ˆ T−t
+0
+�
+log
+�t + r
+r
+��2
+dr.
+Obviously, χ(0) = 0. Note that
+χ′(t) ≤ 2
+ˆ ∞
+0
+1
+1 + r log
+�1 + r
+r
+�
+dr = 2
+ˆ ∞
+0
+x(ex − 1)−1dx = π2/3
+on t ∈ (0, T/2). Thus,
+0 ≤ χ(t) = χ(t) − χ(0) =
+ˆ t
+0
+χ′(s)ds ≤ π2
+3 t
+(5.42)
+on t ∈ (0, T/2). Additionally, χ(t) ≤ N on t ∈ (T/2, T). Therefore,
+H1 ≤ N
+ˆ T/2
+0
+t−1+(−αµ+1/2)pdt + N < ∞,
+where N = N(α, β, p).
+(Case 3.) 0 < α − β < 1/2
+Observe that χ is twice continuously differentiable, and
+χ′(t) = (α − β)−2 �
+−T 2(α−β) − (T − t)2(α−β) + 2T α−β(T − t)α−β�
++ 2(α − β)−1
+ˆ T−t
+0
+(t + r)2(α−β)−1 − (t + r)α−β−1rα−βdr,
+
+STFBES DRIVEN BY MULTIPLICATIVE SPACE-TIME WHITE NOISE
+37
+and
+χ′′(t) = 2(α − β)−1(T − t)2(α−β)−1 − 2(α − β)−1T α−β(T − t)α−β−1
++ 2(α − β)−1T α−β−1(T − t)α−β − 2(α − β)−1t2(α−β)−1
+− 2(α − β)−1(α − β − 1)
+ˆ T−t
+0
+(t + r)α−β−2rα−βdr.
+(5.43)
+In addition, χ(0) = χ′(0) = 0. Then, by using the fundamental theorem of calculus and
+α − β ∈ (0, 1/2), we obtain
+χ(t) =
+ˆ t
+0
+ˆ s
+0
+χ′′(ρ)dρds
+≤ 2(α − β)−1
+ˆ t
+0
+ˆ s
+0
+(T − ρ)2(α−β)−1dρds
+− 2(α − β)−1(α − β − 1)
+ˆ t
+0
+ˆ s
+0
+ˆ T−ρ
+0
+(ρ + r)α−β−2rα−βdrdρds
+≤ N
+ˆ t
+0
+�
+T 2(α−β) − (T − s)2(α−β)�
+ds + N
+ˆ t
+0
+ˆ s
+0
+ˆ T−ρ
+0
+(ρ + r)2(α−β)−2drdρds
+≤ N
+ˆ t
+0
+s2(α−β)ds + N
+ˆ t
+0
+ˆ s
+0
+T 2(α−β)−1drdρds
+≤ Nt2(α−β)+1,
+(5.44)
+where N = N(α, β, T). Thus,
+H1 ≤ N(α, β, p, T)
+ˆ T
+0
+t−1+p(−αµ+α−β+1/2)dt < ∞.
+(Case 4.) α − β = 1/2
+Because χ(0) = χ′(0) = 0, by the fundamental theorem of calculus, we have
+χ(t) =
+ˆ t
+0
+ˆ s
+0
+χ′′(ρ)dρds
+≤ N
+ˆ t
+0
+ˆ s
+0
+(T − ρ)1/2 + N
+ˆ T−ρ
+0
+(ρ + r)−3/2r1/2drdρds
+≤ Nt2(1 + | log t|),
+where N = N(T). Therefore,
+H1 ≤ N(p, T)
+ˆ T
+0
+t−1+p(−αµ+1)(1 + | log t|)p/2dt < ∞.
+(Case 5.) α − β > 1/2
+
+38
+BEOMSEOK HAN
+Similar to before, χ(0) = χ′(0) = 0. Additionally, as in (5.44), we have
+χ(t) =
+ˆ t
+0
+ˆ s
+0
+χ′′(r)drds
+≤ N
+ˆ t
+0
+�
+T 2(α−β) − (T − s)2(α−β)�
+ds + N
+ˆ t
+0
+ˆ s
+0
+ˆ T−ρ
+0
+(ρ + r)2(α−β)−2drdρds
+≤ N
+ˆ t
+0
+sds + N
+ˆ t
+0
+ˆ s
+0
+dρds
+≤ Nt2,
+where N = N(α, β, T). Therefore,
+H1 ≤ N(α, β, p, T)
+ˆ T
+0
+t−1+p(−αµ+1)dt < ∞.
+Thus, we have (5.39). Then, by combining (5.39) and (5.37), we have I41 ≤ N∥g∥p
+Lp(T,l2).
+Next, we deal with I42. Minkowski’s inequality, the fundamental theorem of calculus,
+and Lemma 5.1 yield
+I42 ≤ N
+ˆ T
+0
+t−1−αµp
+�ˆ T−t
+0
+�ˆ
+Rd |∆θ/αqα,β(t + r, y) − ∆θ/αqα,β(r, y)|dy
+�2
+dr
+�p/2
+dt∥g∥p
+Lp(T,l2)
+≤ N
+ˆ T
+0
+t−1−αµp
+�ˆ T−t
+0
+�ˆ t+r
+r
+ˆ
+Rd |∂s∆θ/αqα,β(s, y)|dyds
+�2
+dr
+�p/2
+dt∥g∥p
+Lp(T,l2)
+≤ NH2∥g∥p
+Lp(T,l2),
+where N = N(α, β, d, p, T) and H2 :=
+´ T
+0 t−1−αµp
+�´ T−t
+0
+�´ t+r
+r
+sα−β−θ−1ds
+�2
+dr
+�p/2
+dt.
+Similar to the case of H1 (see (5.38) and (5.39)), we demonstrate that
+H2 < ∞
+by considering five cases for α − β − θ instead of α − β. Then, we have I42 ≤ N∥g∥p
+Lp(T,l2).
+The lemma is proved.
+□
+Proof of Theorem 2.16. It suffices to show that the assertion holds for τ = T. Indeed,
+assume the results holds for Hγ
+p(T), and let τ ≤ T be a bounded stopping time and u ∈
+Hγ
+p(τ). Then, by Definition 2.4 for ε > 0, there exists (f, g) ∈ Hγ−2
+p
+(τ)×Hγ−2+c0
+p
+(τ, l2) such
+that
+∂α
+t u = f + ∂β
+t
+ˆ t
+0
+gkdwk
+t ;
+u(0, ·) = u0(·)
+and
+∥u∥Hγ
+p(τ) + ∥u0∥Uα,γ
+p
++ ∥f∥Hγ−2
+p
+(τ) + ∥g∥Hγ−2+c0
+p
+(τ,l2) ≤ ∥u∥Hγ
+p(τ) + ε.
+Set ¯f := (f − ∆u)1t≤τ and ¯g := g1t≤τ ; thus, u satisfies
+∂α
+t u = ∆u + ¯f + ∂β
+t
+ˆ t
+0
+¯gkdwk
+t ,
+0 < t ≤ τ ;
+u(0, ·) = u0(·).
+(5.45)
+
+STFBES DRIVEN BY MULTIPLICATIVE SPACE-TIME WHITE NOISE
+39
+In contrast, by [25, Theorem 2.18], there exists v ∈ Hγ
+p(T) such that v satisfies
+∂α
+t v = ∆v + ¯f + ∂β
+t
+ˆ t
+0
+¯gkdwk
+t ,
+0 < t ≤ τ ;
+v(0, ·) = u0(·).
+Additionally,
+∥v∥Hγ
+p (T) ≤ N
+��� ¯f
+��
+Hγ−2
+p
+(T) + ∥¯g∥Hγ−2+c0
+p
+(T,l2) + ∥u0∥Uα,γ
+p
+�
+≤ N
+�
+∥u∥Hγ
+p(τ) + ∥u0∥Uα,γ
+p
++ ∥f∥Hγ−2
+p
+(τ) + ∥g∥Hγ−2+c0
+p
+(τ,l2)
+�
+≤ N∥u∥Hγ
+p(τ) + Nε,
+(5.46)
+where N is independent of ε. Because v ∈ Hγ
+p(T), v ∈ C
+�
+[0, T]; Hγ−2ν
+p
+�
+almost surely and
+E sup
+t≤T
+∥v(t, ·)∥p
+Hγ−2ν
+p
+≤ N∥v∥p
+Hγ
+p(T)
+(5.47)
+by the hypothesis. Therefore, due to τ ≤ T, (5.46) and (5.47) yield
+E sup
+t≤τ
+∥v(t, ·)∥p
+Hγ−2ν
+p
+≤ E sup
+t≤T
+∥v(t, ·)∥p
+Hγ−2ν
+p
+≤ N∥v∥p
+Hγ
+p(T) ≤ N∥u∥p
+Hγ
+p(τ) + Nε,
+(5.48)
+where N is independent of ε. Note that ¯u := u − v satisfies
+∂α
+t ¯u = ∆¯u,
+0 < t ≤ τ;
+¯u(0, ·) = 0.
+Then, by the deterministic version of [25, Theorem 2.18], we have u(t, ·) = v(t, ·) for almost
+every (ω, t) ∈ Ω × [0, τ]. Thus, v is an Hγ−2ν
+p
+-valued continuous version of u. Additionally,
+from (5.48)
+E sup
+t≤τ
+∥u(t, ·)∥p
+Hγ−2ν
+p
+≤ N∥u∥p
+Hγ
+p(τ) + Nε.
+In addition, as ε > 0 is arbitrary and N is independent of ε, we have (2.8). Furthermore,
+observe that we have (2.10) similarly. Additionally, notice that (5.46) allows us to prove
+the assertions with ∥f∥Hγ−2
+p
+(T) + ∥g∥Hγ−2+c0
+p
+(T,l2) + ∥u0∥Uα,γ
+p
+instead of ∥u∥Hγ
+p(T).
+Due to Lemma 2.10 (vi), we only consider the case γ = 2ν, where ν ∈ (0, 1) satisfies
+(2.7). Moreover, by using the approximation to the identity, we may assume that u0 is
+infinitely differentiable and compactly supported in x. Furthermore, we also assume that
+f and g = (g1, g2, . . . ) denotes the function of the form satisfying (5.2) (see [26, Theorem
+3.10]). Additionally, it should be remarked that u can be written as
+u(t, x) = T 1
+t u0 + T 2
+t f + T 3
+t g
+since u satisfies
+∂α
+t u = ∆u + f + ∂β
+t
+ˆ t
+0
+gkdwk
+t ;
+u(0, ·) = u0(·)
+for almost every (ω, t) ∈ |(0, τ]] (see [22, 23, 24]).
+
+40
+BEOMSEOK HAN
+First, we prove (i).
+To obtain the continuity of u, notice that T 1
+t u0 is a continuous
+Lp-valued function in t on [0, T]. Indeed, by Remark 5.2 and Lemma 5.3,
+∥T 1
+t u0 − u0∥Lp = N
+����
+ˆ t
+0
+(t − s)α−1T 1
+s ∆u0ds
+����
+Lp
+≤ N
+ˆ t
+0
+(t − s)α−1∥T 1
+s ∆u0∥Lpds
+≤ N
+ˆ t
+0
+(t − s)α−1∥∆u0∥Lpds
+≤ Ntα∥∆u0∥Lp
+for t > 0. Then, we have
+lim
+t↓0 ∥T 1
+t u0 − u0∥Lp → 0.
+(5.49)
+Additionally, for t, h > 0, Lemma 5.3 applies
+��T 1
+t+hu0 − T 1
+t u0
+��
+Lp ≤
+���
+T 1
+t+hu0 − u0
+�
+−
+�
+T 1
+t u0 − u0
+���
+Lp
+≤
+����
+ˆ t+h
+0
+(t + h − s)α−1∆T 1
+s u0ds −
+ˆ t
+0
+(t − s)α−1∆T 1
+s u0ds
+����
+≤
+����
+ˆ t+h
+t
+(t + h − s)α−1∆T 1
+s u0ds
+����
+Lp
++
+����
+ˆ t
+0
+(t + h − s)α−1∆T 1
+s u0ds −
+ˆ t
+0
+(t − s)α−1∆T 1
+s u0ds
+����
+Lp
+≤
+ˆ t+h
+t
+(t + h − s)α−1 ��∆T 1
+s u0
+��
+Lp ds
++
+ˆ t
+0
+�
+(t − s)α−1 − (t + h − s)α−1� ��∆T 1
+s u0
+��
+Lp ds
+≤ α−1 (2hα + tα − (t + h)α) ∥∆u0∥Lp,
+and thus
+lim
+h→0
+��T 1
+t+hu0 − T 1
+t u0
+��
+Lp = 0.
+(5.50)
+Because C∞
+c (Rd) is dense in Lp, (5.49) and (5.50) imply that T 1
+t is continuous on Lp.
+For T 2
+t f and T 3
+t g, by combining (5.17), (5.19), Jensen’s inequality, and the Kolmogorov
+continuity theorem (e.g., [28, Theorem 1.4.8]), we have T 2
+t f ∈ C([0, T]; Hγ−2ν
+p
+) and T 3
+t g ∈
+C([0, T]; Hγ−2ν
+p
+(l2)) almost surely.
+To demonstrate (2.8), we recall that ν is taken as in (2.7). Then, Lemma 5.3 implies that
+E sup
+t≤T
+��T 1
+t u0
+��p
+Hγ−2ν
+p
+≤ NE∥u0∥p
+Lp ≤ NE∥u0∥p
+H
+γ− 2
+αp
+p
+.
+(5.51)
+Additionally, as (2.7) is assumed, Lemma 2.10 (vi) and (5.17) with θ = α − αν, and
+lims↓0 ∥T 2
+s f∥Hγ−2ν
+p
+= 0 yield
+E sup
+t≤T
+��T 2
+t f
+��p
+Hγ−2ν
+p
+≤ N∥f∥p
+Hγ−2
+p
+(T).
+(5.52)
+
+STFBES DRIVEN BY MULTIPLICATIVE SPACE-TIME WHITE NOISE
+41
+Furthermore, due to (2.7), Lemma 2.10 (vi), (5.19) with θ = α(1 − c0/2) − αν, and
+lims↓0 ∥T 3
+s g∥Hγ−2ν
+p
+= 0, we have
+E sup
+t≤T
+��T 3
+t g
+��p
+Hγ−2ν
+p
+≤ N∥g∥p
+Hγ−2+c0
+p
+(T,l2),
+(5.53)
+where c0 is the constant introduced in (2.3). By combining (5.51), (5.52), and (5.53), we
+obtain (2.8).
+Next we prove (ii). Due to (5.15), (5.18), and (5.20) and Theorem 2.16 (i), we have
+u ∈ Cαµ−1/p([δ, T]; Hγ−2ν
+p
+) almost surely. To demonstrate (2.10), choose µ and ν satisfy
+(2.9). Observe that Lemma 5.5 implies that
+E
+sup
+δ≤s<t≤T
+∥T 1
+t u0 − T 1
+s u0∥p
+Hγ−2ν
+p
+|t − s|αµp−1
+≤ NE ∥u0∥p
+Hγ−2ν
+p
+.
+(5.54)
+As (2.7) is assumed, Lemma 2.10 (vi) and (5.18) with θ = α − αν yield the following:
+E
+sup
+0≤s<t≤τ
+∥T 2
+t f − T 2
+2 f∥p
+Hγ−2ν
+p
+|t − s|αµp−1
+≤ N∥f∥p
+Hγ−2
+p
+(τ).
+Furthermore, by (2.7), Lemma 2.10 (vi), and (5.20) with θ = α(1 − c0/2) − αν, we have
+E
+sup
+0≤s<t≤T
+��T 3
+t g − T 3
+s g
+��p
+Hγ−2ν
+p
+(l2)
+|t − s|αµp−1
+≤ N∥g∥p
+Hγ−2+c0
+p
+(τ,l2).
+The theorem is proved.
+□
+Remark 5.8. It should be noted that if u0 = 0, we can consider δ = 0 in Theorem 2.16 (ii)
+since an estimate of T 1
+t u0 (5.54) is not required.
+References
+[1] Garra, Roberto. Fractional-calculus model for temperature and pressure waves in fluid-saturated porous
+rocks. Physical Review E, 84(3):036605, 2011.
+[2] Keller, Jakob J. Propagation of simple non-linear waves in gas filled tubes with friction. Zeitschrift f¨ur
+angewandte Mathematik und Physik ZAMP, 32(2):170–181, 1981.
+[3] Tayyaba Akram, Muhammad Abbas, Muhammad Riaz, Ahmad Ismail, and Norhashidah Ali. An effi-
+cient numerical technique for solving time fractional burgers equation. Alexandria Engineering Journal,
+59(4):2201–2220, 2020.
+[4] Ricardo Almeida. A Gronwall inequality for a general caputo fractional operator. Mathematical Inequal-
+ities Applications, 20(4), 2017.
+[5] Ronald Bagley and Peter Torvik. Fractional calculus in the transient analysis of viscoelastically damped
+structures. AIAA journal, 23(6):918–925, 1985.
+[6] Dumitru Baleanu, Kai Diethelm, Enrico Scalas, and Juan Trujillo. Fractional calculus: models and
+numerical methods, volume 3. World Scientific, 2012.
+[7] Zhen-Qing Chen, Kyeong-Hun Kim, and Panki Kim. Fractional time stochastic partial differential
+equations. Stochastic Processes and their Applications, 125(4):1470–1499, 2015.
+[8] Hongjie Dong and Doyoon Kim. Lp-estimates for time fractional parabolic equations with coefficients
+measurable in time. Advances in Mathematics, 345:289–345, 2019.
+[9] Talaat El-Danaf and Adel Hadhoud. Parametric spline functions for the solution of the one time frac-
+tional burgers’ equation. Applied Mathematical Modelling, 36(10):4557–4564, 2012.
+[10] Alaattin Esen and O34589631333 Tasbozan. Numerical solution of time fractional burgers equation.
+Acta Universitatis Sapientiae, Mathematica, 7(2):167–185, 2015.
+[11] Massimiliano Giona and Hector Eduardo Roman. Fractional diffusion equation for transport phenomena
+in random media. Physica A: Statistical Mechanics and its Applications, 185(1-4):87–97, 1992.
+[12] Loukas Grafakos. Modern fourier analysis, volume 250. Springer, 2009.
+
+42
+BEOMSEOK HAN
+[13] Istv´an Gy¨ongy. Existence and uniqueness results for semilinear stochastic partial differential equations.
+Stochastic Processes and their Applications, 73(2):271–299, 1998.
+[14] Istv´an Gy¨ongy and David Nualart. On the stochastic Burgers’ equation in the real line. The Annals of
+Probability, 27(2):782–802, 1999.
+[15] Beom-Seok Han. Lp-regularity theory for semilinear stochastic partial differential equations with mul-
+tiplicative white noise. Journal of Mathematical Analysis and Applications, page 126366, 2022.
+[16] Beom-Seok Han. A regularity theory for stochastic generalized Burgers’ equation driven by a multiplica-
+tive space-time white noise. Stochastics and Partial Differential Equations: Analysis and Computations,
+pages 1–41, 2022.
+[17] Richard Herrmann. Fractional calculus: an introduction for physicists. World Scientific, 2011.
+[18] Rudolf Hilfer. Applications of fractional calculus in physics. World scientific, 2000.
+[19] Mitsunojo Ichise, Yutaka Nagayanagi, and Tsugio Kojima. An analog simulation of non-integer or-
+der transfer functions for analysis of electrode processes. Journal of Electroanalytical Chemistry and
+Interfacial Electrochemistry, 33(2):253–265, 1971.
+[20] Mustafa Inc. The approximate and exact solutions of the space-and time-fractional Burgers equations
+with initial conditions by variational iteration method. Journal of Mathematical Analysis and Applica-
+tions, 345(1):476–484, 2008.
+[21] Anatoly Kilbas, Oleg Marichev, and Stefan Samko. Fractional integrals and derivatives (theory and
+applications), 1993.
+[22] Ildoo Kim, Kyeong-Hun Kim, and Sungbin Lim. An Lq(Lp)-theory for the time fractional evolution
+equations with variable coefficients. Advances in Mathematics, 306:123–176, 2017.
+[23] Ildoo Kim, Kyeong-Hun Kim, Sungbin Lim. A sobolev space theory for stochastic partial differential
+equations with time-fractional derivatives. The Annals of Probability, 47(4):2087–2139, 2019.
+[24] Kyeong-Hun Kim and Sungbin Lim. Asymptotic behaviors of fundamental solution and its derivatives
+related to space-time fractional differential equations. Journal of the Korean Mathematical Society,
+53(4):929–967, 2015.
+[25] Kyeong-Hun Kim and Daehan Park. A sobolev space theory for the time-fractional stochastic partial
+differential equations driven by levy processes. arXiv preprint arXiv:2006.05050, 2020.
+[26] Nicolai Krylov. An analytic approach to SPDEs. Stochastic partial differential equations: six perspec-
+tives, 64:185–242, 1999.
+[27] Nicolai Krylov. Lectures on elliptic and parabolic equations in Sobolev spaces, volume 96. American
+Mathematical Society, 2008.
+[28] Nicolai Krylov. Introduction to the theory of random processes, volume 43. American Mathematical
+Society, 2002.
+[29] Peter Lewis and David Nualart. Stochastic Burgers’ equation on the real line: regularity and moment
+estimates. Stochastics, 90(7):1053–1086, 2018.
+[30] Dongfang Li, Chengjian Zhang, and Maohua Ran. A linear finite difference scheme for generalized time
+fractional Burgers’ equation. Applied Mathematical Modelling, 40(11-12):6069–6081, 2016.
+[31] Richard Magin. Fractional calculus in bioengineering, part 1. Critical Reviews™ in Biomedical Engi-
+neering, 32(1), 2004.
+[32] Igor Podlubny. Fractional differential equations: an introduction to fractional derivatives, fractional
+differential equations, to Methods of their solution and some of their applications. ISSN. Elsevier Science,
+1998.
+[33] Hun H Sun, Banu Onaral, and Yuan-Ying Tso. Application of the positive reality principle to metal
+electrode linear polarization phenomena. IEEE Transactions on Biomedical Engineering, (10):664–674,
+1984.
+[34] Hans Triebel. Theory of Function Spaces. Modern Birkh¨auser Classics. Springer Basel, 2010.
+[35] John Walsh. An introduction to stochastic partial differential equations. In ´Ecole d’ ´Et´e de Probabilit´es
+de Saint Flour XIV-1984, pages 265–439. Springer, 1986.
+[36] Guang-an Zou and Bo Wang. Stochastic Burgers’ equation with fractional derivative driven by multi-
+plicative noise. Computers & Mathematics with Applications, 74(12):3195–3208, 2017.
+Department of Mathematics, Pohang University of Science and Technology, 77, Cheongam-
+ro, Nam-gu, Pohang, Gyeongbuk, 37673, Republic of Korea
+Email address: hanbeom@postech.ac.kr
+

