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+Hyperparameter Optimization as a Service on INFN
+Cloud
+Matteo Barbetti1,2 and Lucio Anderlini2
+1 Department of Information Engineering, University of Florence,
+via Santa Marta 3, Firenze, Italy
+2 Istituto Nazionale di Fisica Nucleare, Sezione di Firenze,
+via G. Sansonse 1, Sesto Fiorentino (FI), Italy
+E-mail: Matteo.Barbetti@fi.infn.it
+Abstract.
+The simplest and often most effective way of parallelizing the training of complex
+machine learning models is to execute several training instances on multiple machines, possibly
+scanning the hyperparameter space to optimize the underlying statistical model and the learning
+procedure. Often, such a meta learning procedure is limited by the ability of accessing securely
+a common database organizing the knowledge of the previous and ongoing trials. Exploiting
+opportunistic GPUs provided in different environments represents a further challenge when
+designing such optimization campaigns. In this contribution we discuss how a set of RestAPIs
+can be used to access a dedicated service based on INFN Cloud to monitor and possibly
+coordinate multiple training instances, with gradient-less optimization techniques, via simple
+HTTP requests. The service, named Hopaas (Hyperparameter OPtimization As A Service), is
+made of web interface and sets of APIs implemented with a FastAPI back-end running through
+Uvicorn and NGINX in a virtual instance of INFN Cloud. The optimization algorithms are
+currently based on Bayesian techniques as provided by Optuna. A Python front-end is also
+made available for quick prototyping. We present applications to hyperparameter optimization
+campaigns performed combining private, INFN Cloud and CINECA resources.
+1. Introduction
+In the last decade, machine learning (ML) has become an incredibly valuable tool in practically
+every field of application, from scientific research to industry.
+Increasingly complex models
+achieve surprising results in a wide range of applications, such as image generation [1], language
+modelling [2] or medical diagnosis [3]. Most of the ML techniques rely on the optimization of
+an objective function with respect to some internal parameters, describing the performance of
+the algorithm. Usually, when the optimum of the objective function is a minimum, the names
+cost or loss function are adopted.
+The fastest iterative optimization techniques rely on the
+(Stochastic) Gradient Descent technique [4]. Unfortunately, for a wide class of optimization
+problems the gradient of the loss function with respect to the model parameter is extremely
+expensive to compute or cannot be defined at all. For example, optimization problems involving
+noisy loss functions in contexts where analytical derivatives cannot be computed cannot rely on
+gradient-descent techniques, requiring the adoption of slower, often heuristic, methods. A widely
+adopted option is to define a surrogate model describing the variations of the loss function across
+the parameter space together with its uncertainty, driving the optimization algorithm to explore
+those regions where improvements were not statistically excluded from previous evaluations.
+arXiv:2301.05522v1  [cs.DC]  13 Jan 2023
+
+Techniques adopting this approach are referred to as Bayesian optimization (BO) methods and
+have been an active area of research in ML for the last decade [5, 6, 7, 8, 9].
+Tuning the performance of ML models may benefit from the optimization of the
+hyperparameters, defined as all those parameters that are not learned during the model training
+procedure but encode some arbitrariness in the architecture of the model itself or in the procedure
+to train it [5]. In practice, hyperparameter optimization (HPO) studies require training the
+model multiple times to explore the hyperparameter space.
+Since training ML models is
+computationally expensive, HPO campaigns should focus as much as possible on those regions
+of the hyperparameter space where the model performs better to reduce the time needed for
+finding the best configuration. On the other hand, the loss is often a noisy function of the
+hyperparameters as multiple training procedures may result in different performance because of
+the intrinsic randomness of the stochastic gradient-descent techniques.
+Exploring the hyperparameter space requires many independent trainings,
+or trials,
+that can run in parallel on different computing resources.
+In general, accessing more
+resources enables the exploration of larger hyperparameter spaces, possibly resulting in better
+models.
+Opportunistic access to compute resources may provide valuable contribution to
+HPO campagins.
+Unfortunately, coordinating studies on resources from different providers,
+restrictions and regulations challenges the adoption of existing HPO services.
+In this document, we propose Hopaas (Hyperparameter OPtimization As A Service),
+implementing a set of RestAPIs to orchestrate HPO studies across multiple computing instances.
+Computing nodes from multiple HPC centers can concur dynamically to the same optimization
+study, requesting to the Hopaas server a set of hyperparameters to test and then sending
+back the outcome of the training procedure.