8tE2T4oBgHgl3EQfPwar/content/tmp_files/2301.03763v1.pdf.txt

@@ -0,0 +1,2171 @@
+arXiv:2301.03763v1  [quant-ph]  10 Jan 2023
+Quantum Speedups for Zero-Sum Games
+via Improved Dynamic Gibbs Sampling
+Adam Bouland
+Yosheb Getachew
+Yujia Jin
+Aaron Sidford
+Kevin Tian∗
+{abouland,yoshebg,yujiajin,sidford}@stanford.edu, tiankevin@microsoft.com
+Abstract
+We give a quantum algorithm for computing an ǫ-approximate Nash equilibrium of a zero-
+sum game in a m × n payoff matrix with bounded entries. Given a standard quantum oracle
+for accessing the payoff matrix our algorithm runs in time �O(√m + n · ǫ−2.5 + ǫ−3) and outputs
+a classical representation of the ǫ-approximate Nash equilibrium. This improves upon the best
+prior quantum runtime of �O(√m + n·ǫ−3) obtained by [vAG19] and the classic �O((m+n)·ǫ−2)
+runtime due to [GK95] whenever ǫ = Ω((m + n)−1). We obtain this result by designing new
+quantum data structures for efficiently sampling from a slowly-changing Gibbs distribution.
+∗Work partly completed while at Stanford.
+
+1
+Introduction
+There is now a broad family of quantum algorithms for machine learning and fast numerical linear al-
+gebra [BWP+17], built on many quantum algorithmic primitives, e.g. [BHMT02, HHL09, GSLW19].
+More specifically, for a wide range of problems it has been shown how quantum algorithms can (in
+certain parameter regimes) yield faster runtimes.1 These quantum algorithms obtain runtimes which
+improve upon the dimension dependence of classical algorithms, but often at the cost of a worse
+dependence on the error tolerance and/or implicit access to the solution (e.g. query or sampling
+access for solution entries). Consequently, this paper is motivated by the following question.
+To what degree is there an inherent accuracy versus dimension-dependence tradeoff for
+quantum optimization algorithms? What algorithmic techniques improve this tradeoff?
+In this paper we consider this question for the fundamental optimization problem of computing
+ǫ-approximate Nash equilibrium in zero-sum games. Our main result is an improved dependence
+on ǫ for quantum algorithms solving zero-sum games, which is very close to that of its classical
+counterpart. Further, we show that for our algorithms, obtaining a classical representation of the
+solution is obtainable at no additional asymptotic cost. Our work builds upon [vAG19, LCW19],
+which already took a large and important step towards answering the above question by designing
+quantum data structures for efficiently implementing algorithms for solving zero-sum games.
+Interestingly, to obtain our result we provide improved quantum algorithms for solving a dy-
+namic data structure problem of sampling from a slowly-changing Gibbs distribution. Such dynamic
+sampling problems arise as a natural component of stochastic gradient methods for solving zero-sum
+games. We obtain our speedups by improving a Gibbs sampling subroutine developed in [vAG19].
+We design a new dynamic quantum data structure which performs the necessary Gibbs sampling in
+time �O(ǫ− 1
+2), which is faster than the corresponding �O(ǫ−1) runtime achieved by [vAG19]. Beyond
+the intrinsic utility of solving this problem, we hope our improved Gibbs sampler showcases poten-
+tial algorithmic insights that can be gleaned by seeking improved error dependencies for quantum
+optimization algorithms. Moreover, we hope this work encourages the study and design of quantum
+data structures for efficient optimization.
+1.1
+Zero-sum games
+For matrix A ∈ Rm×n its associated zero-sum game is the pair of equivalent optimization problems
+min
+u∈∆m max
+v∈∆n u⊤Av = max
+v∈∆n min
+u∈∆m u⊤Av, where ∆k := {x ∈ Rk
+≥0 : �
+i∈[k] xi = 1}.
+In such a game, we refer to A as the payoff matrix and view the m and n-dimensional simplices, i.e.
+∆m and ∆n, as the space of distributions over [m] and [n] respectively. From this perspective u⊤Av,
+known as payoff or utility of (u, v), is the expected value of Aij when sampling i ∈ [m] and j ∈ [n]
+independently from the distributions corresponding to u and v. Thus, a zero-sum game models a
+two-player game where a minimization player seeks to minimize the payoff while, simultaneously, a
+maximization player seeks to maximize it.
+In this paper, we consider the canonical problem of computing an approximate Nash equilibrium
+of a zero-sum game. Given the payoff matrix A ∈ Rm×n we call a pair (u, v) ∈ ∆m × ∆n an ǫ-
+approximate Nash equilibrium (NE) for ǫ ∈ R>0 if
+�
+max
+v′∈∆n u⊤Av′
+�
+−
+�
+min
+u′∈∆m(u′)⊤Av
+�
+≤ ǫ.
+1Note that quantifying the end-to-end speedups obtained by these methods can be subtle due to I/O overheads,
+different access models [Aar15], and classical de-quantization algorithms [Tan19, CGL+20, GLG22].
+1
+
+We assume that the payoff matrix A and the error-tolerance are given as input to an algorithm, and
+that, for simplicity, ∥A∥max ≤ 1, i.e. the largest entry of A has magnitude at most 1 (this is without
+loss of generality by rescaling A ← ∥A∥���1
+max A and ǫ ← ∥A∥−1
+max ǫ). The main goal of this paper is
+to design improved zero-sum game solvers, i.e. algorithms that compute ǫ-approximate NEs.
+Zero-sum games are foundational to theoretical computer science, optimization, and economics.
+The problem of approximately solving zero-sum games is a natural formulation of approximate linear
+programming (LP) and correspondingly, this problem is a prominent testbed for new optimization
+techniques. Over the past decades there have been numerous advances in the computational com-
+plexity of solving zero-sum games under various assumptions on problem parameter (see Section 1.3
+for a survey). Recent advancements in interior point methods (IPMs) for linear programming, e.g.
+[vdBLL+21] and references therein (discussed in more detail in Section 1.3), solve zero sum-games
+in time �O(mn + min(m, n)2.5).2 Further the linear programming algorithm of [vdB20], shows that
+zero-sum games can be solved deterministically in �O((m+n)ω) time where ω < 2.373 is the current
+matrix multiplication constant [AW21], or �O((m + n)3) without fast matrix multiplication. In this
+paper, we primarily focus on sublinear-time algorithms for approximating NEs.
+A well-known algorithm by [GK95] achieves a runtime of �O((m + n) · ǫ−2), which is the state-
+of-the-art sublinear runtime amongst classical algorithms, without further problem assumptions.
+Recently it has been shown that quantum algorithms can yield strikingly runtime improvements for
+solving zero-sum games and their generalizations [LCW19, vAG19, LWCW21]. In particular, in 2019
+Li, Chakrabati and Wu [LCW19] gave a quantum algorithm for zero sum games in time �O(√m + n·
+ǫ−4), and simultaneously van Apeldoorn and Gilyen [vAG19] gave an algorithm running in time
+�O(√m + n · ǫ−3). These algorithms yield a quadratic improvement in the dimension dependence of
+the best classical algorithm, at the cost of a higher error dependence.
+The algorithms of [LCW19, vAG19, LWCW21] operate using a standard quantum oracle for A
+(formally stated in Section 2), in which one can query the entries of A in superposition. We focus on
+the algorithm of [vAG19] for the rest of this paper, as we focus on improving error dependence. The
+[vAG19] algorithm generalizes the classical algorithm of Grigoriadis and Khachiyan [GK95], and
+obtains a runtime improvement by speeding up a key dynamic Gibbs sampling subroutine required
+by the [GK95] method. As we discuss in greater detail in Section 3, van Apeldoorn and Gilyen give
+a quantum data structure to efficiently perform this sampling in time quadratically faster in the
+dimension, which lies at the core of their algorithmic speedup.
+Our result.
+We give a new quantum algorithm for solving zero-sum games which improves upon
+the runtime of the prior state-of-the-art quantum algorithm, due to [vAG19].
+Theorem 1 (informal, see Theorem 4). Let A ∈ Rm×n with ∥A∥max ≤ 1, and ǫ ∈ (0, 1). Given
+a quantum oracle for A (defined in Section 2), there is an �O(√m + n · ǫ−2.5 + ǫ−3) time algorithm
+which yields a classical output (u, v) ∈ ∆m × ∆n that is an ǫ-approximate NE with high probability.
+Our new algorithm simultaneously improves the best known quantum [vAG19] and classical
+[GK95] algorithms in the parameter regime where IPMs do not dominate sublinear algorithms. In
+particular, it is faster than the classical �O((m+n)·ǫ−2) runtime of [GK95] whenever ǫ−1 = �O(m+n),
+which includes the regime where [GK95] offers advantages over the �O((m + n)ω) runtime of the
+[vdB20] IPM, as ω < 3. This is in contrast to the prior quantum rate of [vAG19], which does
+not achieve an improvement upon [GK95] in the full parameter range where sublinear algorithms
+2We use the �O notation to hide polylogarithmic dependences on problem parameters when convenient for exposi-
+tion; see Section 2 for a more detailed statement of hidden parameters. In informal theorem statements, we use “with
+high probability” to indicate a polylogarithmic dependence on the failure probability.
+2
+
+are currently preferable to IPMs.
+For example, when m ≈ n and (up to logarithmic factors)
+ǫ ∈ [n−c, n− 1
+2 ] where c = 1
+2(ω − 1), the rate of [GK95] is favorable to that of [vAG19] and state-of-
+the-art IPMs [vdB20, CLS21].3
+1.2
+Dynamic Gibbs sampling
+We obtain the improved error dependence in our zero-sum game solver by producing a new, faster
+quantum data structure to perform the Gibbs sampling as used in the algorithm of [vAG19], which
+may be of independent interest. Gibbs sampling is a fundamental algorithmic primitive — the basic
+task is, given vector v ∈ Rn, sample from the probability distribution proportional to exp(v). Gibbs
+sampling is used as a subroutine in many quantum and classical optimization algorithms, e.g. [BS17]
+and follow-up works. In general, quantum algorithms can perform this task more efficiently using
+amplitude estimation, which can boost the acceptance probability of rejection sampling schemes.
+This strategy was implemented in [vAG19], which approximate the maximum entry vmax of v
+using quantum maximum finding [DH96], uniformly sample i ∈ [n], and accept the sample with
+probability exp(vi −vmax) ≤ 1 using quantum rejection sampling. We give a more detailed overview
+of the [vAG19] Gibbs sampler and its complexity analysis in Section 3.2.
+We give a data structure which quadratically improves the error dependence of the [vAG19]
+Gibbs sampling subroutine runtime, from �O(√m + n· ǫ−1) per sample to an amortized �O(√m + n ·
+ǫ− 1
+2) per sample. A key fact which enables this improvement is that the Gibbs distributions one
+samples from in the zero-sum game solver of [GK95] change slowly over time: the base vector v
+receives bounded sparse updates in each iteration. By storing partial information about the Gibbs
+distribution, namely an efficiently-computable overestimate to its entries which remains valid across
+many consecutive iterations, we obtain an improved dynamic Gibbs sampler, which we also provide
+a detailed overview of in Section 3.2.
+We now define our notion of an approximate Gibbs sampler, and then state the dynamic sampling
+problem we consider, which arises naturally in zero-sum game algorithms with sublinear runtimes.
+Definition 1 (Approximate Gibbs oracle). For v ∈ Rn, its associated Gibbs distribution is pv ∈ ∆n
+such that for all i ∈ [n], [pv]i ∝ exp(vi). We say Ogibbs
+v
+is a δ-approximate Gibbs oracle if it samples
+from ˜p ∈ ∆n with ∥˜p − pv∥1 ≤ δ.
+Problem 1 (Sampling maintenance). Let η > 0, δ ∈ (0, 1), and suppose we have a quantum oracle
+for A ∈ Rm×n. Consider a sequence of T Update operations to a dynamic vector x ∈ Rm
+≥0, each
+of the form xi ← xi + η for some i ∈ [m]. In the sampling maintenance problem, in amortized
+Tupdate time per Update we must maintain a δ-approximate Gibbs oracle, Osamp, for A⊤x which is
+queryable in worst-case time Tsamp.
+Our result.
+We provide a quantum algorithm for solving Problem 1, which improves upon the
+runtime implied by the corresponding component in the algorithm of [vAG19].
+Theorem 2 (informal, see Theorem 3). There is a quantum algorithm which solves Problem 1 with
+high probability with max(Tsamp, Tupdate) = �O
+�√n · Tη1.5�
+and an initialization cost of �O
+�
+η3T 3�
+.
+Theorem 2 improves upon the solution to the sampling maintenance Problem 1 implied by
+[vAG19] by a η− 1
+2 factor; in the setting of the [GK95] solver, where T = �O(ǫ−2) and η = Θ(ǫ),
+this is an ǫ− 1
+2-factor improvement.
+At a high level, our improvement is obtained by storing a
+hint consisting of a vector which overestimates the true Gibbs distribution, and an approximate
+3There is evidence that ω = 2 cannot be achieved with current techniques, e.g. [Alm21].
+3
+
+Table 1: Algorithms for computing ǫ-approximate Nash equilibria of zero-sum games.
+Hides polylogarithmic factors and assumes A ∈ Rm×n with ∥A∥max ≤ 1.
+Method
+Query model
+Total runtime
+interior point method [CLS21]
+classical
+max(m, n)ω
+interior point method [vdBLL+21]
+classical
+mn + min(m, n)2.5
+extragradient [Nem04, Nes07]
+classical
+mn · ǫ−1
+stochastic mirror descent (SMD) [GK95]
+classical
+(m + n) · ǫ−2
+variance-reduced SMD [CJST19]
+classical
+mn +
+�
+mn(m + n) · ǫ−1
+[vAG19]
+quantum
+√
+m + n · ǫ−3
+Theorem 1 (our work)
+quantum
+√
+m + n · ǫ−2.5 + ǫ−3
+Table 2: Solutions to Problem 1, T = ǫ−2, η = ǫ. Hides polylogarithmic factors.
+Method
+Query model
+Tsamp
+Tupdate
+explicit updates [GK95]
+classical
+1
+m + n
+max-based rejection sampling [vAG19]
+quantum
+√
+m + n · ǫ−1
+√
+m + n · ǫ−1
+Theorem 2 (our work)
+quantum
+√
+m + n · ǫ− 1
+2
+√
+m + n · ǫ− 1
+2
+normalization factor, which are infrequently updated. Our maintained hint satisfies the desirable
+properties that: (i) it remains valid for a batch of consecutive iterations, and (ii) the degree of
+overestimation is bounded. The former property ensures a fast amortized update time, and the
+latter ensures a fast sample time by lower bounding the acceptance probability of our quantum
+rejection sampler. Our high-level strategy for maintaining improved hints is to repeatedly call our
+sampling access to accurately estimate large entries of the Gibbs distribution, and to exploit stability
+of the distribution under the setting of Problem 1. We discuss our dynamic Gibbs sampler in more
+detail and compare it with previous methods for solving Problem 1 in Section 3.2.
+The initialization cost of Theorem 2 is due to the current state-of-the-art in numerically stable
+implementations of the quantum singular value transformation (SVT) framework of [GSLW19].
+This cost is also the cause of the additive �O(ǫ−3) term in Theorem 1.
+We discuss this cost in
+Appendix D; improvements to numerically stable implementations of [GSLW19] would be reflected
+in the runtimes of Theorems 1 and 2.
+1.3
+Related work
+Quantum optimization and machine learning.
+There are a wide array of quantum algorithms
+for optimization and machine learning which make use of fundamental algorithmic primitives such
+as amplitude amplification [BHMT02], the HHL algorithm [HHL09], and the quantum singular
+value transformation [GSLW19]. For example, a number of works gave HHL-based algorithms for
+a variety of machine learning tasks such as PCA [LMR14], SVMs [RML14], and recommendation
+systems [KP16]. For more details see the survey article of [BWP+17].
+Most relevant to our current work are quantum algorithms for optimization problems.
+For
+example, Brandao and Svore [BS17] gave a quantum algorithm for SDP solving based on the Arora-
+4
+
+Kale algorithm [AK07], which was later improved by [VAGGdW20b]. There have also been quantum
+IPM-based methods for LPs and SDPs [KP20].
+Additionally a series of works have considered
+quantum algorithms for general convex optimization [CCLW20, vAGGdW20a], which make use of
+Jordan’s algorithm for fast gradient estimation [Jor05, GAW19].
+In the area of zero-sum games, in addition to the works previously mentioned [vAG19, LCW19]
+on ℓ1-ℓ1 games (where both players are ℓ1-constrained), there have been several works considering
+different variants of zero-sum games. For example Li, Chakrabati and Wu [LCW19] gave quan-
+tum algorithms for ℓ2-ℓ1 games with quadratic improvement on the dimension. Later Li, Wang,
+Chakrabati and Wu [LWCW21] extended this algorithm to more general ℓq-ℓ1 games with q ∈ (1, 2].
+Zero-sum games.
+Zero-sum games are a canonical modeling tool in optimization, economics and
+machine learning [Neu28]. The classic extragradient (mirror prox) method [Nem04, Nes07] computes
+an ǫ-approximate NE in �O(mn · ǫ−1) time; as discussed previously, the stochastic mirror descent
+method of [GK95] obtains the same accuracy in time �O((m + n) · ǫ−2). An intermediate runtime
+was recently obtained by [CJST19] using variance reduction, described in Table 1.
+Improved runtimes are available under more fine-grained characterizations of the matrix A, such
+as sparsity (e.g. number of nonzero entries per row or column) or numerical sparsity (e.g. rows and
+columns with bounded ℓ1-to-ℓ2 norm ratios) [CJST20]. Notably, the [GK95] algorithm also offers
+runtime improvements under a sparsity assumption, as does the algorithm of [vAG19] in certain
+sparsity-to-accuracy ratio regimes. In this paper, we focus on NE algorithms in the general setting
+(without further sparsity or numerical sparsity assumptions).
+In parallel, a long line of research improving IPMs for solving linear programming [Kar84,
+Ren88, LS14, LS19, vdBLSS20, JSWZ21] has led to a number of different zero-sum game solvers
+with polylogarithmic runtime dependencies on the problem accuracy ǫ. The current state-of-the-
+art variants of IPMs are [CLS21] and [vdBLL+21], which achieve runtimes of �O(max(m, n)ω) and
+�O(mn + min(m, n)2.5) respectively. We refer readers to Table 1 for detailed comparisons. Finally,
+for strongly polynomial runtimes (i.e. with no dependence on ǫ), which are outside the scope of this
+paper, we refer readers to [DNV20] and references therein.
+1.4
+Future work
+Theorem 1’s ǫ dependence is within an ǫ− 1
+2 factor of matching classical counterparts. To the best
+of our knowledge, removing this ǫ− 1
+2 overhead would represent the first quantum algorithm for a
+natural optimization problem which improves upon classical counterparts across all parameters.
+Both our work and [vAG19] solve Problem 1 by leveraging a powerful polynomial approximation-
+based technique developed in [GSLW19], known as the quantum singular value transform (QSVT). In
+both cases, QSVT is used with a polynomial of degree �O(ǫ−1). We note that in closely-related classi-
+cal settings (discussed in [SV14]), Chebyshev polynomial-based approximations yield a quadratically
+smaller degree. However, a boundedness requirement (due to the spectra of quantum gates) pre-
+vents straightforwardly applying these constructions within QSVT. Sidestepping this barrier is a
+natural avenue towards improving our work, which we leave as an open problem.
+More generally, establishing optimal oracle query complexities of dynamic Gibbs sampling (e.g.
+Problem 1) and solving zero-sum games are key problems left open by our work. These questions
+are potentially more approachable than establishing tight time complexity characterizations. For
+example, could max(Tsamp, Tupdate) be improved to �O(√n) in the context of Theorem 1, or can we
+rule out such an improvement in the query model?
+5
+
+1.5
+Organization
+In Section 2 we state the notation used throughout the paper, as well as the (classical and quantum)
+computational models we assume.
+In Section 3, we give a brief technical overview of the core
+components of our algorithm used to prove Theorem 1: the stochastic gradient method our method
+is built on, and an efficient quantum implementation of a key subroutine using a new dynamic Gibbs
+sampler. Finally in Section 4 we give our new quantum sampler, and prove Theorem 2.
+We aim to give a self-contained, but simplified, description of our algorithm in Section 3 to
+improve the readability of the paper for readers with an optimization background unfamiliar with
+quantum computing, and vice versa. In particular, we abstract away the core optimization machin-
+ery (stochastic mirror descent) and quantum machinery (quantum SVT) developed in prior work
+into the statements of Propositions 1 and 2, and focus on how we use these statements black-box
+to build a faster algorithm. The proofs of these statements can be found in Appendices A and B.
+2
+Preliminaries
+General notation.
+�O hides logarithmic factors in problem dimensions (denoted m and n), target
+accuracies (denoted ǫ), and failure probabilities (denoted α). When discussing runtimes for Prob-