+Several trials of one or more studies can be
+tracked and monitored through the web interface provided by the Hopaas service. A reference
+implementation, with a server instance1 deployed on INFN Cloud resources and a simple client
+package [10] wrapping the RestAPIs to Python functions, is also discussed.
+2. Hopaas API specification
+We refer to a trial as a single training attempt with a specific set of hyperparameters to test. A
+study represents an optimization session and includes a collection of trials. In practice, a study
+is unambiguously defined by the set of hyperparameters to optimize, the range of values where
+searching the optimum, and the modality in which this search is carried out (e.g., grid search,
+Bayesian methods [5], or evolutionary algorithms [11]).
+The core activity of the Hopaas service is to manage distributed optimization studies. A set
+of RestAPIs is designed to create trials, finalize them, and update the service on intermediate
+values of the objective function to enable early termination of the trial before the conclusion
+of the training. The APIs named ask, tell and should prune implement these actions upon
+POST HTTP requests, with user authentication based on an API token in the request path.
+A computing node ready to test a set of hyperparameters will query the Hopaas server via
+the ask API, including in the request body all the information needed to define an optimization
+study unambiguously. The Hopaas server will define a new trial, possibly assigning it to an
+existing study, or creating a new one, and providing a unique identifier of the new trial and the
+set of hyperparameters to evaluate as part of the HTTP response.
+Usually, the evaluation of the set of hyperparameters consists of training a model defined by
+those hyperparameters aiming at the resulting value of the objective function. The evaluated
+performance metric may correspond to the loss function computed during the training procedure
+but, in general, it can be any numerical score obtained processing a given set of hyperparameters.
+Once the evaluation is completed, the computing node will finalize the trial using the tell API,
+1 Visit https://hopaas.cloud.infn.it for additional details.
+
+POST REQUEST
+ASK
+POST REQUEST
+TELL
+Hopaas server
+Processing Trial #1
+Study A
+Computing node
+loss
+trials
+Computing node
+Study B
+Processing Trial #1
+Study A
+Study B
+Trial #1
+Trial #2
+Trial #1
+Trial #2
+Trial #3
+Computing node
+Study B
+Processing Trial #3
+/
+Computing node
+Study B
+Processing Trial #2
+Processing Trial #2
+Study A
+Computing node
+On-prem
+Figure 1.
+A Hopaas server orchestrating multiple studies across multiple sites.
+whose body will include the unique identifier of the trial and the final evaluation of the objective
+function.
+The Hopaas server may serve multiple ask requests from different sources, assigning them
+to one or different studies, while updating the surrogate model each time a new evaluation is
+made available by querying the tell API. A schematic representation of the orchestration of
+studies in multiple sites is reported in Figure 1.
+Depending on the specific ML algorithm, intermediate evaluations of the objective function
+can be accessed during the training procedure and used to abort non-promising trials (pruning)
+without wasting computing power to take the training procedure to an end. Optionally, the
+computing node may update the Hopaas server with intermediate evaluations of the objective
+function by querying the should prune API for monitoring and pruning purposes. The body of
+a should prune request will contain the unique identifier of the trial, the intermediate value of
+the loss function and an integer number encoding the progress of the training procedure, named
+the step. The HTTP response will indicate whether the study should be early terminated, or it
+is sufficiently likely to result in an improvement over the previous tests.
+A reference Python front-end was developed aiming at a facilitated access to the Hopaas
+service from Python applications [10]. While Python is a primary choice for many scientific
+applications, it should be noticed that the client simply wraps the RestAPIs into classes and
+functions, as the Hopaas protocol is designed to be language-agnostic, relying on widely adopted
+web communication standards. In addition, the Hopaas client is also framework-agnostic since
+the evaluation of the objective function for a given set of hyperparameters can be implemented
+with any framework and environment.
+3. Implementation
+The reference implementation for the Hopaas service running on INFN Cloud relies on
+containerized applications orchestrated with docker-compose. The web server implementing
+the RestAPIs is a scalable set of Uvicorn instances running an application based on the FastAPI
+framework. The BO algorithms are provided by integrating the back-end with Optuna, while
+
+INPN
+CLOUDCINECAawsCERNINPN
+CLOUDhttp://hopaas.cloud.infn.it/api/ask/user_token
+POST
+http://hopaas.cloud.infn.it/api/tell/user_token
+POST
+200
+empty HTTP response
+Set-up of the optimization study
+(e.g. title, min/max, sampler, search space)
+1.
+Retrieving the trial parameters
+(both constant and optimizable parameters)
+4.
+Objective function computation
+(e.g. closed formula, machine learning score)
+5.
+Creation/loading of the optimization study
+(same set-up allows to load existing optimization study)
+2.