+lem 1, we additionally use �O to hide logarithmic factors in the parameters η, T. For all i ∈ [n] we let
+ei ∈ Rn denote the ith standard basis vector for i ∈ [n] when n is clear. ∥·∥p denotes the ℓp norm of
+a vector. For A ∈ Rm×n, its ith row and jth column are respectively Ai:, A:j. For v ∈ Rn, diag (v)
+is the diagonal n × n matrix with v as the diagonal. Conjugate transposes of A are denoted A∗;
+when the matrix is real we use A⊤. The all-ones and all-zeros vectors of dimension n are 1n and
+0n. Finally, throughout a := ⌈log2 m⌉ and b := ⌈log2 n⌉, so [m] ⊆ [2a] and [n] ⊆ [2b].
+Computation models.
+We assume entries of A are w-bit reals for w = O(log(mn)), and work in
+the word RAM model where w-bit arithmetic operations take O(1) time; for simplicity, we assume
+mathematical operations such as trigonometric functions and radicals can also be implemented ex-
+actly for w-bit words in O(1) time. Throughout, “quantum states” mean unit vectors, and “quantum
+gates” or “oracles” O mean unitary matrices. We follow standard notation and identify a standard
+basis vector ei for i ∈ [n] with |i⟩, an a-qubit state, in which i is represented in binary (i.e. more for-
+mally, |i⟩ = |bin(i)⟩, and bin is omitted for brevity). We consider the standard model of quantum
+access to oracles, in which the oracle O, which is defined by its operation on |s⟩ for all {0, 1}∗-
+valued s (where length is clear from context), can be queried in superposition. If O is queried on
+|v⟩ := �
+s αs|s⟩, the result is O|v⟩ = �
+s αi(O|s⟩). We use |g⟩, |g′⟩, etc. (when clear from context)
+to denote arbitrary sub-unit vectors, which represent garbage states (unused in computations). The
+tensor product of states |u⟩ and |v⟩ on a and b qubits is denoted |u⟩|v⟩, an (a + b)-qubit state. The
+runtime of a quantum circuit is its maximum depth (in arithmetic gates on w-bit words).
+Access model.
+Throughout the paper, we assume a standard quantum oracle for accessing A
+(recall ∥A∥max ≤ 1). In particular, by a quantum oracle for A we mean an oracle OA which, when
+queried with |i⟩|j⟩|s⟩ for i ∈ [m], j ∈ [n], s ∈ {0, 1}w, reversibly writes Aij (in binary) to the third
+register in O(1) time, i.e. OA|i⟩|j⟩|s⟩ = |i⟩|j⟩|s ⊕ Aij⟩ where ⊕ is bitwise mod-2 addition.
+Given a quantum oracle for A, with two queries, by standard constructions one can construct
+an oracle which places the value in the amplitude of the state rather than the register itself. More
+6
+
+formally, one can construct4 an O′
+A, which operates as:
+O′
+A|0⟩|i⟩|j⟩ =
+�
+Aij|0⟩|i⟩|j⟩ +
+�
+1 − |Aij||1⟩|g⟩, for (i, j) ∈ [m] × [n].
+It is standard in the literature to (using ancilla qubits to store the output register where Aij is
+written) construct such an O′
+A from OA under our classical model of computation, see e.g. [GR02].
+For simplicity, we omit discussion of ancilla qubits in the remainder of the paper and assume direct
+access to O′
+A. We also note that there is ambiguity in the implementation of O′
+A in that the square
+root is not unique, and that we have control over the signing used in this implementation. We will
+use this flexibility crucially later in the paper, specifically Corollary 6.
+3
+Overview of approach
+In this section, we give an overview of the approach we take to prove our main results: an improved
+quantum runtime for solving zero-sum games (Theorem 4) and an improved quantum data structures
+for dynamic Gibbs sampling (Theorem 3). We organize this section as follows.
+In Section 3.1, we state Algorithm 1, the optimization method framework we use to solve zero-
+sum games. This framework is a generalization of the classical algorithm of [GK95]. We state its
+guarantees in Proposition 1 and defer the proof to Appendix A. Algorithm 1 assumes access to
+an approximate Gibbs oracle (Definition 1) for sampling from dynamic distributions as stated in
+Problem 1. The bulk of our work is devoted to obtaining an efficient quantum implementation of
+such an oracle (Theorem 3) and using this result we prove Theorem 4 at the end of Section 3.1.
+In Section 3.2, we overview the main technical innovation of this paper, an improved solution to
+Problem 1. Whereas prior work by [vAG19] solves Problem 1 at an amortized ≈ √m + n · ǫ−1 cost
+per iteration, we show how to solve the problem at an amortized ≈ √m + n · ǫ− 1
+2 cost. We remark
+that the only quantum components of our algorithm (quantum SVT and amplitude amplification)
+are abstracted away by Proposition 2, which is proven in Appendix B.
+3.1
+Solving matrix games with a Gibbs sampling oracle
+Our proof of Theorem 4 uses an efficient implementation of the algorithmic framework stated in
+Algorithm 1, based on stochastic mirror descent. In specifying Algorithm 1, we recall our earlier
+Definition 1, which captures the approximate sampling access we require for Algorithm 1’s execution.
+Algorithm 1: MatrixGameSolver(δ, η, T)
+1 Input: A ∈ Rm×n, desired accuracy ǫ ∈ (0, 1), δ-approximate Gibbs oracles for the
+(dynamic) vectors −A⊤xt and Ayt
+2 Parameters: Gibbs sampler parameter δ ∈ (0, 1), step size η > 0, iteration count T
+3 Initialize ˆu ← 0m, ˆv ← 0n, x0 ← 0m, and y0 ← 0n
+4 for t = 0 to T − 1 do
+5
+Independently sample jt, j′
+t ∈ [n] using Ogibbs
+−A⊤xt and it, i′
+t ∈ [m] using Ogibbs
+Ayt
+6
+Update yt+1 ← yt + ηejt and xt+1 ← xt + ηeit
+// Update iterates.
+7
+Update ˆu ← ˆu + 1
+T ei′
+t and ˆv ← ˆv + 1
+T ej′
+t
+// Update output.
+8 return (ˆu, ˆv)
+4This follows e.g. by calling the oracle to obtain the value of Aij in binary (interpreted as a signed number
+between 0 and 1), adding an ancilla qubit, performing arithmetric to compute the rotation angle needed on that
+ancilla, applying a tower of controlled rotation gates to an ancilla qubit using that rotation angle express in binary,
+then calling the standard oracle a second time to uncompute the binary value of Aij. See e.g. [GR02] for details.
+7
+
+The main skeleton of Algorithm 1 (Lines 5-6) using exact oracles is identical to the method of
+[GK95]. However, our framework builds upon [GK95] in the following three ways.
+1. We tolerate total variation error in the sampling procedure via δ-approximate Gibbs oracles.
+2. We provide a high-probability guarantee on the duality gap using martingale arguments.
+3. We subsample the output to obtain a sparse solution yielding a comparable duality gap.
+We remark that several of these improvements have appeared previously, either explicitly or
+implicitly, in the stochastic gradient method literature. For example, an approximation-tolerant
+stochastic gradient method was given in [CJST20], and our proofs of the high-probability guarantees
+are based on arguments in [AL17, CDST19]. For completeness we give a self-contained proof of the
+following guarantee on Algorithm 1 in Appendix A.
+Proposition 1. Let A ∈ Rm×n satisfy ∥A∥max ≤ 1 and ǫ, α ∈ (0, 1). Let δ ≤
+ǫ
+20, η =
+ǫ
+60, and
+T = Θ(ǫ−2 log mn
+α ) for an appropriate constant. With probability ≥ 1 − α, Algorithm 1 outputs an
+ǫ-approximate NE for A.
+Given Proposition 1 to obtain our faster zero-sum game solvers, we simply need to efficiently im-
+plement the Gibbs sampling in Line 5. As introduced in Section 1, Problem 1, describes a dynamic
+approximate Gibbs oracle sampling problem sufficient for this task. Indeed, solving two appropriate
+parameterizations of Problem 1 provides the oracles needed by Algorithm 1. By combining Propo-
+sition 1 with the following Theorem 3 (our solution to Problem 1, discussed in greater detail in
+Section 3.2), we prove our main result Theorem 4.
+Theorem 3. Let α ∈ (0, 1) and δ ≤ η. Given a quantum oracle for A ∈ Rm×n (defined in Section 2)
+with ∥A∥max ≤ 1, we can solve Problem 1 with probability ≥ 1 − α with
+max(Tsamp, Tupdate) = O
+�
+1 + √n · Tη log4 �mn
+δ
+�
+·
+��
+η log
+�nηT
+α
+�
++ η log
+�nηT
+α
+���
+,
+and an additive initialization cost of
+O
+�
+η3T 3 log4
+�nηT
+δ
+�
++ log7
+�nηT
+δ
+��
+.
+Theorem 4. Let A ∈ Rm×n satisfy ∥A∥max ≤ 1, and let ǫ, α ∈ (0, 1). Given a quantum oracle for A
+(defined in Section 2), there is a quantum algorithm which yields a classical output (u, v) ∈ ∆m×∆n
+that is an ǫ-approximate NE for A with probability ≥ 1 − α in time
+O
+�√m + n
+ǫ2.5
+log4 �mn
+ǫ
+�
+log2.5 �mn
+αǫ
+�
++
+√m + n
+ǫ2
+log4 �mn
+ǫ
+�
+log3 �mn
+αǫ
+�
++ 1
+ǫ3 log7 �mn
+ǫ
+��
+.
+Proof. We apply two instances of Theorem 3 to implement the δ-approximate Gibbs oracle for
+the dynamic vectors −A⊤xt and Ayt, to implement each iteration of Algorithm 1 in amortized
+O(1 + Tsamp + Tupdate) time. Using the settings of parameters T, η in Proposition 1 and setting
+δ = Θ(ǫ), which suffices for Algorithm 1 and Theorem 3, we have
+max(Tsamp, Tupdate) = O
+�√m + n
+ǫ
+log4 �mn
+ǫ
+�
+log
+�mn
+αǫ
+� �
+ǫ log
+�mn
+αǫ
+�
++
+�
+ǫ log
+�mn
+αǫ
+���
+.
+The conclusion follows since, by observation, Algorithm 1 costs O(T · (1 + Tsamp + Tupdate)). As
+remarked in the introduction, the additive term in the runtime comes from the cost of stably
+implementing a quantum circuit required in the use of Theorem 3 representing a polynomial trans-
+formation in finite precision, which we discuss in greater detail in Appendix D.
+8
+
+3.2
+Dynamic sampling maintenance via dynamic hint maintenance
+In this section, we overview our proof of Theorem 3, which proceeds in two steps.
+1. We reduce sampling maintenance (Problem 1) to a problem which we call hint maintenance.
+This latter problem is a specialization of the sampling maintenance problem where suitable
+advice, which we call the hint throughout, is provided.
+2. We show how to solve the hint maintenance problem required by Proposition 2 in Theorem 3,
+by recursively calling Proposition 2 in phases, allowing us to maintain hints of suitable quality.
+Reducing sampling maintenance to hint maintenance.
+First, we introduce the following
+data structure for maintaining the x variable in Problem 1, which was used crucially in [vAG19] for
+dynamic Gibbs sampling. This data structure allows efficient queries to subsets of the coordinates
+of x and we use it in our Gibbs sampler as well.
+Lemma 1 (Sampler tree). Let η ∈ R≥0 and m ∈ N. There is a classical data structure, SamplerTree,
+supporting a tree on O(m) nodes such that [m] corresponds to leaves, with the following operations.
+• Init(m, ηfixed): initialize x ← 0m and η ← ηfixed
+• Update(i): xi ← xi + η
+• SubtreeSum(v): return the sum of all xi, where i is in the subtree of v
+The total runtime of T calls to Update is O(T log m), and calls to SubtreeSum cost O(1).
+An implementation of SamplerTree based on propagating subtree sums upon updates is standard
+classical data structure, and we omit further description for brevity. Next, we state our first building
+block towards solving Problem 1, a result which can be thought of as quantum sampling with a hint.
+We defer its proof to Appendix B, as it is primarily based on generalizing dynamic block-encoding
+strategies with bounded-degree polynomial approximations, as pioneered by [GSLW19, vAG19].
+Proposition 2. Let x ∈ Rm
+≥0 correspond to an instance of SamplerTree, and β ≥ ∥x∥1. Let p be
+the Gibbs distribution associated with A⊤x, let Z := �
+j∈[n] exp([A⊤x]j) and �Z ∈ [Z, CZ] for some
+C ≥ 1. Finally, let q ∈ Rn have entries classically queriable in O(1) time, satisfy q ≥ p entrywise,
+qj ∈ [ δ
+n, 1] for all j ∈ [n], and ∥q∥1 = ρ. Suppose �Z, C, ρ, and β are explicitly known. Given
+a quantum oracle for A ∈ Rm×n (defined in Section 2) with ∥A∥max ≤ 1, we can implement a
+δ-approximate Gibbs oracle which has query cost O(√ρC · β log4 � Cmn
+δ
+�
+). The total additional cost
+incurred if x undergoes T Update calls which preserve the invariants on �Z, C, ρ, β is O(T log m).
+Proposition 2 makes use of an overestimating hint vector q and approximate normalization
+constant �Z, which we collectively call the hint. The acceptance probability of our rejection sampling
+is governed by two primary parameters: ρ = ∥q∥1, which reflects the degree of overestimation
+(and can be thought of as a hint quality), and C ≥ 1, which reflects our inability to accept with
+probability pj
+qj when p is implicit (which can be thought of as a normalization quality). In particular,
+the rejection sampling scheme used in Proposition 2 will instead accept with probability
+pj
+Cqj .5
+Here we elaborate briefly on the implementation of Proposition 2 (for more details, see Ap-
+pendix 4). We follow notation of Proposition 2, and also let w := A⊤x such that the unnormalized
+5Exactly computing Z may require time Ω(n) in standard implementations, an obstacle to runtimes ∝ √n.
+9
+
+Gibbs distribution is exp(w), and p = exp(w)
+Z
+. Proposition 2 is a rejection sampler which first loads
+the hint q into superposition, and then applies a filter. Overall, our scheme has the form
+sample j ∼ q
+ρ, then accept with probability exp(wj)
+CZ · qj
+= pj
+Cqj
+,
+(1)
+which results in an accepted sample with probability ≈
+1
+ρC , and hence requires ≈ √ρC trials to suc-
+ceed after applying quantum amplitude amplification, a generalization of Grover search [BHMT02].6
+The latter filtering step is implemented using appropriate block-encoding technology.
+The above discussion suggests that the hint and normalization qualities, parameterized by ρ
+and C, are crucial in controlling the acceptance probability of our scheme. More concretely, in
+our applications of Proposition 2, β = ηT = �O(1
+ǫ ), which is the bound on the ℓ1 norm of the
+xt and yt iterates in Algorithm 1 under the parameter settings of Proposition 1.
+Overall, the
+cost of implementing an approximate Gibbs oracle is then (up to logarithmic factors) √ρC · 1
+ǫ.
+Proposition 2 hence reduces Problem 1 to the problem of maintaining the hint consisting of a vector
+q and a normalization estimate �Z. We mention that Proposition 2 is a strict generalization of a
+corresponding building block in [vAG19], which only used q set to the all-ones vector.
+Approaches for Problem 1.
+We now overview our improved solution to Problem 1 via efficient
+use of Proposition 2. To motivate our solution, we outline three solutions to Problem 1 offering
+different tradeoffs in the overall quality ρC. The first only uses classical information and does not
+use Proposition 2 at all, the second uses Proposition 2 but maintains no history across iterates, and
+the third (building upon the first two) is our approach.
+Solution 1: [GK95]. A standard way to solve Problem 1 is to explicitly update w = A⊤x and
+exp(w), and exactly maintain the normalizing constant Z. This allows us to sample from p in �O(1)
+time. Since w changes by one row of A under a 1-sparse Update operation to x, this is implementable
+in O(n) time per iteration. We can view this as an instance of the scheme (1) with q = p, C = 1,
+and ρ = 1. It yields the (unbalanced) tradeoff for Problem 1 of Tsamp = �O(1) and Tupdate = O(n).
+Solution 2: [vAG19]. A recent work [vAG19] introduced a quantum implementation of the scheme
+(1) with an improved tradeoff. The [vAG19] scheme first uniformly samples, which in the language
+of (1) means q = 1n and ρ = n. It then applies quantum maximum finding [DH96] to obtain an
+approximate maximum entry of w, which they show takes time �O(β · √n); for the sake of simplicity
+here, we assume this exactly yields wmax := maxj∈[n] wj. Finally, the acceptance probability
+pj
+Cqj is
+set to exp(wj − wmax). For q = 1n, this translates to
+pj · exp(wmax − wj) = exp(wmax)
+Z
+≤ 1,
+implying C = 1 suffices.
+We note this bound on C can be tight when w is very non-uniform.
+Overall, the [vAG19] scheme’s update time requires maximum finding, and its sampling time (via
+Proposition 2) requires time �O(β · √ρC) = �O(β · √n).
+For β = �O(1
+ǫ) as in Algorithm 1, this
+yields the balanced tradeoff max(Tsamp, Tupdate) = �O
+�√n · ǫ−1�
+. As discussed earlier, our key in-
+sight is to improve upon this specific choice of hint in [vAG19], for their implicit use of Proposition 2.
+Solution 3: this work. We design better hints for Proposition 2 by executing our algorithm in phases
+corresponding to batches of ≈ 1
+η iterations. At the start of each phase, we use the Gibbs access
+6The β in Proposition 2 comes from loading exp(wj) into a quantum oracle via polynomials of degree ≈ β.
+10
+
+afforded by Proposition 2 to produce a suitable hint for efficiently implementing the next phase. Our
+execution of this strategy, parameterized by an integer k ∈ [n], relies on the following observations.
+1. During ⌈ 1
+η⌉ iterations t ∈ {τ + s}s∈[⌈ 1
+η ⌉] (where τ starts the phase), the dynamic Gibbs
+distribution pt (where t is the iteration index) changes by O(1) multiplicatively, since w
+entrywise changes by O(1) additively. Thus, the quality of a hint vector deteriorates by at
+most a constant in the phase, so it suffices to give a good hint qτ ≥ pτ at the phase start.
+2. By using access to Proposition 2 at the end of the previous phase, we can efficiently estimate
+large entries of pτ.
+More precisely, we sample �O(k) times from pτ, and let the empirical
+distribution of these samples be ˜q. Chernoff bounds show that any large entry [pτ]j = Ω( 1
+k)
+will be accurately reflected in the empirical sample. Hence, we set the hint to
+qj =
+�
+˜qj · O(1)
+˜qj = Ω( 1
+k)
+1
+k · O(1)
+˜qj = O( 1
+k) ,
+for appropriate constants. This yields an improved hint quality of ρ ≈ n
+k , since large entries
+of the hint sum to at most O(1) (as ˜qj ≈ pj), and small entries sum to O(n
+k ).
+3. We show a similar strategy of using empirical concentration, combined with a testing variant
+of Proposition 2, accurately estimates the normalizing factor Z, yielding C = O(1).
+This strategy yields Tsamp = �O(β ·
+�
+n/k) and Tupdate = �O(Tsamp · kη) (since we amortize Tupdate
+over ≈ 1
+η iterations). For the parameter settings of Algorithm 1, optimizing k yields
+max(Tsamp, Tupdate) = �O
+�√n · ǫ− 1
+2
+�
+.
+We prove Theorem 3, our improved solution to Problem 1, in Section 4. Ignoring logarithmic fac-
+tors and assuming η ≪ 1 (as in our setting), Theorem 3 shows we can maintain max(Tsamp, Tupdate) =
+�O(√n · Tη1.5). For the parameter settings T = �O(ǫ−2), η = Θ(ǫ), as stated in Proposition 1, this
+indeed equates to max(Tsamp, Tupdate) = �O(√n · ǫ− 1
+2).
+4
+Gibbs sampling oracle implementation
+In this section, we prove Theorem 3, which gives our solution to Problem 1. To do so, we follow the
+outline given in Section 3.2, wherein we solve Problem 1 in batches of ⌈ 1
+η⌉ iterations, each of which
+we call a “phase.” In Sections 4.1 and 4.2, we only discuss a single phase of Problem 1, consisting
+of the iterations τ + s for s ∈ [⌈ 1
+η⌉] and some initial iteration τ, assuming certain invariants (stated
+below) hold at the start of the phase. We give a complete solution to Problem 1 in Section 4.3.
+Invariant 1 (Approximate normalization access). We explicitly have �Zprev with �Zprev ∈ [Zτ, CZτ]
+for some C = O(1).
+Invariant 2 (Initial sampling maintenance). We have Oτ solving Problem 1 in iteration τ.
+The remainder of this section is then organized as follows.
+• Section 4.1: We show that assuming Invariants 1 and 2 hold at the start of a phase, we can
+perform preprocessing used to construct our hint, consisting of the estimated normalization
+�Z and vector q, in an application of Proposition 2. This gives the cost of Tsamp in Problem 1.
+11
+
+• Section 4.2: We show that at the conclusion of each phase we can maintain Invariants 1 and 2
+for use in the next phase. This gives the cost of Tupdate in Problem 1.
+• Section 4.3: We recursively call the subroutine of Sections 4.1 and 4.2 (which solves Problem 1
+for all the iterations τ + s where s ∈ [⌈ 1
+η⌉] for some τ) ≈ ηT times to prove Theorem 3.
+4.1
+Preprocessing and approximate Gibbs oracle implementation
+In this section, we show how to construct the “hint” q which will be used throughout a phase
+(starting in iteration τ) given access to Oτ, and bound ρ = ∥q∥1 which quantifies the quality of our
+hint, under the assumption that Invariants 1 and 2 hold in the phase. We first show a multiplicative
+stability property of the relevant Gibbs distributions in a phase.
+Lemma 2. For all s ∈ [⌈ 1
+η⌉], we have
+Zτ+s ∈
+�1
+3Zτ, 3Zτ
+�
+, and pτ+s ∈
+�1
+9pτ, 9pτ
+�
+entrywise.
+Proof. Let νt := exp(A⊤xt) for all t, such that pt = νt
+Zt . We have that for any j ∈ [n],
+[ντ+s]j
+[ντ]j
+= exp
+��
+A⊤ (xτ+s − xτ)
+�
+j
+�
+∈ [exp (− ∥A∥max ∥xτ+s − xτ∥1) , exp (∥A∥max ∥xτ+s − xτ∥1)]
+∈ [exp (−ηs) , exp (ηs)] ∈
+�1
+3, 3
+�
+.
+Similarly, Zτ+s ∈ [1
+3Zτ, 3Zτ], and combining yields the conclusion.
+Next, our computation of the overestimating vector q is parameterized by an integer k ∈ [n]
+which will be fixed throughout this section and Section 4.2. We will simply set q to be an upscaled
+variant of an empirical distribution of roughly k draws from Oτ.
+Lemma 3. Let k ∈ [n], α ∈ (0, 1), and suppose δ ≤
+1
+16k. Draw N = Θ(k log nηT
+α ) samples from
+Oτ for an appropriately large constant, and let ˜q ∈ ∆n be the empirical distribution over these N
+samples. Define B := {i ∈ [n] | ˜qi ≥
+1
+2k}. Then for
+qj =
+�
+18˜qj
+j ∈ B
+18
+k
+j ̸∈ B ,
+with probability ≥ 1 −
+α
+2⌈ηT⌉, ∥q∥1 = O(n
+k ) and q ≥ pτ+s entrywise, for all s ≤ 1
+η.
+Proof. The first conclusion ∥q∥1 = O(n
+k ) is immediate from the definition of q, since ∥q∥1 ≤ 18 ∥˜q∥1+
+18n
+k . In light of Lemma 2 (which holds deterministically), to show the second conclusion, it suffices
+to show that with the desired success probability, we have both
+2˜qj ≥ [pτ]j for all j ∈ B
+(2)
+and
+2
+k ≥ [pτ]j for all j ̸∈ B.
+(3)
+Denote α′ :=
+α
+2⌈ηT⌉ for notational convenience, and let ˜p denote the distribution of samples from Oτ,
+and recall that ∥˜p − pτ∥1 ≤
+1
+16k. Because we are taking Θ(k log n
+α′ ) samples from ˜p, we have by a
+standard Chernoff bound that with probability at least 1 − α′ (union bounding over all coordinates
+j ∈ [n]), both of the following hold.
+12
+
+1. For all j ∈ [n] such that ˜pj ≥
+1
+4k, ˜qj ≥ 2˜pj
+3 .
+2. For all j ∈ [n] such that ˜pj ≤
+1
+4k, ˜qj ≤
+1
+2k.
+We condition on these events for the remainder of the proof; we now show (2), (3) in turn.
+Proof of (2). To see (2), the second event above implies that if ˜pj ≤
+1
+4k, then j ̸∈ B. Hence, for
+all j ∈ B, we have ˜qj ≥ 2˜pj
+3 ≥ [pτ]j
+2
+since ∥˜p − pτ∥∞ ≤
+1
+16k ≤ 1
+4 ˜pj for all j ∈ B.
+Proof of (3). To see (3), suppose for contradiction that j ̸∈ B and [pτ]j > 2
+k. This implies that
+˜pj > 1
+k, and hence by the first event above, ˜qj ≥
+1
+2k, contradicting j ̸∈ B.
+Corollary 1. Assume that Invariants 1, 2 hold for the phase consisting of iterations τ +s, s ∈ [⌈ 1
+η⌉].
+We can solve Problem 1 for the phase with probability ≥ 1 −
+α
+2⌈ηT⌉, and
+Tsamp := O
+��n
+k · Tη log4 �mn
+δ
+��
+.
+Proof. We will run the algorithm described in the proof of Lemma 3, and condition on it succeeding,
+giving the failure probability. It then suffices to apply Proposition 2 with q defined in Lemma 3. For
+this q, we parameterize Proposition 2 with C = O(1) (see Invariant 1), ρ = O(n
+k ) (see Lemma 3),
+and β = Tη. It is clear the lower bound on entries of q in Proposition 2 holds.
+4.2
+Maintaining invariants
+We now show how to maintain Invariant 1 at iteration τ ′ := τ + ⌈ 1
+η⌉, for use in the next phase, and
+bound the cost of doing so. We note that Invariant 2 follows immediately from our construction in
+Corollary 1. First, by combining Lemma 2 with Invariant 1,
+Zτ ′ ∈
+� �Zprev
+3C , 3 �Zprev
+�
+.
+(4)
+This suggests that we may use 3 �Zprev = �Z for the next phase; however, this would lead to an
+exponential blowup in the multiplicative range C. To sidestep this, we develop a tester for a hidden
+parameter governing a success probability, which will be used to give a refined estimate �Z. We
+require the following corollary of Proposition 2, whose proof we defer to Appendix B.
+Corollary 2. Following notation of Proposition 2, let R :=
+�Z
+Z . There is a quantum oracle Otest
+which can be implemented under T Update calls to x in O(T log m) time, and has query cost
+O
+��
+ρC · β log4
+�Cmn
+ℓδ
+��
+.
+Furthermore, for explicitly known constants Cℓ and Cu, Otest returns “success” with probability p for
+Cℓ
+√Rρ ≤ p ≤
+Cu
+√Rρ.
+Corollary 2 differs from Proposition 2 in that it returns a Boolean-valued answer (as opposed to
+a sample from an approximate Gibbs distribution), and has a success probability parameterized by
+explicit constants. We now show how to use Corollary 2 to maintain Invariant 1.
+13
+
+Lemma 4. Assume Invariants 1, 2 hold for the phase consisting of iterations τ + s, s ∈ [⌈ 1
+η⌉], and
+suppose C ≥ 4C2
+u
+C2
+ℓ
+for C = O(1), where Cu and Cℓ are the constants from Corollary 2. Further,
+suppose we have obtained q satisfying the conclusion of Lemma 3 (i.e. that the algorithm in Lemma 3
+succeeded). We can determine �Z such that �Z ∈ [Zτ ′, CZτ ′] with probability ≥ 1 −
+α
+2⌈ηT⌉, in time
+O
+��n
+k · Tη log4 �mn
+δ
+�
+log
+�ηT
+α
+��
+.
+Proof. Define �Z0 := 3 �Zprev, R0 :=
+�Z0
+Zτ′ , and note that �Z0 ∈ [Zτ ′, 9CZτ ′] by Invariant 1 and Lemma 2.
+Next, assuming the success of Lemma 3, we have that the success probability p of Otest from
+Corollary 2 using the estimate �Z0 satisfies (for the unknown R0 ∈ [1, 9C], and known Cℓ, Cu, ρ)
+Cℓ
+√R0ρ ≤ p ≤
+Cu
+√R0ρ.
+For N := 27 log 4⌈ηT⌉
+α
+· 3√Cρ
+Cℓ , we first run Otest N times and check the number of successes, denoted
+by S, which fits within the runtime budget by Corollary 2. By a Chernoff bound, we have that with
+probability ≥ 1 −
+α
+2⌈ηT⌉, we have
+54 log 4⌈ηT⌉
+α
+·
+�
+C
+R0
+≤ 2
+3pN ≤ S ≤ 4
+3pN ≤ 108 log 4⌈ηT⌉
+α
+· Cu
+Cℓ
+·
+�
+C
+R0
+.
+Hence, we can determine the quantity R0 up to a multiplicative factor of 4C2
+u
+C2
+ℓ
+≤ C, which also
+implies the same multiplicative approximation factor for Zτ ′, as desired.
+4.3
+Proof of Theorem 3
+Theorem 3. Let α ∈ (0, 1) and δ ≤ η. Given a quantum oracle for A ∈ Rm×n (defined in Section 2)
+with ∥A∥max ≤ 1, we can solve Problem 1 with probability ≥ 1 − α with
+max(Tsamp, Tupdate) = O
+�
+1 + √n · Tη log4 �mn
+δ
+�
+·
+��
+η log
+�nηT
+α
+�
++ η log
+�nηT
+α
+���
+,
+and an additive initialization cost of
+O
+�
+η3T 3 log4
+�nηT
+δ
+�
++ log7
+�nηT
+δ
+��
+.
+Proof. We first claim that for any k ∈ [n], we can solve Problem 1 with probability ≥ 1 − α and
+Tsamp = O
+��n
+k · Tη log4 �mn
+δ
+��
+,
+Tupdate = O
+���n
+k · Tη log4 �mn
+δ
+��
+· kη log
+�nηT
+α
+��
+.
+This follows from combining Lemma 3 (amortized over ⌈ 1
+η⌉ iterations), Corollary 1, and Lemma 4,
+and taking a union bound over at most ⌈ηT⌉ phases. Here we note that the cost of log m per
+iteration to support Update costs to x in Lemma 1, Proposition 2, and Corollary 2 is not dominant.
+By choosing k = Θ(max(1, (η log mn
+αǫ )−1)), we balance the costs of Tsamp and Tupdate, yielding the
+conclusion. We finally note that by picking an appropriate constant in the definition of k, we have
+δ ≤ η =⇒ δ ≤
+1
+16k as required by Lemma 3, the only component specifying a bound on δ.
+14
+
+Acknowledgments
+We thank András Gilyén for communication regarding the prior work [vAG19]. AB was supported
+in part by the DOE QuantISED grant DE-SC0020360, by the AFOSR under grant FA9550-21-
+1-0392, and by the U.S. DOE Office of Science under Award Number DE-SC0020266.
+YG was
+supported in part by the Stanford MS&E DE&I Research program. YJ was supported in part by a
+Stanford Graduate Fellowship and a Danzig-Lieberman Graduate Fellowship. AS was supported in
+part by a Microsoft Research Faculty Fellowship, NSF CAREER Award CCF1844855, NSF Grant
+CCF-1955039, a PayPal research award, and a Sloan Research Fellowship. KT thanks Ewin Tang
+for her expertise on quantum linear algebra and for fielding many of our questions.
+References
+[Aar15]
+Scott Aaronson. Read the fine print. Nature Physics, 11(4):291–293, 2015.
+[AK07]
+Sanjeev Arora and Satyen Kale. A combinatorial, primal-dual approach to semidefi-
+nite programs. In Proceedings of the thirty-ninth annual ACM symposium on Theory
+of computing, pages 227–236, 2007.
+[AL17]
+Zeyuan Allen-Zhu and Yuanzhi Li. Follow the compressed leader: Faster online
+learning of eigenvectors and faster MMWU. In Doina Precup and Yee Whye Teh,
+editors, Proceedings of the 34th International Conference on Machine Learning,
+ICML 2017, Sydney, NSW, Australia, 6-11 August 2017, volume 70 of Proceedings
+of Machine Learning Research, pages 116–125. PMLR, 2017.
+[Alm21]
+Josh Alman. Limits on the universal method for matrix multiplication. Theory
+Comput., 17:1–30, 2021.
+[AW21]
+Josh Alman and Virginia Vassilevska Williams. A refined laser method and faster
+matrix multiplication. In Dániel Marx, editor, Proceedings of the 2021 ACM-SIAM
+Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10
+- 13, 2021, pages 522–539. SIAM, 2021.
+[BHMT02]
+Gilles Brassard, Peter Høyer, Michele Mosca, and Alain Tapp. Quantum amplitude
+amplification and estimation. Quantum Computation and Quantum Information,
+305:53–74, 2002.
+[BS17]
+Fernando GSL Brandao and Krysta M Svore.
+Quantum speed-ups for solving
+semidefinite programs.
+In 2017 IEEE 58th Annual Symposium on Foundations
+of Computer Science (FOCS), pages 415–426. IEEE, 2017.
+[Bub15]
+Sébastien Bubeck. Convex optimization: Algorithms and complexity. Foundations
+and Trends in Machine Learning, 8(3-4):231–357, 2015.
+[BWP+17]
+Jacob Biamonte, Peter Wittek, Nicola Pancotti, Patrick Rebentrost, Nathan Wiebe,
+and Seth Lloyd. Quantum machine learning. Nature, 549(7671):195–202, 2017.
+[CCLW20]
+Shouvanik Chakrabarti, Andrew M Childs, Tongyang Li, and Xiaodi Wu. Quantum
+algorithms and lower bounds for convex optimization. Quantum, 4:221, 2020.
+15
+
+[CDST19]
+Yair Carmon, John C. Duchi, Aaron Sidford, and Kevin Tian. A rank-1 sketch
+for matrix multiplicative weights. In Alina Beygelzimer and Daniel Hsu, editors,
+Conference on Learning Theory, COLT 2019, 25-28 June 2019, Phoenix, AZ, USA,
+volume 99 of Proceedings of Machine Learning Research, pages 589–623. PMLR,
+2019.
+[CGL+20]
+Nai-Hui Chia, András Gilyén, Tongyang Li, Han-Hsuan Lin, Ewin Tang, and Chun-
+hao Wang. Sampling-based sublinear low-rank matrix arithmetic framework for
+dequantizing quantum machine learning. In Proceedings of the 52nd Annual ACM
+SIGACT symposium on theory of computing, pages 387–400, 2020.
+[CJST19]
+Yair Carmon, Yujia Jin, Aaron Sidford, and Kevin Tian. Variance reduction for
+matrix games. In Hanna M. Wallach, Hugo Larochelle, Alina Beygelzimer, Florence
+d’Alché-Buc, Emily B. Fox, and Roman Garnett, editors, Advances in Neural Infor-
+mation Processing Systems 32: Annual Conference on Neural Information Process-
+ing Systems 2019, NeurIPS 2019, December 8-14, 2019, Vancouver, BC, Canada,
+pages 11377–11388, 2019.
+[CJST20]
+Yair Carmon, Yujia Jin, Aaron Sidford, and Kevin Tian. Coordinate methods for
+matrix games. In Sandy Irani, editor, 61st IEEE Annual Symposium on Foundations
+of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020,
+pages 283–293. IEEE, 2020.
+[CLS21]
+Michael B Cohen, Yin Tat Lee, and Zhao Song. Solving linear programs in the
+current matrix multiplication time. Journal of the ACM (JACM), 68(1):1–39, 2021.
+[DH96]
+Christoph Dürr and Peter Høyer. A quantum algorithm for finding the minimum.
+CoRR, quant-ph/9607014, 1996.
+[DNV20]
+Daniel Dadush, Bento Natura, and Làszlò A Vègh. Revisiting tardos’s framework
+for linear programming: faster exact solutions using approximate solvers. In Sandy
+Irani, editor, 61st IEEE Annual Symposium on Foundations of Computer Science,
+FOCS 2020, Durham, NC, USA, November 16-19, 2020, pages 931–942. IEEE,
+2020.
+[GAW19]
+András Gilyén, Srinivasan Arunachalam, and Nathan Wiebe. Optimizing quantum
+optimization algorithms via faster quantum gradient computation. In Proceedings of
+the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1425–
+1444. SIAM, 2019.
+[GK95]
+Michael D. Grigoriadis and Leonid G. Khachiyan. A sublinear-time randomized
+approximation algorithm for matrix games. Operation Research Letters, 18(2):53–
+58, 1995.
+[GLG22]
+Sevag Gharibian and François Le Gall. Dequantizing the quantum singular value
+transformation: Hardness and applications to quantum chemistry and the quantum
+pcp conjecture. In Proceedings of the 54th Annual ACM SIGACT Symposium on
+Theory of Computing, pages 19–32, 2022.
+[GR02]
+Lov Grover and Terry Rudolph. Creating superpositions that correspond to effi-
+ciently integrable probability distributions. CoRR, abs/quant-ph/0208112, 2002.
+16
+
+[GSLW19]
+András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe. Quantum singular
+value transformation and beyond: exponential improvements for quantum matrix
+arithmetics. In Moses Charikar and Edith Cohen, editors, Proceedings of the 51st
+Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix,
+AZ, USA, June 23-26, 2019, pages 193–204. ACM, 2019.
+[Haa19]
+Jeongwan Haah. Product decomposition of periodic functions in quantum signal
+processing. Quantum, 3:190, 2019.
+[HHL09]
+Aram W Harrow, Avinatan Hassidim, and Seth Lloyd.
+Quantum algorithm for
+linear systems of equations. Physical review letters, 103(15):150502, 2009.
+[Jor05]
+Stephen P Jordan.
+Fast quantum algorithm for numerical gradient estimation.
+Physical review letters, 95(5):050501, 2005.
+[JSWZ21]
+Shunhua Jiang, Zhao Song, Omri Weinstein, and Hengjie Zhang. A faster algorithm
+for solving general lps. In Proceedings of the 53rd Annual ACM SIGACT Symposium
+on Theory of Computing, STOC 2021, 2021, pages 823–832, 2021.
+[Kar84]
+Narendra Karmarkar. A new polynomial-time algorithm for linear programming.
+In Proceedings of the sixteenth annual ACM symposium on Theory of computing,
+pages 302–311, 1984.
+[KP16]
+Iordanis Kerenidis and Anupam Prakash.
+Quantum recommendation systems.
+arXiv preprint arXiv:1603.08675, 2016.
+[KP20]
+Iordanis Kerenidis and Anupam Prakash. A quantum interior point method for lps
+and sdps. ACM Transactions on Quantum Computing, 1(1):1–32, 2020.
+[LCW19]
+Tongyang Li, Shouvanik Chakrabarti, and Xiaodi Wu. Sublinear quantum algo-
+rithms for training linear and kernel-based classifiers. In International Conference
+on Machine Learning, pages 3815–3824. PMLR, 2019.
+[LMR14]
+Seth Lloyd, Masoud Mohseni, and Patrick Rebentrost. Quantum principal compo-
+nent analysis. Nature Physics, 10(9):631–633, 2014.
+[LS14]
+Yin Tat Lee and Aaron Sidford. Path finding methods for linear programming:
+Solving linear programs in o (vrank) iterations and faster algorithms for maximum
+flow. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science,
+pages 424–433. IEEE, 2014.
+[LS19]
+Yin Tat Lee and Aaron Sidford. Solving linear programs with sqrt (rank) linear
+system solves. arXiv preprint arXiv:1910.08033, 2019.
+[LWCW21]
+Tongyang Li, Chunhao Wang, Shouvanik Chakrabarti, and Xiaodi Wu. Sublinear
+classical and quantum algorithms for general matrix games. In Proceedings of the
+AAAI Conference on Artificial Intelligence, volume 35, pages 8465–8473, 2021.
+[Nem04]
+Arkadi Nemirovski. Prox-method with rate of convergence O(1/t) for variational in-
+equalities with lipschitz continuous monotone operators and smooth convex-concave
+saddle point problems. SIAM Journal on Optimization, 15(1):229–251, 2004.
+17
+
+[Nes07]
+Yurii Nesterov. Dual extrapolation and its applications to solving variational in-
+equalities and related problems. Mathematical Programing, 109(2-3):319–344, 2007.
+[Neu28]
+John Von Neumann. Zur theorie der gesellschaftsspiele. Mathematische Annalen,
+100:295–320, 1928.
+[NJLS09]
+Arkadi Nemirovski, Anatoli B. Juditsky, Guanghui Lan, and Alexander Shapiro.
+Robust stochastic approximation approach to stochastic programming. SIAM J.
+Optim., 19(4):1574–1609, 2009.
+[Ren88]
+James Renegar. A polynomial-time algorithm, based on newton’s method, for linear
+programming. Mathematical programming, 40(1):59–93, 1988.
+[RML14]
+Patrick Rebentrost, Masoud Mohseni, and Seth Lloyd. Quantum support vector
+machine for big data classification. Physical review letters, 113(13):130503, 2014.
+[SV14]
+Sushant Sachdeva and Nisheeth K. Vishnoi. Faster algorithms via approximation
+theory. Found. Trends Theor. Comput. Sci., 9(2):125–210, 2014.
+[Tan19]
+Ewin Tang. A quantum-inspired classical algorithm for recommendation systems. In
+Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing,
+pages 217–228, 2019.
+[vAG19]
+Joran van Apeldoorn and András Gilyén. Quantum algorithms for zero-sum games.
+CoRR, abs/1904.03180, 2019.
+[vAGGdW20a] Joran van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf. Convex
+optimization using quantum oracles. Quantum, 4:220, 2020.
+[VAGGdW20b] Joran Van Apeldoorn, András Gilyén, Sander Gribling, and Ronald de Wolf. Quan-
+tum sdp-solvers: Better upper and lower bounds. Quantum, 4:230, 2020.
+[vdB20]
+Jan van den Brand. A deterministic linear program solver in current matrix mul-
+tiplication time. In Proceedings of the Thirty-first Annual ACM-SIAM Symposium
+on Discrete Algorithms, SODA 2020, 2020, pages 259–278, 2020.
+[vdBLL+21]
+Jan van den Brand, Yin Tat Lee, Yang P. Liu, Thatchaphol Saranurak, Aaron
+Sidford, Zhao Song, and Di Wang. Minimum cost flows, mdps, and ℓ1-regression
+in nearly linear time for dense instances. In Proceedings of the 53rd Annual ACM
+SIGACT Symposium on Theory of Computing, STOC 2021, 2021, pages 859–869,
+2021.
+[vdBLSS20]
+Jan van den Brand, Yin Tat Lee, Aaron Sidford, and Zhao Song. Solving tall dense
+linear programs in nearly linear time. In Proceedings of the 52nd Annual ACM
+SIGACT Symposium on Theory of Computing, pages 775–788, 2020.
+18
+
+A
+Solving matrix games with a Gibbs sampling oracle
+In this section, we prove Proposition 1, which shows how to solve a zero-sum matrix game using
+an approximate Gibbs sampling oracle (via Algorithm 1). To briefly motivate the algorithm we use
+and our proof of its guarantees, we recall the problem we consider is of the form
+min
+v∈∆n max
+u∈∆m f(u, v) := u⊤Av,
+where
+∥A∥max ≤ 1,
+(5)
+and we define the associated gradient operator as
+g(u, v) = (−Av, A⊤u).
+(6)
+Taking (stochastic) mirror descent steps on the gradient operator in (5) is well-known to yield an
+approximate NE to the matrix game [Bub15]. We show that an approximate implementation of this
+strategy, combined with appropriate subsampling, efficiently yields an approximate NE. We begin
+by making the following observation.
+Lemma 5. Let u, ˜u ∈ ∆m have ∥u − ˜u∥1 ≤ δ. Let ˜g := Ai: where i ∼ ˜u, and g := A⊤u. Then,
+∥g − E˜g∥∞ ≤ δ.
+Proof. Note that E˜g = A⊤˜u, and
+��A⊤(u − ˜u)
+��
+∞ ≤ ∥u − ˜u∥1 ≤ δ since ∥A∥max ≤ 1.
+We next present a variant of the classical mirror descent analysis, which bounds the expected
+approximation quality of iterates of Algorithm 1 prior to subsampling.
+Proposition 3. Let δ ≤
+ǫ
+20, η =
+ǫ
+15 and T ≥ 6 log(mn)
+ηǫ
+in Algorithm 1. Let the iterates of Algorithm 1
+be {xt, yt}T−1
+t=0 , and denote ut :=
+exp(Ayt)
+∥exp(Ayt)∥1 , vt :=
+exp(−A⊤xt)
+∥exp(−A⊤xt)∥1
+for all 0 ≤ t < T. For (¯u, ¯v) :=
+1
+T
+�T−1
+t=0 (ut, vt), we have
+E
+�
+max
+u∈∆m u⊤A¯v − min
+v∈∆n ¯u⊤Av
+�
+≤ ǫ.
+(7)
+Proof. By definition of the updates, at every iteration 0 ≤ t ≤ T − 1, we have
+ut+1 = argminu∈∆m
+
+
+η⟨−A:jt, u⟩ +
+�
+i∈[m]
+[u]i log [u]i
+[ut]i
+
+
+ ,
+vt+1 = argminv∈∆n
+
+
+η⟨Ait:, v⟩ +
+�
+j∈[n]
+[v]j log [v]j
+[vt]j
+
+
+ .
+Consequently, by the optimality conditions of ut+1 and vt+1 respectively, we have for any u ∈ ∆m,
+v ∈ ∆n, and letting Vx(x′) := �
+k[x′]k log [x′]k
+[x]k be the KL divergence between simplex variables of
+appropriate dimension,
+⟨−A:j, ut − u⟩ + ⟨Ai:, vt − v⟩ ≤ 1
+η
+�
+Vut(u) − Vut+1(u) + Vvt(v) − Vvt+1(v)
+�
++
+�
+⟨−A:j, ut − ut+1⟩ − 1
+η Vut(ut+1)
+�
++
+�
+⟨Ai:, vt − vt+1⟩ − 1
+η Vvt(vt+1)
+�
+≤ 1
+η
+�
+Vut(u) − Vut+1(u) + Vvt(v) − Vvt+1(v)
+�
++ η
+2 ∥A:j∥2
+∞ + η
+2 ∥Ai:∥2
+∞ ,
+(8)
+19
+
+where for the last inequality we use Hölder’s inequality and the fact that V is 1-strongly convex in
+the ℓ1 norm (by Pinsker’s inequality). Averaging the above for 0 ≤ t < T, and denoting wt := (ut, vt)
+and ˜gt := (−A:jt, Ait:), we obtain for any w = (u, v) ∈ ∆m × ∆n,
+1
+T
+T−1
+�
+t=0
+⟨˜gt, wt − w⟩ ≤ 1
+ηT (Vu0(u) + Vv0(v)) + η.
+(9)
+In the above, we further recalled the bound ∥A∥max ≤ 1 by assumption. In order to bound the
+deviation of the left-hand side from its expectation, we use a “ghost iterate” argument following
+[NJLS09, CJST19]. In particular, we define iterates ˜ut, ˜vt as follows: let ˜u0 ← u0, ˜v0 ← v0, and
+then for each 0 ≤ t < T, define
+˜ut+1 := argminu∈∆m
+
+
+η⟨−Avt + A:jt, ¯u⟩ +
+�
+i∈[m]
+[u]i log [u]i
+[˜ut]i
+
+
+ ,
+˜vt+1 := argminv∈∆n
+
+
+η⟨A⊤ut − A:it, ¯v⟩ +
+�
+j∈[n]
+[v]j log [v]j
+[˜vt]j
+
+
+ ,
+where i, j above are the same coordinates as were used in defining the updates to ut+1 and vt+1.
+By an analogous bound to (8), where we note
+��A:jt − A⊤vt
+��
+∞ , ∥Aut − Ait:∥∞ ≤ 2,
+�
+−A⊤vt + A:jt, ˜ut − u
+�
++ ⟨Aut − Ait:, ˜vt − v⟩ ≤ 1
+η
+�
+V˜ut(u) − V˜ut+1(u) + V˜vt(v) − V˜vt+1(v)
+�
++ 4η.
+Averaging the above for 0 ≤ t < T, and denoting ˜wt := (˜ut, ˜vt) and gt := g(wt) (see (5)), we obtain
+for any w = (u, v) ∈ ∆m × ∆n,
+1
+T
+�
+t∈[T]−1
+⟨gt − ˜gt, ˜wt − w⟩ ≤ 1
+ηT (Vu0(u) + Vv0(v)) + 4η.
+(10)
+Summing inequalities (9) and (10), and maximizing over w = (u, v) ∈ ∆m × ∆n, we have
+max
+w∈∆m×∆n
+1
+T
+T−1
+�
+t=0
+⟨gt, wt − w⟩ ≤
+max
+u∈∆n,v∈∆m
+2
+ηT (Vu0(u) + Vv0(v))
++ 5η + 1
+T
+T−1
+�
+t=0
+⟨gt − ˜gt, wt − ˜wt⟩.
+(11)
+Taking expectations over the above, we have
+E
+�
+max
+w∈∆m×∆n
+1
+T
+T−1
+�
+t=0
+⟨gt, wt − w⟩