+Parameters sampled according to study set-up
+(the prepared set of parameters defines an optimization trial)
+3.
+Optimization study updated with trial results
+(parallel tests enabled by gradientless optimization strategies)
+6.
+Processing...
+Ready for testing a new trial set
+(both from the same or a new optimization study)
+7a.
+Ready for providing a new trial set
+(score-driven suggestions enabled by default)
+7b.
+Framework-agnostic optimization campaign
+Computing instance
+Hopaas server
++
+Authorized users only!
+200
+HTTP response with trial parameters
+Figure 2.
+Workflow of an optimization study with a client-server approach based on RestAPIs.
+future extensions to additional frameworks are planned. The access to the Uvicorn instances
+from the Internet is mediated by an NGINX reverse proxy accessed via the encrypted HTTPS
+protocol. A PostgreSQL instance is part of the docker-compose configuration to provide shared
+persistency to the multiple instances of the web application back-end.
+The workflow of the
+interaction between the Hopaas server and computing nodes is depicted in Figure 2.
+The same Hopaas server is designed to serve web-based user access. A web application,
+developed in HTML, CSS and JavaScript, is shipped to the client browser as defined by a set
+of web-specific APIs in Uvicorn. The web pages of the front-end provide dynamic visualizations
+by fetching data from specialized APIs at regular intervals. Plots showing the evolution of the
+loss reported by different studies and trials are obtained with the Chartist library.
+The user authentication and authorization procedure of the web application is managed
+relying on access tokens as defined by the OAuth2 standard, using the INFN GitLab instance
+as identity provider.
+Support for INDIGO IAM is also planned for the future [12].
+Once
+authenticated, users can generate multiple API tokens through the web application. Each API
+token has a validity period defined at generation and can be revoked at any time. Tokens with
+shorter validity are more appropriate for usage in public or untrusted contexts.
+4. Tuning the LHCb ultra-fast simulation models with Hopaas and Marconi 100
+Machine Learning is an important research area in High Energy Physics (HEP), with first
+applications dating back to the 1990s. Recent years have witnessed an explosion in the use of
+ML techniques in HEP to face the computational challenges raised by the upcoming and future
+runs of the Large Hadron Collider (LHC) [13]. With an increasing number, complexity and
+range of applications of the ML models, HPO is becoming popular in HEP [14], and specialized
+frameworks targeting distributed computing are being developed [7].
+The reference implementation of the Hopaas service presented here has been successfully used
+for HEP applications and, in particular, to optimize the parameterizations for Lamarr [15], a
+novel LHCb ultra-fast simulation framework. Most of the parameterizations of Lamarr rely
+
+scikitdmlc
+XGBoostA
+wwWINPN
+CLOUDuvicornFastAPion Generative Adversarial Networks (GANs) [16], advanced algorithms taken from Computer
+Vision that were demonstrated to be able to well reproduce the distributions obtained from
+standard simulation techniques [17, 18]. Adversarial models are particularly sensitive to the
+choice of the hyperparameter configuration and require intensive optimization campaigns to
+model accurately the target distributions.
+Several optimization studies have been orchestrated by the Hopaas service using diverse
+computing instances, from scientific providers (like INFN, CERN and CINECA) and from
+commercial cloud provider (like GCP or AWS). Most of the resources have been provided by
+the CINECA supercomputer Marconi 100, with a custom network configuration to enable
+the communication with the Hopaas server [19]. Hopaas was able to coordinate dozens of
+optimization studies with hundreds of trials on each study from more than twenty concurrent
+and diverse computing nodes.
+5. Conclusion and future work
+Hyperparameter tuning and Bayesian methods for gradient-less optimization provide an effective
+and simple mean of exploiting opportunistic compute resources to improve ML models.
+Unfortunately, environment variability and constraints set by different resource providers make
+the application of existing HPO services challenging.
+With Hopaas, we propose a solution
+designed to require the addition of the thinnest possible layer in the model training application,
+querying a central service via HTTPS and minimal RestAPIs. A reference implementation with
+a server instance running on INFN Cloud and a Python client was presented and tested in a real-
+world application to coordinate hyperparameter optimization campaigns on multiple resource
+providers including INFN, CERN, and CINECA. In the future we will improve the quality of
+the Web User Interface, for example enabling custom model documentation and sharing among
+multiple users, and introduce support to multi-objective optimizations.
+Acknowledgments
+We would like to thank Doina Cristina Duma and the rest of the INFN Cloud group for the
+technical support in the deployment and test of Hopaas. We acknowledge enlightening and
+motivating discussions with Diego Ciangottini, Stefano Dal Pra, Piergiulio Lenzi and Daniele
+Spiga, especially on future applications and developments.