+�
+≤
+max
+u∈∆n,v∈∆m
+2
+ηT [Vu0(u) + Vv0(v)]
++ 5η + E
+�
+1
+T
+T−1
+�
+t=0
+⟨gt − ˜gt, wt − ˜wt⟩
+�
+(i)
+≤ 2 log(mn)
+ηT
++ 5η + 1
+T
+�
+t∈[T]−1
+⟨gt − E˜gt, wt − ¯wt⟩,
+(ii)
+≤ 2 log(mn)
+ηT
++ 5η + 4δ
+(iii)
+≤ ǫ.
+20
+
+In the above, (i) used the diameter bound of the KL divergence from the uniform distribution, i.e.
+maxu∈∆m Vu0(u) = log m (and a similar bound for Vv0(v)). Further, (ii) uses that ˜gt is conditionally
+independent of wt and ˜wt, and by the assumption on the Gibbs sampler ∥gt − E˜gt∥∞ ≤ δ (via
+Lemma 5), and Hölder, and (iii) uses our choices of T, η and δ.
+Finally, we note that the desired claim follows by linearity: for any w = (u, v),
+1
+T
+T−1
+�
+t=0
+⟨gt, wt − w⟩ =
+�
+g
+�
+1
+T
+T−1
+�
+t=0
+wt
+�
+, 1
+T
+T−1
+�
+t=0
+wt − w
+�
+= u⊤A¯v − ¯u⊤Av.
+By using a simple martingale argument (inspired by those in [AL17, CDST19]) to bound the
+error term in (11), we show that the guarantee of Proposition 3 holds with high probability.
+Corollary 3. Let α ∈ (0, 1), and let δ ≤
+ǫ
+20, η =
+ǫ
+20 and T ≥ 8 log(mn)
+ηǫ
++
+2048 log 1
+α
+ǫ2
+in Algorithm 1.
+Then with probability at least 1−α, following notation of Proposition 3, (¯u, ¯v) are an ǫ-approximate
+NE for A.
+Proof. Consider the filtration given by Ft = σ(u0, v0, ˜g0, · · · , ˜gt, ut+1, vt+1).
+We will bound the
+terms �T−1
+t=0 ⟨gt − ˜gt, wt − ¯wt⟩ in (7). To do so, we define a martingale difference sequence of the
+form Dt := ⟨gt − ˜gt, wt − ¯wt⟩ − ⟨gt − E [˜gt|Ft−1] , wt − ¯wt⟩ which is adapted to the filtration Ft. We
+first note that Dt ≤ ∥gt−1 − ˜gt−1∥∞ ∥wt−1 − ¯wt−1∥1 ≤ 8 with probability 1. Consequently, applying
+the Azuma-Hoeffding inequality yields
+T−1
+�
+t=0
+Dt ≤
+�
+128T log 1
+α with probability ≥ 1 − α.
+Plugging this back into (11) and using the KL divergence range bound, Lemma 5 with our definition
+of Ogibbs, and choices of parameters, we thus have with probability 1 − α,
+max
+w∈∆m×∆n
+1
+T
+T−1
+�
+t=0
+⟨gt, wt − w⟩ ≤ 2 log mn
+ηT
++ 5η + 4δ +
+�
+128 log 1
+α
+T
+≤ ǫ.
+(12)
+The remainder of the proof follows analogously to Proposition 3.
+The Gibbs sampling oracles implicitly maintain access to ut ∝ exp(Ayt) and vt ∝ exp(−A⊤xt),
+which by averaging gives (¯u, ¯v) =
+1
+T
+�T−1
+t=0 (ut, vt) as one approximate equilibrium as guaranteed
+in Corollary 3. To turn the implicitly maintained iterates into an actual classic output, we subsample
+the iterates. Below we formally show one can take the empirical average of independent samples
+from distributions close to ¯u and ¯v to also obtain an approximate equilibrium (with the same
+approximation factor up to constant factors) with high probability.
+Lemma 6. Suppose ¯u = 1
+T
+�T−1
+t=0 ut for {ut}T−1
+t=0 ⊂ ∆m and ¯v = 1
+T
+�T−1
+t=0 vt for {vt}T−1
+t=0 ⊂ ∆n are an
+ǫ-approximate NE for A. Further suppose that for some δ ∈ (0, 1), {˜ut}T−1
+t=0 ⊂ ∆m, {˜vt}T−1
+t=0 ⊂ ∆n,
+and for all 0 ≤ t < T − 1, we have ∥˜ut − ut∥1 ≤ δ and ∥˜vt − vt∥1 ≤ δ. Let ˆu = 1
+T
+�T−1
+t=0 eit where
+each eit ∈ Rm is sampled independently according to ˜ut; similarly, let ˆv = 1
+T
+�T−1
+t=0 ejt where each
+ejt ∈ Rn is sampled independently according to ˜vt. Suppose T ≥
+16 log mn
+α
+ǫ2
+. Then with probability at
+least 1 − α, (ˆu, ˆv) are a (2ǫ + 2δ)-approximate NE for A.
+21
+
+Proof. First, let ˜uavg =
+1
+T
+�T−1
+t=0 ˜ut and ˜vavg =
+1
+T
+�T−1
+t=0 ˜vt.
+By convexity of norms, we have
+∥˜uavg − ¯u∥1 ≤ δ and ∥˜vavg − ¯v∥1 ≤ δ, and hence under the NE approximation guarantee of (¯u, ¯v)
+and Hölder’s inequality,
+max
+u∈∆m u⊤A˜vavg − min
+v∈∆m ˜u⊤
+avgAv ≤ ǫ + 2δ.
+Let z be a fixed vector in [−1, 1]n. By Hoeffding’s inequality, since each random variable ⟨z, ejt⟩ lies
+in the range [−1, 1] and Eˆv = ˜vavg, we have that
+Pr
+�
+|⟨z, ˆv − ˜vavg⟩| ≥ ǫ
+2
+�
+≤ 2 exp
+�
+−Tǫ2
+8
+�
+≤
+α
+m + n.
+(13)
+Next, note that maxu∈∆m u⊤A˜vavg is achieved by a basis vector u = ei. Hence, applying a union
+bound over (13) for all z = Ai: shows that with probability at least 1 −
+αm
+m+n,
+max
+u∈∆m u⊤Aˆv ≤ max
+u∈∆m u⊤A˜vavg + ǫ
+2.
+By symmetry, with probability at least 1 −
+αn
+m+n,
+min
+v∈∆n ˆu⊤Av ≥ min
+v∈∆n ˜u⊤
+avgAv − ǫ
+2.
+The conclusion follows from a union bound, and combining the above three displays.
+Finally, we put these pieces together to give a complete guarantee.
+Proposition 1. Let A ∈ Rm×n satisfy ∥A∥max ≤ 1 and ǫ, α ∈ (0, 1). Let δ ≤
+ǫ
+20, η =
+ǫ
+60, and
+T = Θ(ǫ−2 log mn
+α ) for an appropriate constant. With probability ≥ 1 − α, Algorithm 1 outputs an
+ǫ-approximate NE for A.
+Proof. We follow notation of Proposition 3. By applying Corollary 3 (up to constant factors), we
+have that with probability at least 1 − α
+2 , ¯u := 1
+T
+�T−1
+t=0 ut and ¯v := 1
+T
+�T−1
+t=0 vt satisfy
+max
+u∈∆m u⊤A¯v − min
+v∈∆n ¯u⊤Av ≤ ǫ
+3.
+Finally, Lemma 6 (with failure probability α
+2 ) and a union bound yields the desired conclusion.
+B
+Quantum rejection sampling with a hint
+In this section, we prove Proposition 2, which gives a dynamic quantum rejection sampling subrou-
+tine and bounds its cost of implementation. Our result is an extension of analogous developments
+in [vAG19], but are stated more generally to allow for the use of an appropriate “hint” vector in the
+rejection sampling procedure. We build up to our main result in several pieces.
+Amplitude amplification.
+First, for a quantum decision algorithm which applies unitary U and
+then measures, yielding an accepting state with probability α, quantum amplification [BHMT02]
+shows we can apply U ≈ α− 1
+2 times to obtain an accepting state with high probability.
+Proposition 4 (Theorem 3, [BHMT02]). Let S ⊆ {0, 1}s, let U be a s-qubit quantum oracle, and
+let α be the probability that measuring the result of applying U yields an accepting state. There
+is a (quantum) algorithm using O(α− 1
+2 log 1
+δ ) queries to U and O(log s log 1
+δ) additional time that
+returns s with s ∈ S with probability ≥ 1 − δ.
+22
+
+Loading from trees.
+Given a dynamic vector x ∈ Rm
+≥0 which is supported in an appropriate
+efficient data structure SamplerTree (see Lemma 1), and a known bound β ≥ ∥x∥1, we recall a result
+of [GR02] which allows us to form a superposition of the entries in x (suitably rescaled).
+Lemma 7. Let x ∈ Rm
+≥0 correspond to an instance of SamplerTree, and β ≥ ∥x∥1. We can maintain
+a quantum oracle OSamplerTree which takes O(log m) time to apply, such that the total cost of building
+OSamplerTree after T calls to Update is O(T log m), and
+OSamplerTree|0⟩⊗(a+1) =
+�
+i∈[m]
+�xi
+β |0⟩|i⟩ +
+�
+1 − ∥x∥1
+β
+|1⟩|g⟩.
+Proof. This is implicit in [GR02]. We first apply a 1-qubit gate to condition on selecting from the
+tree (with probability ∥x∥1
+β ), and then apply the [GR02] procedure conditioned on the first qubit
+being |0⟩, which controls for one qubit at a time while propagating subtree sums (provided by
+SamplerTree via SubtreeSum). The cost to build the circuit follows because on an Update we need
+to change the gates corresponding to the relevant leaf-to-root path.
+Corollary 4. Let x ∈ Rm
+≥0 correspond to an instance of SamplerTree, and let β ≥ ∥x∥1, and suppose
+A ∈ Rm×n has ∥A∥max ≤ 1. We can maintain a quantum oracle OA⊤x which takes O(log m) time
+to apply, with total building cost O(T log m) after T calls to Update, such that for any j ∈ [n],
+OA⊤x|0⟩⊗(a+2)|j⟩ = |0⟩
+
+ �
+i∈[m]
+�
+Aijxi
+β
+|0⟩|i⟩ + |1⟩|g⟩
+
+ |j⟩.
+Proof. We apply O′
+A (see Section 2) to the output of OSamplerTree, ignoring the additional qubit.
+We remark here that the additional qubit in Corollary 4 will shortly become useful in constructing
+an appropriate block-encoding of a scaling of diag
+�
+A⊤x
+�
+.
+Polynomial approximation.
+In order to give approximate Gibbs samplers for the types of dy-
+namic vectors Algorithm 1 encounters, we further require some tools from polynomial approximation
+theory. We first state a helper result on boundedly approximating the exponential, a variant of which
+was also used in [vAG19]. We provide a proof in Appendix C.
+Lemma 8 (Lemma 7, [vAG19]). Let β ≥ 1, ξ ≤
+1
+10. There is a polynomial Pβ,ξ of degree O(β log 1
+ξ )
+such that maxx∈[−1,1] |Pβ,ξ(x)| ≤ 3 and maxx∈[−1,0] |Pβ,ξ(x) − exp(βx)| ≤ ξ.
+Next, we state a further corollary of Lemma 8 to be used in our rejection sampler.
+Corollary 5. Let B, δ ≥ 0 and suppose v ∈ Rn has ∥v∥∞ ≤ B. Further, suppose for some c ≥ 0,
+−c ≤ maxj∈[n] vj ≤ 0. Let q ∈ Rn
+≥0 satisfy qj ∈ [ℓ, 1] entrywise. Finally, define uj := vj
+2B entrywise.
+There is a degree-∆ polynomial P, for ∆ = O(B · (c + log n
+ℓδ)), such that for wj := P(uj)2qj and
+zj := exp(2Buj)qj entrywise,
+����
+w
+∥w∥1
+−
+z
+∥z∥1
+����
+1
+≤ δ.
+(14)
+Moreover, maxx∈[−1,1] |P(x)| ≤ 1
+2, and ∥w∥1 ≥ 1−δ
+36 ∥z∥1.
+23
+
+Proof. Assume δ ≤ 2 else the statement is clearly true. First, uj ∈ [− 1
+2, 0] entrywise by the stated
+assumptions (since vj ∈ [−B, 0] entrywise). Let Pβ,ξ(·) be the polynomial given by Lemma 8 which
+ξ-approximates exp(β·) on [− 1
+2, 0]. We define
+P(u) := 1
+6PB,ξ (u) , for ξ :=
+δℓ
+6n exp(c).
+The degree bound and absolute value bound of this polynomial follows immediately from Lemma 8,
+so it remains to show the distance bound. The guarantees of Lemma 8 then imply for all j ∈ [n],
+|6P(uj) − exp (Buj)| ≤ ξ.
+(15)
+We further have that uj ≤ 0, so exp(Buj) ≤ 1. Hence, we also have
+|6P(uj) + exp (Buj)| ≤ 2 + ξ ≤ 3.
+Combining yields for all j ∈ [n],
+��36P(uj)2 − exp (2Buj)
+�� ≤ 3ξ.
+(16)
+Next, let yj := 36wj for all j ∈ [n], and note that
+y
+∥y∥1 =
+w
+∥w∥1. We bound
+����
+w
+∥w∥1
+−
+z
+∥z∥1
+����
+1
+=
+�
+j∈[n]
+����
+yj
+∥y∥1
+−
+zj
+∥z∥1
+���� ≤
+�
+j∈[n]
+����
+yj
+∥y∥1
+−
+yj
+∥z∥1
+���� +
+�
+j∈[n]
+����
+yj
+∥z∥1
+−
+zj
+∥z∥1
+����
+≤
+����1 − ∥y∥1
+∥z∥1
+���� + ∥y − z∥1
+∥z∥1
+≤ 2 ∥y − z∥1
+∥z∥1
+.
+(17)
+By using the definitions of y, z and (16), as well as the assumed ranges on q,
+∥y − z∥1 ≤ 3nξ, ∥z∥1 ≥ ℓ exp(−c).
+The second inequality used that some vj = 2Buj is at least −c by assumption. Combining the above
+display with (17) and the definition of ξ concludes the proof of (14). Finally, using the bounds on
+∥y − z∥1 , ∥z∥1 above shows that
+∥w∥1 = 1
+36∥y∥1 ≥ 1 − δ
+36 ∥z∥1.
+Block-encoding.
+Our approximate Gibbs oracle follows an implementation strategy pioneered by
+[GSLW19] termed “block-encoding.” Specifically, we follow [GSLW19] and say that U, an (a + ℓ)-
+qubit quantum gate, is an ℓ-bit block-encoding of M if the top-left 2a �� 2a submatrix of U is M.
+Block-encoded matrices admit efficient composable operations, such as the application of linear
+combinations and bounded polynomials. We summarize these properties in the following.
+Proposition 5 (Lemma 52, [GSLW19]). Let U1 and U2 be ℓ-bit block-encodings of M1, M2 of the
+same size. There is an O(ℓ)-bit block-encoding of 1
+2M1 + 1
+2M2 which takes the same asymptotic
+time to apply as applying U1 and U2.
+Proposition 6 (Theorem 56, [GSLW19]). Let U be an ℓ-bit block-encoding of M, and P : [−1, 1] →
+[− 1
+2, 1
+2] be a degree-∆ polynomial. There is an O(ℓ)-bit block-encoding of P(M) which can be applied
+in O(∆) applications of U and U† and O(ℓ∆) additional time.
+24
+
+We also demonstrate that an application of Corollary 4 yields a simple block-encoding of
+diag
+�
+A⊤x
+β
+�
+. A similar construction previously appeared in [vAG19].
+Corollary 6. Let x ∈ Rm
+≥0 correspond to an instance of SamplerTree, and β ≥ ∥x∥1. Let M :=
+diag
+�
+A⊤x
+β
+�
+and U := O∗
+A⊤x(SWAP12 ⊗ I)OA⊤x, where SWAP12 swaps the first two qubits and
+OA⊤x is from Corollary 4. Then U is a block-encoding of M, and can be applied in time O(log m),
+with total building cost O(T log m) after T calls to Update.
+Proof. Define wij := Aijxi
+β
+for convenience. By the definition of OA⊤x, we have that
+(SWAP12 ⊗ I) OA⊤x
+�
+|0⟩⊗(a+2)|j⟩
+�
+=
+
+|00⟩
+�
+i∈[m]
+√wij|i⟩ + |10⟩|g⟩
+
+ |j⟩.
+Hence, for j, j′ ∈ [n], we compute ⟨j′|⟨0|⊗(a+2)U|0⟩⊗(a+2)|j⟩ as:
+⟨j′|
+
+|00⟩
+�
+i∈[m]
+√wij|i⟩ + |01⟩|g⟩
+
+
+∗ 
+|00⟩
+�
+i∈[m]
+√wij|i⟩ + |10⟩|g⟩
+
+ |j⟩
+=
+��
+i∈[m] wij = [A⊤x]j
+β
+j = j′
+0
+j ̸= j′ .
+In particular the |01⟩ and |10⟩ terms disappear, and |j⟩, |j′⟩ are orthogonal unless j = j′. In the
+above, we required that √wij∗√wij = wij, which is only true if wij is nonnegative. To bypass this
+issue, we will implement the two copies of OA⊤x in slightly different ways, to obtain the correct
+signing. For notational clarity, we let OL be the oracle which is conjugated on the left and OR
+be the oracle on the right, such that U = (OL)∗(SWAP12 ⊗ I)(OR). Note that x is entrywise
+nonnegative and β > 0, and hence the only factor determining the sign of wij is Aij.
+When
+Aij ≥ 0, we will define the oracles O′
+A used to load
+�
+Aij for OL and OR in a consistent way
+(i.e. use the same-signed square root), so that √wij2 = wij. When Aij < 0 we will define them
+in an inconsistent way, so that after the conjugation operation, −√wij√wij = wij. We have thus
+shown that ⟨0|⊗(a+2)U|0⟩⊗(a+2) = M which implies the first conclusion. To see the second, all our
+gates are reversible (arithmetic circuits are reversible, and OA is its own inverse), and hence the
+complexity of applying O∗
+A⊤x is the same as OA⊤x.
+Finally, we put together the pieces and prove Proposition 2, which we use repeatedly throughout
+the paper to implement our Gibbs sampling oracles.
+Proposition 2. Let x ∈ Rm
+≥0 correspond to an instance of SamplerTree, and β ≥ ∥x∥1. Let p be
+the Gibbs distribution associated with A⊤x, let Z := �
+j∈[n] exp([A⊤x]j) and �Z ∈ [Z, CZ] for some
+C ≥ 1. Finally, let q ∈ Rn have entries classically queriable in O(1) time, satisfy q ≥ p entrywise,
+qj ∈ [ δ
+n, 1] for all j ∈ [n], and ∥q∥1 = ρ. Suppose �Z, C, ρ, and β are explicitly known. Given
+a quantum oracle for A ∈ Rm×n (defined in Section 2) with ∥A∥max ≤ 1, we can implement a
+δ-approximate Gibbs oracle which has query cost O(√ρC · β log4 � Cmn
+δ
+�
+). The total additional cost
+incurred if x undergoes T Update calls which preserve the invariants on �Z, C, ρ, β is O(T log m).
+Proof. Throughout the proof, let δ ← min(1
+2, δ) and B := 4(β + log(Cn
+δ )).
+Also define ℓ :=
+δ
+n (following notation of Corollary 5).
+We first observe that since maxj∈[n][A⊤x]j ≤ log Z ≤
+25
+
+maxj∈[n][A⊤x]j + log n,
+− log(Cn) ≤ max
+j∈[n][A⊤x]j − log
+�
+�Zqj
+�
+≤ 0.
+Here, the upper bound used that for all j ∈ [n], exp([A⊤x]j − �Zqj) = pj
+qj · Z
+�Z ≤ 1 by assumption.
+Hence, for v := A⊤x − log( �Zq) entrywise,
+−c ≤ max
+j∈[n] vj ≤ 0 for c := log(Cn).
+Next, we note log( �Zq) is entrywise bounded in magnitude by B
+2 :
+log( �Zqj) ≤ log(CZ) ≤ log
+�
+n · max
+j∈[n] exp([A⊤x]j)
+�
++ log C ≤ B
+2 ,
+log( �Zqj) ≥ log Z + log δ
+n ≥ min
+j∈[n][A⊤x]j − log n
+δ ≥ −B
+2 .
+Define M1 := diag
+�
+A⊤x
+2B
+�
+and M2 := diag
+�
+− 1
+2B log( �Zq)
+�
+. By the calculations above, we have
+∥M2∥op ≤ 1
+2, and similarly it is clear that ∥M1∥op ≤
+1
+2 because
+��A⊤x
+��
+∞ ≤ β. Moreover, by
+using Corollary 6 with β ← B, we obtain U1, a block-encoding of M1 applicable in O(log m) time.
+Using a similar construction as Corollary 6, since q, B, and �Z are all efficiently classically queriable,
+we obtain U2, a block-encoding of M2 applicable in O(1) time. Hence, Proposition 5 yields U, a
+block-encoding of
+M1 + M2 = diag
+� v
+2B
+�
+,
+which can be applied in O(log mn) time. Next, let P be the degree-∆ = O(B log Cn
+δ ) polynomial
+from Corollary 5, parameterized by B, v, c, q, ℓ as defined earlier.
+Corollary 5 shows that P :
+[−1, 1] → [− 1
+2, 1
+2]. Thus, Proposition 6 then yields U′, a block-encoding of diag
+�
+P( v
+2B )
+�
+which can
+be applied in O(∆ · log mn) time. Furthermore, since q and ρ are efficiently classically queriable,
+we can define a gate Oq which is applicable in O(1) time and acts as
+Oq|0⟩⊗(b+1) = |0⟩
+�
+j∈[n]
+�qj
+ρ |j⟩ + |1⟩|g⟩.
+Applying U′ to the output of Oq with appropriate ancilla qubits then yields
+|0⟩⊗O(1) �
+j∈[n]
+�
+qjP(uj)2
+ρ
+|j⟩|gj⟩ + |g′⟩, where uj := vj
+2B for all j ∈ [n].
+Post-selecting on the first register being the all-zeroes state and measuring on the register corre-
+sponding to j, we see that we obtain a sample j ∈ [n] with probability proportional to qjP(uj)2. By
+Corollary 5, conditioned on the sample succeeding, the resulting distribution is δ-close in ℓ1 to the
+distribution proportional to q ◦ exp(v) ∝ exp(A⊤x), and hence the result is a δ-approximate Gibbs
+oracle. Finally, we bound the query cost of the oracle. Define wj := P(uj)2qj and zj := exp(vj)qj
+as in Corollary 5. By definition of v, �Z,
+∥z∥1 =
+�
+j∈[n]
+exp
+��
+A⊤x
+�
+j
+�
+�Z
+∈
+�
+C−1, 1
+�
+.
+26
+
+Moreover, the last conclusion in Corollary 5 shows ∥w∥1 ≥
+1
+72 ∥z∥1 ≥ (72C)−1. Hence,
+�
+j∈[n]
+qjP(uj)2
+ρ
+= ∥w∥1
+ρ
+≥
+1
+72Cρ.
+In other words, we have an oracle which we can apply in time O(∆·log mn) which correctly returns
+a sample with probability α ≥
+1
+72Cρ. By applying Proposition 4 to improve the success probability,
+we obtain the desired conclusion at a O(√Cρ log 1
+δ ) overhead.
+Corollary 2. Following notation of Proposition 2, let R :=
+�Z
+Z . There is a quantum oracle Otest
+which can be implemented under T Update calls to x in O(T log m) time, and has query cost
+O
+��
+ρC · β log4
+�Cmn
+ℓδ
+��
+.
+Furthermore, for explicitly known constants Cℓ and Cu, Otest returns “success” with probability p for
+Cℓ
+√Rρ ≤ p ≤
+Cu
+√Rρ.
+Proof. Our oracle Otest is the oracle from Proposition 2, except we will choose a sufficiently small
+constant value of δ. It returns “success” when the sample is accepted by the rejection sampler after
+boosting by amplitude amplification. Before boosting, the success probability from Proposition 2
+is Θ( 1
+Rρ) where the constants in the upper and lower bounds are explicit. Further, the constants
+from Proposition 4 are explicit, and hence boosting by amplitude amplification improves the success
+probability to Θ(
+1
+√Rρ) with known constant bounds as required by the corollary statement.
+C
+Bounded approximation to exp on [−1, 1]
+Here, we give a proof of a lemma (with slightly different constants) used in the prior work [vAG19].
+This section builds entirely off prior results on polynomial approximation in [GSLW19]; we include
+it for completeness because a proof was not given in [vAG19]. As a reminder, we stated and used
+the following result earlier when constructing our rejection sampler in Appendix B.
+Lemma 8 (Lemma 7, [vAG19]). Let β ≥ 1, ξ ≤
+1
+10. There is a polynomial Pβ,ξ of degree O(β log 1
+ξ )
+such that maxx∈[−1,1] |Pβ,ξ(x)| ≤ 3 and maxx∈[−1,0] |Pβ,ξ(x) − exp(βx)| ≤ ξ.
+To obtain the lemma, we will utilize the following result from [GSLW19].
+Proposition 7 (Corollary 66, [GSLW19]). Let x0 ∈ [−1, 1], r ∈ (0, 2], δ ∈ (0, r]. Let f : [x0 − r −
+δ, x0 + r + δ] → C be such that f(x0 + x) = �
+ℓ≥0 aℓxℓ for all x ∈ [−r − δ, r + δ]. Suppose B > 0 is
+such that �
+ℓ≥0(r + δ)ℓ|aℓ| ≤ B and let ǫ ∈ (0,
+1
+2B ]. There is a polynomial P (see Appendix D for
+its numerically stable implementation) of degree O
+� 1
+δ log B
+ǫ
+�
+such that
+max
+x∈[x0−r,x0+r] |f(x) − P(x)| ≤ ǫ and
+max
+x∈[−1,1] |P(x)| ≤ ǫ + B.
+Proof of Lemma 8. We apply Proposition 7 with f(x) := exp(βx) which has a convergent Taylor
+series everywhere, and the parameter settings x0 = −1, r = 1, δ =
+1
+β, B = e.
+We have that
+27
+
+f(x0 + x) = �
+ℓ≥0 exp(−β)βℓ·xℓ
+ℓ!
+= �
+ℓ≥0 aℓxℓ with aℓ = exp(−β)βℓ
+ℓ! for any integer ℓ ≥ 0. We also
+check that our choice of B is valid, via
+�
+ℓ≥0
+(r + δ)ℓ|aℓ| = exp(−β)
+�
+ℓ≥0
+�
+1 + 1
+β
+�ℓ βℓ
+ℓ! = exp(−β)
+�
+ℓ≥0
+(β + 1)ℓ
+ℓ!
+= exp(β + 1 − β) = e.
+Hence by Proposition 7, we have for any ξ ≤
+1
+2e, there is a polynomial P of degree O(β log 1
+ξ )
+such that maxx∈[−2,0] | exp(βx) − P(x)| ≤ ǫ and maxx∈[−1,1] | ˜P(x)| ≤ e + 1
+6 + ξ ≤ 3.
+D
+Numerically stable implementation of polynomial approximation
+Throughout this section, let ∆ = O(1
+ǫ log2(mn
+ǫ )) be the degree of the polynomial used in the proof
+of Proposition 2 in Appendix B (specifically, constructed in the proof of Proposition 2, where we
+have C = O(1) and δ = O(ǫ) in our applications).
+The polynomial we use is constructed via
+a decomposition in the Fourier basis (see Lemmas 57 and 65, [GSLW19]).
+It is not immediate
+that this polynomial transform can be implemented stably in finite-precision arithmetic, within
+the quantum singular value transformation framework of [GSLW19], which is used in the proof
+of Proposition 2. However, [Haa19] shows that given such a decomposition in the Fourier basis,
+we can obtain a numerically-stable implementation of the polynomial transformation required as a
+quantum circuit up to additive error ξ, in time
+O
+�
+∆3 log
+�∆
+ξ
+��
+.
+In our setting (in the proof of Proposition 2), it is straightforward to check that ξ = poly(m, n, ǫ−1).
+This construction results in the additive term in Theorem 4.
+28
+

9tFJT4oBgHgl3EQfoyxM/content/2301.11597v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:79e625ab7976c129b39eeb312becb30ffc16cfba061f383270ca932118dd4de5
+size 2380508

9tFJT4oBgHgl3EQfoyxM/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:b61ca1da7b6dec3b13aa0182ce54f1b0c4e26de338201553d91bb175319cc977
+size 2883629

9tFJT4oBgHgl3EQfoyxM/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:53ddd3b99cd93382787d6b21f9b26b7d12aa05a675b89a31ed741f248b809bab
+size 103487

AtAzT4oBgHgl3EQfhv1C/content/tmp_files/2301.01488v1.pdf.txt

@@ -0,0 +1,1850 @@
+Informed Down-Sampled Lexicase Selection:
+Identifying productive training cases for
+efficient problem solving
+Ryan Boldi∗
+rbahlousbold@umass.edu
+University of Massachusetts, Amherst, MA 01003, USA
+Martin Briesch∗
+briesch@uni-mainz.de
+Johannes Gutenberg University, Mainz, 55128, Germany
+Dominik Sobania
+dsobania@uni-mainz.de
+Johannes Gutenberg University, Mainz, 55128, Germany
+Alexander Lalejini
+lalejina@gvsu.edu
+Grand Valley State University, Allendale, MI 49401, USA
+Thomas Helmuth
+thelmuth@hamilton.edu
+Hamilton College, Clinton, NY, 13323, USA
+Franz Rothlauf
+rothlauf@uni-mainz.de
+Johannes Gutenberg University, Mainz, 55128, Germany
+Charles Ofria
+ofria@msu.edu
+Michigan State University, East Lansing, MI 48824, USA
+Lee Spector
+lspector@amherst.edu
+Amherst College, Amherst, MA 01002, USA
+Abstract
+Genetic Programming (GP) often uses large training sets and requires all individuals
+to be evaluated on all training cases during selection. Random down-sampled lexicase
+selection evaluates individuals on only a random subset of the training cases allow-
+ing for more individuals to be explored with the same amount of program executions.
+However, creating a down-sample randomly might exclude important cases from the
+current down-sample for a number of generations, while cases that measure the same
+behavior (synonymous cases) may be overused despite their redundancy. In this work,
+we introduce Informed Down-Sampled Lexicase Selection.
+This method leverages
+population statistics to build down-samples that contain more distinct and therefore
+informative training cases. Through an empirical investigation across two different GP
+systems (PushGP and Grammar-Guided GP), we find that informed down-sampling
+significantly outperforms random down-sampling on a set of contemporary program
+synthesis benchmark problems. Through an analysis of the created down-samples, we
+find that important training cases are included in the down-sample consistently across
+independent evolutionary runs and systems. We hypothesize that this improvement
+can be attributed to the ability of Informed Down-Sampled Lexicase Selection to main-
+tain more specialist individuals over the course of evolution, while also benefiting from
+reduced per-evaluation costs.
+Keywords
+Genetic programming, parent selection algorithms, selection schemes, lexicase selec-
+tion, down-sampling, informed down-sampling
+∗Both authors contributed equally.
+©2022 by the Massachusetts Institute of Technology
+Preprint
+arXiv:2301.01488v1  [cs.NE]  4 Jan 2023
+
+R. Boldi, M. Briesch, D. Sobania, A. Lalejini, T. Helmuth, F. Rothlauf, C. Ofria and L. Spector
+1
+Introduction
+In Evolutionary Computation, we often use large sets of training data to evaluate the
+quality of candidate solutions. For instance, most Genetic Programming (GP) systems
+evaluate programs using input/output examples (training cases) that specify the ex-
+pected behavior of a correct program. Many GP selection strategies aggregate each
+program’s performance across all training cases to produce one fitness score that can be
+used for selection. In contrast, lexicase selection (Spector, 2012; Helmuth et al., 2015)
+avoids aggregation and considers each training case separately, which has been shown
+to improve diversity maintenance (Helmuth et al., 2016; Dolson and Ofria, 2018) and
+problem-solving success across a wide range of domains (Moore and Stanton, 2017;
+Metevier et al., 2019; Aenugu and Spector, 2019; Ding and Spector, 2021; Lalejini et al.,
+2022).
+However, standard lexicase selection has the drawback that we have to evaluate all
+individuals on all training cases, which can be computationally expensive when eval-
+uation is non-trivial. To reduce lexicase selection’s computational cost, recent work in-
+troduced down-sampled lexicase selection (Moore and Stanton, 2017; Hernandez et al.,
+2019; Ferguson et al., 2020). In down-sampled lexicase selection, the training set is ran-
+domly down-sampled, reducing the number of test case evaluations required to assess
+the quality of each candidate solution. This in turn reduces the cost of evaluating an
+entire set of individuals, allowing us to reallocate computational resources to other as-
+pects of an evolutionary search (e.g., increasing search time or population size), which
+can substantially improve problem-solving success (Helmuth and Spector, 2020, 2021;
+Hernandez et al., 2019). However, a naive random down-sample can leave out poten-
+tially important test cases, resulting in a loss of diversity (Ferguson et al., 2020; Helmuth
+et al., 2020; Hernandez et al., 2022).
+In order to put more computational effort towards evaluating individuals on im-
+portant training cases, we propose informed down-sampling (IDS), which uses runtime
+population statistics to build a down-sample that contains more distinct cases.
+Given a set of solutions, two training cases are distinct from each other if the sub-
+sets of solutions that solve each of the two test cases have little-to-no overlap. Two
+training cases are synonymous if the opposite is true: there is substantial overlap be-
+tween the subsets of solutions that solve each case*. Consequently, Informed down-
+sampling favors the distinct training cases over synonymous cases when building a
+down-sample to use for selection. We expect these informed down-samples to better
+maintain unique individuals, increasing overall population diversity while also putting
+more selection pressure on individuals whose descendants are more likely to solve the
+problem. These unique individuals are often viewed as the stepping-stones for evolu-
+tion to use in finding a perfect solution program (Helmuth et al., 2020).
+To assess the performance of Informed Down-Sampled Lexicase Selection, we
+compare lexicase selection without down-sampling (standard lexicase), with random
+down-sampling, and with informed down-sampling across eight problems from the
+first and second program synthesis benchmark suites (Helmuth and Spector, 2015; Hel-
+muth and Kelly, 2021). We conduct our experiments in two independent GP frame-
+works, Grammar-Guided Genetic Programming (G3P) (Whigham et al., 1995; Forsten-
+lechner et al., 2016, 2017) and PushGP (Spector and Robinson, 2002; Spector et al., 2004).
+We find that building a down-sample based on information we collect from the
+*Synonymous cases can also be thought of as cases that have different inputs and outputs yet measure
+a very similar functionality such that there is a high correlation between individuals’ performance on these
+cases.
+2
+Preprint
+
+Informed Down-Sampled Lexicase Selection
+population is a valuable way to improve the success rates of evolutionary runs at a
+fixed computational cost. Furthermore, simply tracking which cases are distinct, and
+ensuring they are placed in a down-sample, can significantly improve problem solving
+performance. Our results provide evidence that informed down-sampling improves
+the success rate of search in the two GP systems used. By analyzing the composition
+of down-samples, we also verify that informed down-sampling builds down-samples
+that contain more informative test cases (i.e. edge cases) than random down-sampling.
+2
+Related Work
+In most GP applications, parent selection uses the performance of candidate solutions
+on a set of training cases to pick individuals that contribute genetic material to the next
+generation. Most selection algorithms aggregate the scores on these training cases to get
+a single score per candidate and then select the most fit candidates using tournament
+selection (Brindle, 1980), implicit fitness sharing (Smith et al., 1993), fitness proportion-
+ate selection (Holland, 1992), or another selection strategy. The fitness aggregation pro-
+cedure for these methods often results in a loss of semantic information about which
+training cases the individual performs well on (Krawiec et al., 2016), motivating the
+development of selection strategies that consider each individual’s performance on all
+training cases encountered (Vanneschi et al., 2014; Goings et al., 2012; Deb et al., 2002;
+Horn et al., 1994).
+In contrast, lexicase selection does not aggregate fitness or performance measures
+(Spector, 2012). For each parent selection event, the lexicase selection procedure first
+places all individuals in the population into a “parent pool” (i.e., the pool of individ-
+uals eligible to be selected). To select a parent, lexicase selection shuffles the training
+cases into a random ordering, and each training case is considered in sequence. For
+each training case, the parent pool is filtered down to just the individuals that have the
+best (or tie for the best) performance, removing all but the best candidates from further
+consideration. If there is only one individual that remains in the pool during this filter-
+ing process, this individual is selected. If the training cases are exhausted and there are
+still individuals in the pool, one of these individuals is selected at random.
+Meanwhile, many variants of lexicase selection have been proposed for use in dif-
+ferent problems or domains. For example, epsilon lexicase selection (La Cava et al.,
+2016; Moore and Stanton, 2017), batch lexicase selection (Aenugu and Spector, 2019;
+Sobania and Rothlauf, 2022), gradient lexicase selection (Ding and Spector, 2021), lexi-
+case selection for GAs (Metevier et al., 2019), weighted shuffle lexicase selection (Troise
+and Helmuth, 2017), and fast lexicase selection (Ding et al., 2022).
+One of the most promising variants of lexicase selection is down-sampled lexicase
+selection, which was first proposed for expensive evolutionary robotics runs by Moore
+and Stanton (2017) and later formalized by Hernandez et al. (2019) for GP runs. So far,
+down-sampled lexicase selection increased the success and generalization rates for a
+variety of problems (Ferguson et al., 2020). Down-sampled lexicase selection works by
+randomly sampling once in each generation the training set to create a smaller set of
+cases. These cases are then used to perform all selection events in the population for
+that one generation. This limitation on the number of test cases reduces the computa-
+tional costs of evaluating the individuals, which is usually one of the most expensive
+operations in evolutionary runs. These savings could be used to perform computation-
+ally cheaper GP runs, increase the population size, or run evolution for more genera-
+tions.
+Down-sampled lexicase selection has also been found to significantly outperform
+Preprint
+3
+
+R. Boldi, M. Briesch, D. Sobania, A. Lalejini, T. Helmuth, F. Rothlauf, C. Ofria and L. Spector
+regular lexicase selection in a variety of program synthesis benchmarks (Hernandez
+et al., 2019; Ferguson et al., 2020; Helmuth and Spector, 2020, 2021; Helmuth and Ab-
+delhady, 2020). However, creating a down-sample randomly can exclude important
+training cases from the current down-sample for a number of generations (Hernandez
+et al., 2022), while synonymous cases may be overused. As a first attempt at chang-
+ing the composition of cases in the down-sample, Boldi et al. (2022) explored using
+a rolling down-sample and a disjoint down-sample for lexicase selection runs. While
+the results were neutral-if-not-negative, they highlighted the presence of synonymous
+cases in practice and suggest that an attempt at mediating the time put into evaluating
+individuals on these synonymous cases might improve search performance.
+Work in the EC literature that is related to informed down-sampling primarily
+includes the co-evolution of fitness predictors and maximizers (Schmidt and Lipson,
+2005, 2008; ˇSikulov´a and Sekanina, 2012). That work attempts to evolve a smaller set
+of training cases, or fitness predictors, to evaluate the fitness of individuals instead of
+using the entire training set. While our studied methods do not involve co-evolution,
+they both result in a compressed training set that is roughly as informative as the set
+of all available data. Another example is the use of random down-sampling to im-
+prove performance of AutoML runs that use Genetic Programming (Zogaj et al., 2021).
+In the broader machine learning community, random down-sampling is used to gen-
+erate mini-batches for stochastic gradient descent (Ruder, 2017), and forms of non-
+random down-sampling are used to detect hard or informative parts of the training
+data (Loshchilov and Hutter, 2015; Bachem et al., 2017; Paul et al., 2021; Chrysakis and
+Moens, 2020).
+3
+Informed Down-Sampling
+Informed down-sampling addresses randomly down-sampled lexicase’s drawback of
+sometimes including many synonymous training cases in a down-sample, which is
+computationally inefficient and can result in a failure to accurately assess candidate so-
+lution quality. For example, down-sampled lexicase selection might fail to select candi-
+date solutions that specialize on training cases absent from a particular random down-
+sample, resulting in the loss of potentially important genetic material from the popu-
+lation. Instead of down-sampling randomly, informed down-sampling creates down-
+samples composed of more distinct training cases than a random sample would contain
+using runtime population statistics. As a result, we expect informed down-sampling
+lexicase selection to maintain more diverse populations, while reducing computation
+spent on evaluating individuals on synonymous training cases.
+We suggest two methods of building an informed down-sample. First, we explore
+the idealized effectiveness of informed down-sampling by presenting it with full infor-
+mation. This method requires evaluating the entire population on all training cases,
+performing the same number of program executions per generation as normal lexicase
+selection. Therefore, informed down-sampling with full information cannot capital-
+ize on the computational savings afforded by random down-sampling. However, the
+full information approach provides useful intuition for building an informed down-
+sample, allowing us to measure the problem-solving success of our sampling approach
+under idealized conditions.
+Next, we present an approach for creating an informed down-sample that reduces
+the number of per-generation evaluations required for selection (relative to standard
+lexicase selection). This second approach, referred to as the “sparse information” ap-
+proach, estimates the distinctness of training cases based on a sample of individuals
+4
+Preprint
+
+Informed Down-Sampled Lexicase Selection
+I1
+I2
+I3
+I4
+I5
+I6
+�
+�����
+�
+�����
+S1
+0
+1
+0
+1
+1
+0
+S2
+1
+1
+0
+0
+1
+1
+S3
+1
+0
+1
+1
+0
+1
+S4
+0
+1
+0
+0
+1
+1
+S5
+0
+1
+0
+1
+1
+0
+Figure 1: Example of the data structure that is used to determine distances between
+cases. c1,...,5 are cases, with their respective solve vectors S1,...,5, and I1,...,6 are indi-
+viduals. The entry at Sj and Ii represents whether the ith individual solved the jth test
+case or not. The binary solve vectors Sj can be read off as the respective row for the
+jth case. The distance between two cases, D(cx, cy), is the Hamming distance between
+their respective solve vectors. For example, D(c1, c2) = 3 and D(c2, c3) = 4.
+from the parent population. Indeed, building an informed down-sample using sparse
+information results in nearly the same per-generation evaluation savings as when using
+random down-sampling.
+3.1
+Building an Informed Down-Sample with Full Information
+In our informed down-sampling approach with full information, we create one down-
+sample of training cases per generation, and we use candidate solution performances
+on only the sampled training cases to choose parents with lexicase selection. To con-
+struct an informed down-sample with full information, we evaluate all members of the
+population on all training cases. In this work, each of these evaluations is on a pass/fail
+basis. Next, we construct the “solve vector” Sj for each training case cj, which is a vec-
+tor of binary values that specifies which individuals in the population have solved the
+training case. We then calculate the Hamming distance between solve vectors for all
+pairs of training cases, allowing us to measure how distinct training cases are relative
+to one another.
+We begin constructing the down-sample by randomly selecting an initial training
+case to include. Then we find the training case whose solve vector is maximally distant
+from the closest training case already included in the down-sample, and add it to the
+down-sample. We repeatedly add training cases to the down-sample in this way until
+reaching a parameterized sample size.
+Figure 1 provides an example set of binary solve vectors for a set of five training
+cases and a population of six individuals.
+The columns in this matrix Ii describe the performance of the ith individual on
+all cases. A value of 1 at (Ii, cj) implies that the ith individual solved the jth test case
+(error = 0), or Si
+j = 1. Since all members of a population of size p are evaluated on all
+test cases (at least initially), we can say that ∥Sj∥ = p for all cases, cj. Thus, the number
+of columns corresponds to the population size.
+We define the distance between two training cases D(cx, cy) := Hamming(Sx, Sy)
+where Hamming(·, ·) is the Hamming distance between two vectors. For binary vec-
+tors, the distance function is defined as: D(cx, cy) = �p
+i=1 |Si
+x − Si
+y|. Thus, two training
+cases that are solved by the same set of individuals are deemed to have D(c1, c2) = 0
+Preprint
+5
+
+R. Boldi, M. Briesch, D. Sobania, A. Lalejini, T. Helmuth, F. Rothlauf, C. Ofria and L. Spector
+and are called “synonymous cases”. For example, for the cases in Figure 1, c1 and c5
+have identical solve vectors, and therefore are synonymous (D(c1, c5) = 0).
+We think of this distance function as indicating the joint information contained in
+a pair of cases. Two cases that have exactly the same individuals solving them (i.e. are
+synonymous) have little to no joint information because having both of the cases in
+the sample would be about as informative as just having one of them. Two cases that
+have a high distance from each other, due to being solved by different subsets of the
+population, have high joint information as each case is responsible for informing the
+system about the performance of one set of individuals. Having both of these cases, as
+opposed to one alone, would be a more faithful approximation of using the full training
+set.
+Once we have a method to evaluate the pairwise distance between two cases, we
+can use it to select a down-sample of the training set for use in the current generation.
+In this work, we apply a variant of Farthest First Traversal to select the down-sample
+(Hochbaum and Shmoys, 1985). The creation of the down-sample starts with the selec-
+tion of one random case to include. Then, at each step, we scan each unselected test
+case and measure it’s minimum distance to any test in the current down-sample. We
+select the case that has the largest minimum distance. In other words, we successively
+add the test case that is furthest from the current down-sample at its nearest point.
+Our Farthest First Traversal algorithm is shown in algorithm 1. Starting with an
+empty down-sample, we first add a random case to the down-sample (line 4), and
+then iteratively add the cases that are maximally far from the closest case to it (5-9). If
+there are multiple cases with the same maximum minimum distance, ties are broken
+randomly. The MinDisti value stores the distance from a given case, ci to the closest
+case to it in the down-sample. The cases.popMaxMinDistCase() function removes
+and returns the case in cases that has the maximum value for MinDisti. Note here
+that it is often the case that the minimum distances all go to zero at a point during the
+down-sample formation. At this point, every case left over in the training set has a
+synonymous case in the down-sample already. When this happens, the farthest first
+procedure will automatically select cases at random from the training set to fill up the
+required down-sample size. Figure 2 shows an example of performing informed down-
+sampling with full information using the case solve vectors from Figure 1.
+Algorithm 1 Farthest First Traversal Down-Sample Selection
+Data: D(·, ·) : D(ci, cj) = D(cj, ci) = distance from case i to case j,
+r = down-sample rate
+1: cases ← set of all cases in training set
+2: ds ← empty set
+▷ the down-sample
+3: size ← r × |cases|
+▷ desired size of down-sample
+4: ds.add(cases.popRandomCase())
+5: while ∥ds∥ < size do
+6:
+for every case c in cases do
+7:
+MinDisti ← minimum distance from ci to any case in ds
+8:
+end for
+9:
+ds.add(cases.popMaxMinDistCase())
+10: end while
+11: return ds
+6
+Preprint
+
+Informed Down-Sampled Lexicase Selection
+D =
+c1
+c2
+c3
+c4
+c5
+�
+�����
+�
+�����
+c1
+0
+3
+4
+2
+0
+c2
+3
+0
+4
+1
+3
+c3
+4
+4
+0
+5
+5
+c4
+2
+1
+5
+0
+2
+c5
+0
+3
+5
+2
+0
+Random
+�
+��
+�
+ds = {c1}
+→
+c3 had max. distance to c1
+�
+��
+�
+ds = {c1, c3}
+→
+c2 had max. min. distance to {c1, c3}
+�
+��
+�
+ds = {c1, c3, c2}
+Figure 2: Example running procedure of informed down-sampling with full informa-
+tion to pick a down-sample of size 3 (or r =
+3
+5). We have a tabular representation
+of the distance function D generated by computing the Hamming distance between
+each pair of cases’ solve vectors. Beginning with a randomly selected case c1, we se-
+quentially add the cases that are at the maximum distance to their closest case in the
+down-sample. The first step is simply finding the case (c3) in the training set with the
+maximum distance to c1. To select the next case, we need to find, for c2, c4 and c5,
+which of c1 and c3 is closest to them, respectively, and then which of those cases is far-
+thest away. In this example, c2 was added as it had a higher distance (3) to its closest
+case than did c4 or c5 (2 and 0, respectively). Notice that the cases that were left out, c4
+and c5, are synonymous or nearly synonymous with cases already in the down-sample:
+c2 and c1, respectively.
+3.2
+Building an Informed Down-Sample with Sparse Information
+Down-sampled lexicase selection’s problem-solving benefits stem from the computa-
+tional savings gained by not evaluating the entire population on the whole training set
+for every generation. For a fixed computational budget, down-sampling allows more
+computational resources to be allocated to other aspects of evolutionary search, such
+as running for more generations or increasing population size. As a result, a larger
+portion of the search space can be explored (Helmuth and Spector, 2021). Informed
+down-sampling with full information requires the evaluation of all individuals on all
+training cases in order to construct the down-sample to use in selection. This entire pro-
+cess is counter productive, as we could have just used the initial population evaluation
+to select individuals and circumvent the entire down-sampling process. The benefit of
+down-sampling comes from its ability to use sparse information in the individual selec-
+tion process. Since our aim is to improve on random down-sampling, we must reduce
+the number of necessary program executions in order to calculate distances between
+training cases, so that we can benefit from sparse evaluations in both our individual
+selections and our down-sample creation.
+We present two methods to decrease the number of evaluations required for the
+pairwise distance calculation procedure. The first method, parent sampling, samples a
+proportion ρ of the parents to evaluate the distances for every generation. These parent-
+samples are evaluated on the entire training set. In our runs with a population size of
+1000, if we were to randomly sample 0.01 (or ρ = 0.01) of these parents to become
+the parent sample, these 10 parents would be evaluated on all training cases. This
+results in case solve vectors of length 10 that are used to calculate the distances between
+Preprint
+7
+
+R. Boldi, M. Briesch, D. Sobania, A. Lalejini, T. Helmuth, F. Rothlauf, C. Ofria and L. Spector
+cases. Distances between cases are determined purely based on these parent-sample
+evaluations. We use the distance matrix generated from these parents to estimate the
+joint informativeness.
+The second method, scheduled case distance computation, involves recomputing the
+distance matrix from the current population every k generations, as opposed to every
+generation. This schedule reduces the amount of computation required for the evalua-
+tion of case distances even further by not performing it every generation. While such
+a schedule does not update the distances between cases as often, we still re-sample the
+down-sample based on these distances every generation. Due to the stochastic nature of
+the down-sample selection process (specifically the random selection of the first case),
+it is likely that the same down-sample will not be used to evaluate the population in
+consecutive generations.
+In combination, parent sampling and scheduled case distance computation allow
+us to select a down-sample using far less information about individuals while losing
+only a small amount of information about cases and their similarity. This technique
+enables informed down-sampling to explore nearly as many individuals as random
+down-sampling does. Putting it all together, the informed down-sampling with sparse
+information algorithm is detailed in Algorithm 2. This algorithm walks through a sin-
+gle generation’s selection events, returning the parents for the next generation.
+Algorithm 2 Informed Down-Sampling with Sparse Information
+Data:
+P : population,
+cases: set of all training cases,
+k : scheduled case distance computation parameter,
+ρ : parent sampling rate,
+G : current generation counter,
+D : case distance matrix.
+▷ all distances initialized to be maximally far
+Result: A list of selected parents
+1: if G%k == 0 then
+2:
+ˆP ← sample ρ×|P| parents from P
+3:
+evaluate ˆP on cases
+4:
+calculate D from case solve vectors from solutions in ˆP on cases
+5: end if
+6: D(·, ·) ← distance function derived from indexing into D
+7: ds ← create downsample using farthest first traversal down-sampling (See Algo 1)
+8: P ← select |P| new parents using lexicase selection from P using ds as cases
+9: return P
+4
+Experimental Methods
+We conducted a series of experiments to study the performance of informed down-
+sampled lexicase selection. We compared the performance of informed down-sampled,
+random down-sampled, and standard lexicase selection on a series of program synthe-
+sis benchmark problems. We performed all experiments in two independent genetic
+programming systems to show that the findings are robust across different program
+representations: PushGP and Grammar Guided Genetic Programming (G3P).
+This section introduces the benchmark problems and genetic programming sys-
+tems used in our experiments and describes our experimental design.
+8
+Preprint
+
+Informed Down-Sampled Lexicase Selection
+Table 1: Program synthesis benchmark problems selected from the first and second gen-
+eral program synthesis benchmark suite, along with their respective input and output
+types and multiplicities.
+Problem
+Suite
+Input Type
+Output Type
+Count Odds
+PSB1
+Vector of Integer
+Integer
+Find Pair
+PSB2
+Vector of Integer
+Two Integers
+Fizz Buzz
+PSB2
+Integer
+String
+Fuel Cost
+PSB2
+Vector of Integer
+Integer
+GCD
+PSB2
+Two Integers
+Integer
+Grade
+PSB1
+Five Integers
+String
+Scrabble Score
+PSB1
+String
+Integer
+Small or Large
+PSB1
+Integer
+String
+4.1
+Program Synthesis Benchmark Problems
+We evaluate each system using eight program synthesis benchmark problems from the
+first and second general program synthesis benchmark suites (Helmuth and Spector,
+2015; Helmuth and Kelly, 2021). These problems are well-studied and are commonly
+used to compare parent selection algorithms in a GP context (Sobania et al., 2022b,a).
+These two benchmark suites include a variety of introductory program synthesis prob-
+lems that require the manipulation of multiple data types with complex looping or
+conditional structures.
+Each benchmark problem is defined by a set of input/output examples (referred
+to as cases) that specify the desired behavior of a correct program. For each problem,
+we split the input/output examples into a training set and a testing set. During evolu-
+tion, we assessed program quality using only the training set. We used the testing set
+to measure how well a program generalized on examples unseen during evolution. We
+consider each input/output example on a pass/fail basis; that is, a program passes a
+test case if it produces the correct output when run with the associated input. A pro-
+gram is a solution if it passes all of the training cases; it generalizes if it passes all training
+and all testing cases. We refer to runs as “success” if they result in the production of
+a generalizing solution. We used the same training and testing data sets across both
+PushGP and G3P for each problem to ensure the data available is not biasing perfor-
+mance.
+Table 1 shows the eight program synthesis benchmark problems that we have cho-
+sen, along with their input and output types. We selected these particular problems to
+allow us to test informed down-sampling on a set of easy, medium, and hard problems
+as established by published success rates using PushGP and random down-sampled
+lexicase selection (Helmuth and Spector, 2021; Helmuth and Kelly, 2022). We also en-
+sured that these problems require qualitatively different programmatic paradigms to
+solve, such as looping and conditional execution (Helmuth and Kelly, 2022).
+4.2
+Genetic Programming Systems
+PushGP is a system that evolves computer programs in the Push programming lan-
+guage, a stack-based language specifically invented for use in genetic programming
+(Spector and Robinson, 2002; Spector et al., 2004). Push literals are pushed onto one
+of a set of datatype specific stacks while instructions are also stored on a stack dur-
+ing interpretation. These instructions usually act on data from the stacks and leave
+Preprint
+9
+
+R. Boldi, M. Briesch, D. Sobania, A. Lalejini, T. Helmuth, F. Rothlauf, C. Ofria and L. Spector
+Table 2: General and System-Specific Evolution Parameters
+General Parameter
+Value
+runs per problem
+100
+population size
+1,000
+size of training set
+200
+size of test set
+1,000
+program execution limit
+60 million
+maximum number (base) of generations
+300
+PushGP Parameter
+Value
+variation operator
+UMAD
+UMAD rate
+0.1
+G3P Parameter
+Value
+crossover operator
+subtree crossover
+crossover probability
+0.95
+mutation operator
+subtree mutation
+mutation steps
+1
+maximum tree depth
+17
+elite size
+5
+initialisation
+position-independent grow
+maximum initial tree depth
+10
+their return value on the stacks. Instructions take values from and return results to
+the appropriately typed stack, including from and to the instruction stack, allowing for
+programs to use multiple data types and complex conditional execution paradigms. In
+this work, we used the propeller implementation of PushGP†.
+G3P uses a context-free grammar in Backus-Naur form to evolve individuals in a
+desired programming language and supports the use of different data types and con-
+trol structures (Whigham et al., 1995; Forstenlechner et al., 2016, 2017). To prevent the
+generation of many invalid solutions during search, we use a tree-based representation
+instead of the common genotype-phenotype mapping known from classical grammat-
+ical evolution (Ryan et al., 1998; Sobania and Rothlauf, 2020). For the implementation
+of G3P, our code‡ is based on the PonyGE2 framework (Fenton et al., 2017).
+Table 2 shows the system-specific parameters for PushGP and G3P, and the general
+parameters that are used in both systems. The “runs per problem” parameter refers to
+the number of independent evolutionary runs that were conducted for each problem
+and experimental configuration. The PushGP system uses the uniform mutation by ad-
+dition and deletion (UMAD) mutation operator (Helmuth et al., 2018). This UMAD op-
+erator works with a 0.1 mutation rate. For G3P, we use subtree mutation and crossover,
+with a crossover probability of 0.95. The initialization for G3P is position-independent
+grow (Fagan et al., 2016). We use grammars based on those provided by the PonyGE2
+framework with small adjustments to make them better comparable to the PushGP
+instructions.
+†https://github.com/ryanboldi/propeller/releases/tag/Informed-Downsampling
+‡https://gitlab.rlp.net/mbriesc/informed-down-sampled-lexicase-selection
+10
+Preprint
+
+Informed Down-Sampled Lexicase Selection
+4.3
+Evaluation and Generation Limits
+In order to make a fair comparison between methods that perform different numbers of
+program executions per generation, we use the recommendation from the PSB2 bench-
+mark suite to limit each GP run to 60 million program executions (Helmuth and Kelly,
+2021). Since program executions typically take up the majority of the computational
+requirements of a GP run, this ensures runs receive similar amounts of computation re-
+gardless of whether they use down-sampling. In standard runs using all training cases,
+the 60 million executions are used by at most 300 generations of a population size of
+1000 individuals evaluated on 200 cases. With random down-sampling, we increase
+the maximum number of generations by the same factor as the down-sampling. For
+example, if one tenth of the training data is used in each sample, we can run evolu-
+tion for ten times the number of generations while keeping the number of individual
+program executions constant.
+More generally, if we let G be the maximum number of generations for a run using
+all training cases, we allow our random down-sampling runs a limit of ˆG generations
+where ˆG is given by
+ˆG = G
+r ,
+where r is the down-sample rate. For informed down-sampled lexicase selection the
+generational limit is calculated by
+ˆG =
+G
+r + ρ(1−r)
+k
+,
+where ρ is the parent sampling rate and k is the parameter for the scheduled case dis-
+tance computation. The exact generational limits for each experimental configuration
+are shown in table 3.§
+4.4
+Experimental Configurations
+We explore 11 different configurations of lexicase selection for each problem: standard
+lexicase selection (Lex), random down-sampled lexicase selection (Rnd), IDS lexicase
+selection with full information, as well as three sparse information configurations. To
+better match previous literature, all down-sampling methods were performed both
+with r ∈ {0.05; 0.1}.
+Table 3 shows the configurations of the different runs performed in this work.
+These runs, due to different generational computational costs, have different genera-
+tional limits as explained in section 4.3.
+Full information down-sampling is simply using a parent-sample rate of 1, which
+means that the distances between training cases are determined by all parents’ perfor-
+mance on every test case. With this, the quality of the distance metric between two
+cases is not limited by the parent-sampling or generational gaps we are using to reduce
+computational load. Full information down-sampling is included as a control exper-
+iment to compare with using all cases for selection in standard lexicase selection. It
+is important to note that we run for the same number of generations as with regular
+lexicase selection because we need to evaluate all parents on all test-cases in order to
+determine the distances between the cases.
+§As our implementations evaluate the fitness of individuals in the parent sample twice, we run the IDS
+with sparse information runs for slightly (< 40) fewer generations to compensate the additional computa-
+tional effort.
+Preprint
+11
+
+R. Boldi, M. Briesch, D. Sobania, A. Lalejini, T. Helmuth, F. Rothlauf, C. Ofria and L. Spector
+Table 3: Different settings conducted in our experiments for standard lexicase selection
+(Lex), random down-sampled lexicase selection (Rnd) and informed down-sampled
+lexicase selection (IDS). The variable r denotes the down-sampling rate, ρ is the parent
+sampling rate, k is generational interval at which we update the distance matrix and ˆG
+specifies the maximum number of generations.
+Method
+Lex
+Rnd
+IDS
+Rnd
+IDS
+r
+-
+0.05
+0.05
+0.1
+0.1
+ρ
+-
+-
+1
+0.01
+0.01
+0.01
+-
+1
+0.01
+0.01
+0.01
+k
+-
+-
+1
+1
+10
+100
+-
+1
+1
+10
+100
+ˆG
+300
+6000
+300
+5042
+5888
+5988
+3000
+300
+2752
+2973
+2997
+Finally, the six informed down-sampling methods we have chosen for this work
+include, for both the 0.05 and 0.1 down-sample rate (r), 0.01 parent sampling (ρ) rate
+with a few different distance calculation scheduling (k) parameters. Through a set of
+preliminary experiments, the value of ρ = 0.01 for the parent sampling rate was de-
+termined to be effective while not resulting in too many extra program executions¶.
+In conjunction, these hyper-parameters mean that every k generations, 10 parents are
+used to determine the distances between all training cases, where k ∈ {1, 10, 100}.
+5
+Results and Discussion
+We discuss the success rates achieved by both GP systems using standard lexicase se-
+lection, random down-sampling, and different configurations of IDS. Further, we study
+how the composition of the down-samples found by IDS change over the number of
+generations.
+5.1
+Informed Down-Sampling Improves Problem-solving Success
+Tables 4 and 5 show the success rates for PushGP and G3P respectively on the chosen
+program synthesis benchmark problems for different parameter configurations. The
+success rate is defined as the number of runs that result in a program that passes the
+complete training set as well as the entire unseen test set.
+For random down-sampling and IDS, we measured solutions on only the down-
+samples during the actual run. As such, we execute these runs to the maximum gener-
+ational limit, and then conduct a post-hoc analysis to see if any solutions passed all of
+the training cases. If so, this is the solution that we then evaluate on the unseen test set
+to determine whether it generalizes or not.
+For all studied configurations, we report success rates based on 100 runs. For each
+benchmark problem, we highlight in bold the best success rate at each of the down-
+sample sizes. Problem names in bold are those where an informed down-sampling
+run outperformed random at both down-sample rates on that problem. Problem names
+that are underlined are those where a random down-sampling run outperformed an
+informed down-sampling run at both down-sample rates. Asterisks signify results
+that are significantly better than random down-sampling at the same down-sample size.
+¶As we are trying to approach the computational savings of random down-sampled lexicase selection,
+the smaller the value of ρ, the better. We found that the relatively small value of ρ = 0.01 resulted in sampling
+that was good enough to determine the joint case information.
+12
+Preprint
+
+Informed Down-Sampled Lexicase Selection
+Standard lexicase selection was not included in our statistical analyses, as IDS is pre-
+sented to improve upon random down-sampling at a fixed down-sample size. We per-
+formed significance analysis with a two proportion z-test and Bonferroni-Holm correc-
+tion. Shown with * are those significant at the α = 0.1 level, ** the α = 0.05 level, and
+*** the α = 0.01 level.
+For the PushGP results, let us consider the Fizz Buzz problem. Standard lexicase
+selection had 13 successful runs. Using random down-sampling at the 0.05 down-
+sampling rate improved this result to 64, in line with the findings of Helmuth and
+Spector (2021). Using the same down-sampling rate with IDS, a 0.01 parent rate, and
+k = 100 yielded 95 successful runs. This is significantly better than random down-
+sampling at the 0.01 level. This is an important result as IDS is significantly improving
+on random down-sampling, which in turn improves on lexicase selection. Another set
+of PushGP IDS runs where we observed significant improvements were those of the
+Count Odds problem. While standard lexicase selection achieves 24 successes, random
+down-sampling at either down-sample rate (r = 0.05 or r = 0.1) does not produce
+more than 26 successful runs. The failure to meaningfully improve success rates by
+random down-sampling seemed to be addressed by informed down-sampling. This
+is clear as informed down-sampling at all configurations ensures that close to if-not-
+all 100 runs successfully generalize to the held out test set. This and similar results
+hint that while randomly down-sampled lexicase selection works well usually, there
+are some problems where important cases might be being dropped out, resulting in a
+similar performance to standard lexicase selection despite the increased search gener-
+ations. Informed down-sampling has the ability to improve success rates both when
+random down-sampling improves upon standard lexicase selection, and when it does
+not.
+Only one configuration of G3P resulted in a significant improvement on random
+down-sampling at the same down-sample rate. For the Grade problem at the 0.05
+down-sample rate, we see significantly more successes when using IDS with ρ = 0.01
+and k = 10. For this problem, using this informed down-sample configuration re-
+sulted in 57% of the runs yielding a generalizing solution, where, using random down-
+sampling resulted in only 39% of the runs yielding a success. The fact that only a single
+configuration of IDS resulted in a significant improvement suggests that the problem-
+solving benefits of using IDS are representation- and problem-dependent, motivating
+future work to continue improving IDS to achieve more universal improvements to
+problem-solving success.
+We have a number of hypotheses explaining this improved performance. The first
+of these is that the informed down-sampling procedure increases the number of spe-
+cialists (individuals exceptional on a few cases, but have a high total error) that survive
+over the course of evolutionary time. These individuals could be better maintained
+with IDS as the cases they are exceptional on are still placed in the down-samples
+throughout evolution, preventing them from being lost as could happen when ran-
+domly down-sampling.
+Another hypothesis for IDS’s improved performance is that it reduces the compu-
+tation used to evaluate individuals on synonymous cases. When two cases are fully
+synonymous, all individuals that solve one case solve the other as well. When using
+lexicase selection, having both of these cases in the down-sample would result in little
+difference in the probability of selecting each individual compared to having only one
+case in the down-sample. After one of the two cases has been used to filter the pool
+of candidate solutions, the other will have no filtering pressure because all remaining
+Preprint
+13
+
+R. Boldi, M. Briesch, D. Sobania, A. Lalejini, T. Helmuth, F. Rothlauf, C. Ofria and L. Spector
+Table 4: Number of generalizing solutions (successes) out of 100 runs achieved by PushGP on the test set.
+Method
+Lex
+Rnd
+IDS
+Rnd
+IDS
+r
+-
+0.05
+0.1
+ρ
+-
+-
+1
+0.01
+0.01
+0.01
+-
+1
+0.01
+0.01
+0.01
+k
+-
+-
+1
+1
+10
+100
+-
+1
+1
+10
+100
+Count Odds
+24
+25
+43***
+99***
+100***
+98***
+26
+55***
+95***
+99***
+97***
+Find Pair
+5
+27
+9
+32
+32
+36
+15
+7
+19
+19
+21
+Fizz Buzz
+13
+64
+2
+85***
+94***
+95***
+45
+3
+75
+78*
+81**
+Fuel Cost
+41
+72
+1
+83
+85
+83
+76
+7
+69
+72
+70
+GCD
+20
+74
+4
+76
+67
+69
+54
+6
+56
+63
+62
+Grade
+0
+0
+0
+0
+1
+0
+1
+0
+0
+1
+1
+Scrabble Score
+8
+8
+6
+69***
+64***
+75***
+16
+9
+55***
+74***
+64***
+Small or Large
+34
+93
+37
+69
+69
+69
+69
+39
+60
+66
+54
+14
+Preprint
+
+Informed Down-Sampled Lexicase Selection
+Table 5: Number of generalizing solutions (successes) out of 100 runs achieved by G3P on the test set.
+Method
+Lex
+Rnd
+IDS
+Rnd
+IDS
+r
+-
+0.05
+0.1
+ρ
+-
+-
+1
+0.01
+0.01
+0.01
+-
+1
+0.01
+0.01
+0.01
+k
+-
+-
+1
+1
+10
+100
+-
+1
+1
+10
+100
+Count Odds
+65
+66
+45
+53
+62
+63
+67
+58
+60
+58
+72
+Find Pair
+0
+0
+0
+1
+0
+0
+1
+0
+0
+1
+0
+Fizz Buzz
+62
+83
+50
+84
+78
+85
+78
+53
+81
+89
+72
+Fuel Cost
+33
+34
+17
+28
+27
+29
+29
+21
+21
+25
+33
+GCD
+0
+1
+0
+0
+0
+1
+0
+0
+0
+0
+0
+Grade
+36
+39
+29
+51
+57*
+44
+44
+37
+46
+51
+48
+Scrabble Score
+6
+10
+1
+11
+10
+10
+14
+0
+6
+3
+3
+Small or Large
+41
+52
+49
+54
+63
+63
+59
+52
+57
+55
+63
+Preprint
+15
+
+R. Boldi, M. Briesch, D. Sobania, A. Lalejini, T. Helmuth, F. Rothlauf, C. Ofria and L. Spector
+individuals perform identically on the synonymous cases. Having a synonymous case
+does increase the chance that one of the two cases appears earlier in the shuffled case
+ordering, producing a minor (though perhaps undesired) change in selection proba-
+bility. Synonymous (or near synonymous) cases additionally take spots in the down-
+sample that cannot be allocated to other, more-informative cases. When using IDS, we
+ensure that the first few cases added to the down-sample measure relatively different
+behaviors. This may allow IDS to select a larger variety of individuals than random
+down-sampling, instead approximating the variety that could be selected by full lexi-
+case selection.
+These results, in general, make it clear that informed down-sampling by farthest
+first traversal is significantly outperforming randomly down-sampled lexicase selec-
+tion on a portion of these program synthesis benchmark problems for the PushGP
+evolutionary framework. The G3P results are less clearly in favor of informed down-
+sampling, but still point to minor improvements in success rates. It is important to
+note that all of our down-sampled runs (besides full-information) consistently and sig-
+nificantly outperform standard lexicase selection, which has in turn been shown to
+significantly outperform other selection strategies. This result agrees with that of Hel-
+muth and Abdelhady (2020), showing down-sampled lexicase selection being, before
+this work, the state of the art in program synthesis with genetic programming. Our in-
+formed down-sampling runs outperform random down-sampling (higher success rate
+for both down-sample rates) on 6/8 of the problems we studied for PushGP, with 3/8
+of them being statistically significant. For G3P, informed down-sampling improves on
+3/8 problems, with 1/8 being significant.
+Random down-sampling outperformed informed down-sampling (across both
+down-sampling levels) on only one problem (Small or Large) for PushGP, and none for
+G3P. For Small or Large with PushGP, we see that the worse performance with informed
+down-sampling can be attributed to a lower generalization rate (and not worse perfor-
+mance on the training sets). The generalization rates can be found in Appendix Figure 6
+for PushGP and Appendix Figure 7 for G3P. Future work should explore the effect that
+informed down-sampling has on generalization in more depth.
+5.2
+Using Smaller Informed Down-Samples Tends to Improve Success Rates
+In general, our IDS runs at a 0.05 down-sample rate have a higher success rate than
+their equivalent counterparts at the 0.1 down-sample rate. This difference is likely due
+to the fact that the runs at a 0.1 down-sample rate have a substantially lower genera-
+tional limit, meaning that we are exploring a smaller portion of the space of possible
+solution programs. With 200 training cases, our down-sample contains 10 and 20 cases
+respectively for the 0.05 and 0.1 down-sample rates. A possible reason for the improved
+performance at 0.05 is that a larger proportion of these cases are indeed our distinct, or
+informative, cases. Note that once the Farthest First Traversal process selects a rep-
+resentative case for every synonymous group in the down-sample, every remaining
+solution’s minimum distances to the current sample will be equal to 0, so the selections
+are performed randomly to fill the rest of the cases. Since we are using the same prob-
+lems, with the same number of behavioral niches, we will see the runs with 20 cases in
+the down-sample having more synonymous cases in the down-sample. Due to the fact
+that the content of the training cases is not notably more informative to make up for the
+decreased generational limit, we see a lower success rate. We will analyze the specific
+cases that compose the down-samples in section 5.3.
+The exceptions to this trend are the full information down-sampling runs. For
+16
+Preprint
+
+Informed Down-Sampled Lexicase Selection
+these runs, the larger down-samples tend to perform better. This result is likely due
+to the fact that the generational limit was set to 300 for both sampling levels (as they
+both evaluate all individuals on all test cases), and so having a smaller down-sample
+size would not change the number of evaluations. With more cases in the sample, the
+GP method can take into account more information when performing selection, which
+could result in more informed search. The magnitude of the differences for success rate
+across sample size for the full IDS runs suggests that there are diminishing returns for
+including more cases in the sample.
+5.3
+Informed Down-Sampling Automatically Discovers Important Training Cases
+To gain a deeper insight into how IDS composes down-samples, we visualize how the
+selected training cases (used for a down-sample) develop over the generations of an
+evolutionary run.
+Figures 3 and 4 show the composition of down-samples for every problem at every
+generation using PushGP (Fig. 3) and G3P (Fig. 4) with down-sample rate r = 0.05. We
+present results for a full information configuration (ρ = 1 and k = 1) as well as a
+sparse information configuration (ρ = 0.01 and k = 10). We chose to analyze both a full
+information and sparse information run in order to see whether our sparse information
+configurations are finding the same training cases to be informative as if we had used
+all parents to evaluate the distances between training cases.
+The plots show how often certain training cases are included in the down-sample
+at every generation, averaged over all active runs. Each row represents a case in the
+training data, ordered by its position in the training set. The training sets used were
+generated by first adding some human-expert defined edge cases, and filling the rest
+with cases that were randomly generated by an function that already implements our
+desired program (oracle function). For each figure, there is a single marker on the y-
+axis that shows where exactly the expert-case cutoff for the training set was. Thus, the
+rows above the marker in the visuals are representing cases that humans determined
+to be important based on the problem definition.
+Brighter colors imply that a case is included more often, darker colors imply a
+lower number of inclusions.
+For PushGP (Figure 3), we see that the configurations with sparse information of-
+ten include the same cases in the down-sample as the runs with full information. This
+result means that by using a parent sampling rate of ρ = 0.01 and a case distance
+evaluation schedule parameter of k = 10, we can significantly reduce the number of
+evaluations needed to calculate distances between cases, while still maintaining a good
+approximation to the ground truth (full information, where we use all parents every
+generation to calculate distances). However, the composition for our sparse informa-
+tion runs are slightly more noisy than that for full information, suggesting that using
+parent sampling could introduce some extra stochasticity to the down-sample creation
+process.
+For all studied benchmark problems, we see that IDS has a strong bias toward
+specific training cases that are included substantially more often in the down-sample.
+These selected training cases are mainly consistent with the human-defined edge cases
+that exist at the beginning of the training set. This result shows that informed down-
+sampling is indeed often finding the same cases to be informative as those that a human
+expert would, without any knowledge of the problem definition. However, with IDS,
+we can draw further comparisons of informativeness within this expert-defined groups
+of cases. This can be seen as some cases are selected more often that others within the
+Preprint
+17
+
+R. Boldi, M. Briesch, D. Sobania, A. Lalejini, T. Helmuth, F. Rothlauf, C. Ofria and L. Spector
+Full Information
+Sparse Information
+Count Odds
+Cases
+Find Pair
+Cases
+Fizz Buzz
+Cases
+Generations
+Fuel Cost
+Cases
+Generations
+Figure 3: Down-sample composition over generations for PushGP with 0.05 down-
+sample rate for a full information (ρ = 1 and k = 1) and a sparse information configu-
+ration (ρ = 0.01 and k = 10).
+first several cases.
+We then look at the labels of the specific training cases that are found to be impor-
+tant. We see that these training cases make sense to be included more often than others
+in the down-samples. Note that the labels of the specific training cases are not included
+18
+Preprint
+
+Informed Down-Sampled Lexicase Selection
+Full Information
+Sparse Information
+GCD
+Cases
+Grade
+Cases
+Scrabble Score
+Cases
+Generations
+Small or Large
+Cases
+Generations
+Figure 3: Continued.
+in the plots for simplicity, but can be queried based on their specific index in the data
+sets provided in our code implementation.
+For example, for the Small or Large problem, cases around the decision boundaries
+as well as numbers between 0 and 1000 are more often included. For the Grade problem,
+those edge cases with very close decision boundaries are included while the ones with
+far away boundaries are not taken into account for the down-sample. For Fuel Cost,
+Preprint
+19
+
+R. Boldi, M. Briesch, D. Sobania, A. Lalejini, T. Helmuth, F. Rothlauf, C. Ofria and L. Spector
+Full Information
+Sparse Information
+Count Odds
+Cases
+Find Pair
+Cases
+Fizz Buzz
+Cases
+Generations
+Fuel Cost
+Cases
+Generations
+Figure 4: Down-sample composition over generations for G3P with 0.05 down-sample
+rate for a full information (ρ = 1 and k = 1) and a sparse information configuration
+(ρ = 0.01 and k = 10).
+nearly all of the human defined edge cases are found to be important, while for the
+GCD problem the first two cases in particular make it in nearly every down-sample,
+while the rest are selected less often.
+20
+Preprint
+
+Informed Down-Sampled Lexicase Selection
+Full Information
+Sparse Information
+GCD
+Cases
+Grade
+Cases
+Scrabble Score
+Cases
+Generations
+Small or Large
+Cases
+Generations
+Figure 4: Continued.
+For the Scrabble Score problem, we see that the first edge cases, which specify the
+score for each letter, does not seem to be informative at all. This result is not surprising,
+as this information is already available to PushGP through a vector with these scores
+on the vector stack. However, the three edge cases after them with empty strings and
+special characters as input are included a lot. For Count Odds, the edge cases denot-
+ing empty lists, or lists with zero or a single odd number were found to be important,
+Preprint
+21
+
+R. Boldi, M. Briesch, D. Sobania, A. Lalejini, T. Helmuth, F. Rothlauf, C. Ofria and L. Spector
+indicating that those contain all the important information to learn what are odd and
+even numbers as well as how to handle a list. For Fizz Buzz, all edge cases seem im-
+portant while for the Find Pair problem only those edge cases with lists of length 3 are
+consistently included. Those lists of length 2 in the edge cases are represented in the
+down-sample less often.
+Lastly, we see that the composition of the down-sample stays rather stable during
+the evolutionary run for the PushGP system, explaining why there is only a small dif-
+ference in our experiments between calculating the distances every k = 1 and k = 100
+generations (see Table 4).
+For G3P (Fig 4), we see similar results as with PushGP. However, for the prob-
+lems that require iterative structures to be solved (Count Odds, Find Pair) we see that
+the down-sample quickly dissolves into random noise instead of any form of struc-
+ture. This dynamic occurs despite the fact that the same edge cases as with PushGP are
+initially identified in the first few generations. This result is not surprising as finding it-
+erative structures is known to be challenging for grammar-guided approaches, as such
+structures are difficult to be built step-by-step guided by the performance on a set of
+training cases. (Sobania and Rothlauf, 2020; Sobania et al., 2022b). Another difference
+between the case compositions are that, while IDS with G3P tends to discover the same
+cases as those found with PushGP, their use is less consistent, resulting in lines that
+are more faint than those for PushGP. Both of these hypotheses could help explain the
+relatively worse improvement that IDS yields for G3P than for PushGP.
+However, for the problems that require conditionals, like Small or Large and Grade,
+we see that the important cases are identified and used during evolution. This result is
+also reflected in the success rates compared to random down-sampling (see Table 5).
+Interestingly, IDS identifies many of the same cases as important for G3P as well as
+PushGP. This result suggests that the structure of the problem itself determines which
+cases are important rather than the considered representation. This dynamic makes
+IDS potentially useful across many different systems and approaches.
+6
+Conclusion and Future work
+In this work, we proposed a novel approach to construct down-samples in an informed
+manner during evolution when using down-sampled lexicase selection. We find that
+changing the composition of down-samples to include cases that are more “informa-
+tive” helps improve problem solving performance with a fixed computational bud-
+get. Informativeness, we hypothesize, is linked to how distinct the cases in the down-
+sample are. Cases that are solved by the same subset of the population are likely testing
+for the same behavior, and thus need not be included in the down-sample at the same
+time. Cases that test for different behaviors likely maintain different behavioral groups
+of individuals, which could promote and maintain higher levels of diversity in the pop-
+ulation.
+In our empirical comparisons of these down-sampling methods, we find evidence
+to support the conclusion that selecting cases in an informed manner increases the suc-
+cess rate of GP runs. These results were confirmed across two independent GP systems
+by using well studied benchmark problems. We find that using IDS often increases the
+proportion of informative cases in the down-sample as verified by improved success
+rates as well as by directly inspecting the content of the down-samples. IDS improves
+upon the state of the art selection method across the majority of the program synthesis
+problems explored in this work.
+This work is a first exploration into changing the case composition of down-
+22
+Preprint
+
+Informed Down-Sampled Lexicase Selection
+samples for lexicase selection runs. As such, it opens many potential directions for
+future research. Due to the modular nature of the informed down-sampling system,
+different methods could be used for either the pairwise information measurement, or
+for the down-sample creation portions of the algorithm. An exploration into differ-
+ent down-sampling levels, and the effects levels have on the informational content of
+down-samples is also a promising direction for future work. Additionally, IDS intro-
+duces new hyperparameters for the parent sampling rate and generational schedule;
+it would be beneficial to create a method for automatically setting these dependant on
+the problem and the state of the GP search. Finally, even though there are reasons to
+believe that IDS and down-sampling in general work well with lexicase selection, there
+is nothing that ties them to a particular selection method; it may be informative to
+explore the effects of IDS on other parent selection methods such as tournament selec-
+tion. Finally, comparing the extent to which different down-sampling strategies blunt
+lexicase’s ability to maintain specialists could also yield important insights into why
+informed down-sampling improves success rates as much as it does.
+7
+Acknowledgements
+This material is based upon work supported by the National Science Foundation un-
+der Grant No. 1617087. Any opinions, findings, and conclusions or recommendations
+expressed in this publication are those of the authors and do not necessarily reflect the
+views of the National Science Foundation.
+This work was performed in part using high performance computing equipment
+obtained under a grant from the Collaborative R&D Fund managed by the Mas-
+sachusetts Technology Collaborative.
+Parts of this research were conducted using the supercomputer Mogon and/or ad-
+visory services offered by Johannes Gutenberg University Mainz (hpc.uni-mainz.de),
+which is a member of the AHRP (Alliance for High Performance Computing in
+Rhineland Palatinate, www.ahrp.info) and the Gauss Alliance e.V.
+The authors would like to thank Anil Saini, Austin Ferguson, Cooper Sigrist, Con-
+stantin Weiser, Edward Pantridge, Jose Hernandez, Li Ding and the Members of the
+PUSH lab at Amherst College for discussions that helped shape this work.
+References
+Aenugu, S. and Spector, L. (2019). Lexicase selection in learning classifier systems. In Proceedings
+of the Genetic and Evolutionary Computation Conference, GECCO ’19, page 356–364, New York,
+NY, USA. Association for Computing Machinery.
+Bachem, O., Lucic, M., and Krause, A. (2017). Practical coreset constructions for machine learn-
+ing. arXiv: Machine Learning.
+Boldi, R., Helmuth, T., and Spector, L. (2022).
+Exploring Environmental Change for Down-
+Sampled Lexicase Selection. volume Why it Didn’t Work-Shop of ALIFE 2022: The 2022 Con-
+ference on Artificial Life.
+Brindle, A. (1980). Genetic algorithms for function optimization. PhD thesis, University of Alberta.
+Chrysakis, A. and Moens, M.-F. (2020). Online continual learning from imbalanced data. In III,
+H. D. and Singh, A., editors, Proceedings of the 37th International Conference on Machine Learning,
+volume 119 of Proceedings of Machine Learning Research, pages 1952–1961. PMLR.
+Deb, K., Pratap, A., Agarwal, S., and Meyarivan, T. (2002). A fast and elitist multiobjective genetic
+algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation, 6(2):182–197.
+Preprint
+23
+
+R. Boldi, M. Briesch, D. Sobania, A. Lalejini, T. Helmuth, F. Rothlauf, C. Ofria and L. Spector
+Ding, L., Boldi, R., Helmuth, T., and Spector, L. (2022). Lexicase selection at scale. In Genetic
+and Evolutionary Computation Conference Companion (GECCO ’22 Companion), July 9–13, 2022,
+Boston, MA, USA.
+Ding, L. and Spector, L. (2021). Optimizing neural networks with gradient lexicase selection. In
+International Conference on Learning Representations.
+Dolson, E. and Ofria, C. (2018). Ecological theory provides insights about evolutionary compu-
+tation. In Proceedings of the Genetic and Evolutionary Computation Conference Companion, GECCO
+’18, page 105–106, New York, NY, USA. Association for Computing Machinery.
+Fagan, D., Fenton, M., and O’Neill, M. (2016). Exploring position independent initialisation in
+grammatical evolution. In 2016 IEEE Congress on Evolutionary Computation (CEC), pages 5060–
+5067.
+Fenton, M., McDermott, J., Fagan, D., Forstenlechner, S., Hemberg, E., and O’Neill, M. (2017).
+Ponyge2: Grammatical evolution in python. In Proceedings of the Genetic and Evolutionary Com-
+putation Conference Companion, pages 1194–1201.
+Ferguson, A. J., Hernandez, J. G., Junghans, D., Lalejini, A., Dolson, E., and Ofria, C. (2020). Char-
+acterizing the effects of random subsampling on lexicase selection. In Banzhaf, W., Goodman,
+E., Sheneman, L., Trujillo, L., and Worzel, B., editors, Genetic Programming Theory and Practice
+XVII, pages 1–23. Springer International Publishing, Cham.
+Forstenlechner, S., Fagan, D., Nicolau, M., and O’Neill, M. (2017). A grammar design pattern
+for arbitrary program synthesis problems in genetic programming. In European Conference on
+Genetic Programming, pages 262–277. Springer.
+Forstenlechner, S., Nicolau, M., Fagan, D., and O’Neill, M. (2016). Grammar design for derivation
+tree based genetic programming systems. In European Conference on Genetic Programming, pages
+199–214. Springer.
+Goings, S., Goldsby, H., Cheng, B. H., and Ofria, C. (2012).
+An ecology-based evolutionary
+algorithm to evolve solutions to complex problems. In Artificial Life 13, pages 171–177. MIT
+Press.
+Helmuth, T. and Abdelhady, A. (2020). Benchmarking parent selection for program synthesis by
+genetic programming. In Proceedings of the 2020 Genetic and Evolutionary Computation Conference
+Companion, pages 237–238, Canc´un Mexico. ACM.
+Helmuth, T. and Kelly, P. (2021). PSB2: The second program synthesis benchmark suite. In 2021
+Genetic and Evolutionary Computation Conference, GECCO ’21, Lille, France. ACM.
+Helmuth, T. and Kelly, P. (2022).
+Applying genetic programming to psb2: The next gen-
+eration program synthesis benchmark suite.
+Genetic Programming and Evolvable Machines,
+23(3):375–404.
+Helmuth, T., McPhee, N. F., and Spector, L. (2016). Effects of lexicase and tournament selection
+on diversity recovery and maintenance. In Proceedings of the 2016 on Genetic and Evolution-
+ary Computation Conference Companion, GECCO ’16 Companion, page 983–990, New York, NY,
+USA. Association for Computing Machinery.
+Helmuth, T., McPhee, N. F., and Spector, L. (2018). Program synthesis using uniform mutation
+by addition and deletion. In Proceedings of the Genetic and Evolutionary Computation Conference,
+GECCO ’18, page 1127–1134, New York, NY, USA. Association for Computing Machinery.
+Helmuth, T., Pantridge, E., and Spector, L. (2020). On the importance of specialists for lexicase
+selection. Genetic Programming and Evolvable Machines, 21(3):349–373.
+Helmuth, T. and Spector, L. (2015). General program synthesis benchmark suite. In GECCO
+’15: Proceedings of the 2015 conference on Genetic and Evolutionary Computation Conference, pages
+1039–1046, Madrid, Spain. ACM.
+24
+Preprint
+
+Informed Down-Sampled Lexicase Selection
+Helmuth, T. and Spector, L. (2020). Explaining and exploiting the advantages of down-sampled
+lexicase selection. In Artificial Life Conference Proceedings, pages 341–349. MIT Press.
+Helmuth, T. and Spector, L. (2021). Problem-solving benefits of down-sampled lexicase selection.
+Artificial Life, pages 1–21.
+Helmuth, T., Spector, L., and Matheson, J. (2015). Solving uncompromising problems with lexi-
+case selection. IEEE Transactions on Evolutionary Computation, 19(5):630–643.
+Hernandez, J. G., Lalejini, A., Dolson, E., and Ofria, C. (2019). Random subsampling improves
+performance in lexicase selection. In GECCO ’19: Proceedings of the Genetic and Evolutionary
+Computation Conference Companion, pages 2028–2031, Prague, Czech Republic. ACM.
+Hernandez, J. G., Lalejini, A., and Ofria, C. (2022). An Exploration of Exploration: Measuring the
+Ability of Lexicase Selection to Find Obscure Pathways to Optimality. In Banzhaf, W., Trujillo,
+L., Winkler, S., and Worzel, B., editors, Genetic Programming Theory and Practice XVIII, pages
+83–107. Springer Nature Singapore, Singapore.
+Hochbaum, D. S. and Shmoys, D. B. (1985). A best possible heuristic for the k-center problem.
+Math. Oper. Res., 10:180–184.
+Holland, J. H. (1992). Adaptation in Natural and Artificial Systems: An Introductory Analysis with
+Applications to Biology, Control and Artificial Intelligence. MIT Press, Cambridge, MA, USA.
+Horn, J., Nafpliotis, N., and Goldberg, D. (1994). A niched Pareto genetic algorithm for multi-
+objective optimization. In Proceedings of the First IEEE Conference on Evolutionary Computation.
+IEEE World Congress on Computational Intelligence, pages 82–87, Orlando, FL, USA. IEEE.
+Krawiec, K., Swan, J., and O’Reilly, U.-M. (2016).
+Behavioral Program Synthesis: Insights and
+Prospects, pages 169–183. Springer International Publishing, Cham.
+La Cava, W., Spector, L., and Danai, K. (2016). Epsilon-lexicase selection for regression. In Pro-
+ceedings of the Genetic and Evolutionary Computation Conference 2016, GECCO ’16, page 741–748,
+New York, NY, USA. Association for Computing Machinery.
+Lalejini, A., Dolson, E., Vostinar, A. E., and Zaman, L. (2022). Artificial selection methods from
+evolutionary computing show promise for directed evolution of microbes. eLife, 11:e79665.
+Loshchilov, I. and Hutter, F. (2015). Online batch selection for faster training of neural networks.
+ArXiv, abs/1511.06343.
+Metevier, B., Saini, A. K., and Spector, L. (2019). Lexicase selection beyond genetic programming.
+In Banzhaf, W., Spector, L., and Sheneman, L., editors, Genetic Programming Theory and Practice
+XVI, pages 123–136. Springer International Publishing, Cham.
+Moore, J. M. and Stanton, A. (2017). Lexicase selection outperforms previous strategies for in-
+cremental evolution of virtual creature controllers. In Knibbe, C., Beslon, G., Parsons, D. P.,
+Misevic, D., Rouzaud-Cornabas, J., Bred`eche, N., Hassas, S., 0001, O. S., and Soula, H., ed-
+itors, Proceedings of the Fourteenth European Conference Artificial Life, ECAL 2017, Lyon, France,
+September 4-8, 2017, pages 290–297. MIT Press.
+Paul, M., Ganguli, S., and Dziugaite, G. K. (2021). Deep learning on a data diet: Finding impor-
+tant examples early in training. Advances in Neural Information Processing Systems, 34:20596–
+20607.
+Ruder, S. (2017). An overview of gradient descent optimization algorithms. arXiv:1609.04747
+[cs].
+Ryan, C., Collins, J. J., and Neill, M. O. (1998). Grammatical evolution: Evolving programs for an
+arbitrary language. In European conference on genetic programming, pages 83–96. Springer.
+Schmidt, M. and Lipson, H. (2005). Co-evolution of fitness maximizers and fitness predictors.
+In Rothlauf, F., editor, Late breaking paper at Genetic and Evolutionary Computation Conference
+(GECCO’2005), Washington, D.C., USA.
+Preprint
+25
+
+R. Boldi, M. Briesch, D. Sobania, A. Lalejini, T. Helmuth, F. Rothlauf, C. Ofria and L. Spector
+Schmidt, M. D. and Lipson, H. (2008). Coevolution of fitness predictors. IEEE Transactions on
+Evolutionary Computation, 12:736–749.
+Smith, R. E., Forrest, S., and Perelson, A. S. (1993). Population diversity in an immune system
+model: Implications for genetic search. In WHITLEY, L. D., editor, Foundations of Genetic Algo-
+rithms, volume 2 of Foundations of Genetic Algorithms, pages 153–165. Elsevier.
+Sobania, D., Briesch, M., and Rothlauf, F. (2022a). Choose your programming copilot: a com-
+parison of the program synthesis performance of github copilot and genetic programming. In
+Proceedings of the Genetic and Evolutionary Computation Conference, pages 1019–1027.
+Sobania, D. and Rothlauf, F. (2020). Challenges of program synthesis with grammatical evolution.
+In European Conference on Genetic Programming (Part of EvoStar), pages 211–227. Springer.
+Sobania, D. and Rothlauf, F. (2022). Program synthesis with genetic programming: The influence
+of batch sizes. In Genetic Programming: 25th European Conference, EuroGP 2022, Held as Part of
+EvoStar 2022, Madrid, Spain, April 20–22, 2022, Proceedings, page 118–129, Berlin, Heidelberg.
+Springer-Verlag.
+Sobania, D., Schweim, D., and Rothlauf, F. (2022b). A comprehensive survey on program syn-
+thesis with evolutionary algorithms. IEEE Transactions on Evolutionary Computation.
+Spector, L. (2012). Assessment of problem modality by differential performance of lexicase selec-
+tion in genetic programming: A preliminary report. In Proceedings of the 14th Annual Conference
+Companion on Genetic and Evolutionary Computation, GECCO ’12, page 401–408, New York, NY,
+USA. Association for Computing Machinery.
+Spector, L., Perry, C., Klein, J., and Keijzer, M. (2004). Push 3.0 programming language descrip-
+tion. Technical Report HC-CSTR-2004-02, School of Cognitive Science, Hampshire College,
+USA.
+Spector, L. and Robinson, A. (2002). Genetic programming and autoconstructive evolution with
+the push programming language. Genetic Programming and Evolvable Machines, 3(1):7–40.
+Troise, S. A. and Helmuth, T. (2017). Lexicase selection with weighted shuffle. In Banzhaf, W.,
+Olson, R. S., Tozier, W., and Riolo, R., editors, Genetic Programming Theory and Practice XV,
+Genetic and Evolutionary Computation, pages 89–104, University of Michigan in Ann Arbor,
+USA. Springer.
+Vanneschi, L., Castelli, M., and Silva, S. (2014). A survey of semantic methods in genetic pro-
+gramming. Genetic Programming and Evolvable Machines, 15(2):195–214.
+Whigham, P. A. et al. (1995). Grammatically-based genetic programming. In Proceedings of the
+workshop on genetic programming: from theory to real-world applications, volume 16, pages 33–41.
+Citeseer.
+Zogaj, F., Cambronero, J. P., Rinard, M. C., and Cito, J. (2021). Doing more with less: characteriz-
+ing dataset downsampling for AutoML. Proceedings of the VLDB Endowment, 14(11):2059–2072.
+ˇSikulov´a, M. and Sekanina, L. (2012).
+Coevolution in Cartesian Genetic Programming.
+In
+Moraglio, A., Silva, S., Krawiec, K., Machado, P., and Cotta, C., editors, Genetic Programming,
+Lecture Notes in Computer Science, pages 182–193, Berlin, Heidelberg. Springer.
+26
+Preprint
+
+Informed Down-Sampled Lexicase Selection
+A
+Generalization Rates
+Table 6: Generalization rate for PushGP. These data indicate the proportion of the runs
+that passed the training set that also passed the held out test set.
+Method
+Lex
+Rnd
+IDS
+Rnd
+IDS
+r
+-
+0.05
+0.1
+ρ
+-
+-
+1
+0.01
+0.01
+0.01
+-
+1
+0.01
+0.01
+0.01
+k
+-
+-
+1
+1
+10
+100
+-
+1
+1
+10
+100
+Count Odds
+1.00
+0.96
+0.98
+0.99
+1.00
+0.99
+0.96
+1.00
+0.98
+0.99
+0.99
+Find Pair
+1.00
+0.82
+0.82
+0.73
+0.74
+0.80
+0.50
+0.88
+0.79
+0.68
+0.75
+Fizz Buzz
+0.93
+0.96
+1.00
+0.93
+0.95
+0.99
+1.00
+1.00
+0.96
+0.96
+0.96
+Fuel Cost
+1.00
+1.00
+1.00
+0.99
+0.99
+0.99
+1.00
+1.00
+1.00
+1.00
+1.00
+GCD
+0.91
+0.93
+1.00
+0.93
+0.83
+0.87
+0.82
+0.75
+0.80
+0.89
+0.87
+Grade
+-
+-
+-
+-
+1.00
+-
+1.00
+-
+-
+1.00
+1.00
+Scrabble Score
+1.00
+1.00
+1.00
+1.00
+1.00
+1.00
+1.00
+1.00
+0.98
+1.00
+1.00
+Small or Large
+0.71
+0.95
+0.80
+0.78
+0.74
+0.71
+0.81
+0.77
+0.69
+0.73
+0.64
+Table 7: Generalization rate for G3P. These data indicate the proportion of the runs that
+passed the training set that also passed the held out test set.
+Method
+Lex
+Rnd
+IDS
+Rnd
+IDS
+r
+-
+0.05
+0.1
+ρ
+-
+-
+1
+0.01
+0.01
+0.01
+-
+1
+0.01
+0.01
+0.01
+k
+-
+-
+1
+1
+10
+100
+-
+1
+1
+10
+100
+Count Odds
+0.94
+0.96
+0.96
+0.88
+1.00
+0.96
+1.00
+0.92
+0.95
+0.91
+0.95
+Find Pair
+-
+-
+-
+1.00
+-
+-
+1.00
+-
+-
+1.00
+-
+Fizz Buzz
+0.79
+0.87
+0.85
+0.84
+0.78
+0.85
+0.83
+0.82
+0.82
+0.89
+0.73
+Fuel Cost
+1.00
+0.97
+1.00
+0.97
+0.96
+1.00
+1.00
+0.96
+0.96
+1.00
+1.00
+GCD
+-
+0.17
+-
+-
+-
+0.25
+-
+-
+-
+-
+-
+Grade
+0.42
+0.45
+0.50
+0.53
+0.59
+0.45
+0.47
+0.54
+0.47
+0.54
+0.49
+Scrabble Score
+1.00
+1.00
+1.00
+1.00
+0.92
+0.83
+1.00
+-
+0.86
+1.00
+0.60
+Small or Large
+0.47
+0.57
+0.65
+0.56
+0.64
+0.66
+0.68
+0.59
+0.60
+0.579
+0.65
+Preprint
+27
+

C9E4T4oBgHgl3EQfFwy_/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:793cd509b59ddeeab47be342607d90d26ed2afeacd9c10d5fbc0945581a0c471
+size 129855

CtE0T4oBgHgl3EQfQQAs/content/2301.02189v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:bb54594044be1479cd21fcecb615347004f6880d38076ab335d1986ec0daa478
+size 6212423

CtE0T4oBgHgl3EQfQQAs/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:f160ec6ed32e013d0d8fcd69127b9576bb23a4f041f1be7ff47842e49ffaf282
+size 13500461

DdAzT4oBgHgl3EQfwf7y/content/2301.01725v1.pdf

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:0f4e85485cea3a2027a6a08de3756967183384d11cb9f57cbc2ff04d4d617897
+size 1073396

DdAzT4oBgHgl3EQfwf7y/vector_store/index.faiss

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:76f32f4c2c97031a087b24d8de8d4798db441b1fb007e226f485cf350a32d993
+size 7012397

FtAzT4oBgHgl3EQfHPtI/vector_store/index.pkl

@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:8cd2787875cd2837f1997a6cbf8392caf764b53e33de56a19319ae335fd63ae0
+size 160400