+References
+[1] Ramesh A et al. 2021 PMLR’21 pp 8821–8831
+[2] Brown T et al. 2020 NeurIPS’20 pp 1877–1901
+[3] Richens J G, Lee C M and Johri S 2020 Nat. Comm. 11 3923
+[4] Orr G and M¨uller K 2003 Neural Networks: Tricks of the Trade (Springer Berlin Heidelberg)
+[5] Bergstra J et al. 2011 NeurIPS’11 pp 2546–2554
+[6] Bergstra J, Yamins D and Cox D 2013 PMLR’13 pp 115–123
+[7] Liaw R et al. 2018 (Preprint 1807.05118)
+[8] Akiba T et al. 2019 KDD’19 pp 2623–2631 (Preprint 1907.10902)
+[9] Head T et al. 2021 URL https://doi.org/10.5281/zenodo.5565057
+[10] Barbetti M and Anderlini L 2023 URL https://doi.org/10.5281/zenodo.7528502
+[11] Tani L et al. 2021 EPJ C 81 170
+[12] Spiga D et al. 2020 EPJ Web Conf. 245 07020
+[13] Albertsson K et al. 2018 J. Phys.: Conf. Ser. 1085 022008
+[14] Wulff E, Girone M and Pata J 2022 (Preprint 2203.01112)
+[15] Anderlini L et al. 2022 PoS ICHEP2022 233
+[16] Goodfellow I et al. 2014 NeurIPS’14 pp 2672–2680 (Preprint 1406.2661)
+[17] Anderlini L et al. (LHCb) 2022 (Preprint 2204.09947)
+[18] Ratnikov F et al. 2023 Nucl. Instrum. Meth. A 1046 167591
+[19] Mariotti M, Spiga D and Boccali T 2021 PoS ISGC2021 002
+

-dE5T4oBgHgl3EQfRg6t/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf,len=158
+page_content='Hyperparameter Optimization as a Service on INFN  Cloud  Matteo Barbetti1,2 and Lucio Anderlini2  1 Department of Information Engineering, University of Florence,  via Santa Marta 3, Firenze, Italy  2 Istituto Nazionale di Fisica Nucleare, Sezione di Firenze,  via G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Sansonse 1, Sesto Fiorentino (FI), Italy  E-mail: Matteo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='Barbetti@fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='it  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  The simplest and often most effective way of parallelizing the training of complex  machine learning models is to execute several training instances on multiple machines, possibly  scanning the hyperparameter space to optimize the underlying statistical model and the learning  procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Often, such a meta learning procedure is limited by the ability of accessing securely  a common database organizing the knowledge of the previous and ongoing trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Exploiting  opportunistic GPUs provided in different environments represents a further challenge when  designing such optimization campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' In this contribution we discuss how a set of RestAPIs  can be used to access a dedicated service based on INFN Cloud to monitor and possibly  coordinate multiple training instances, with gradient-less optimization techniques, via simple  HTTP requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' The service, named Hopaas (Hyperparameter OPtimization As A Service), is  made of web interface and sets of APIs implemented with a FastAPI back-end running through  Uvicorn and NGINX in a virtual instance of INFN Cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' The optimization algorithms are  currently based on Bayesian techniques as provided by Optuna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' A Python front-end is also  made available for quick prototyping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' We present applications to hyperparameter optimization  campaigns performed combining private, INFN Cloud and CINECA resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Introduction  In the last decade, machine learning (ML) has become an incredibly valuable tool in practically  every field of application, from scientific research to industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Increasingly complex models  achieve surprising results in a wide range of applications, such as image generation [1], language  modelling [2] or medical diagnosis [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Most of the ML techniques rely on the optimization of  an objective function with respect to some internal parameters, describing the performance of  the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Usually, when the optimum of the objective function is a minimum, the names  cost or loss function are adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  The fastest iterative optimization techniques rely on the  (Stochastic) Gradient Descent technique [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Unfortunately, for a wide class of optimization  problems the gradient of the loss function with respect to the model parameter is extremely  expensive to compute or cannot be defined at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' For example, optimization problems involving  noisy loss functions in contexts where analytical derivatives cannot be computed cannot rely on  gradient-descent techniques, requiring the adoption of slower, often heuristic, methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' A widely  adopted option is to define a surrogate model describing the variations of the loss function across  the parameter space together with its uncertainty, driving the optimization algorithm to explore  those regions where improvements were not statistically excluded from previous evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='05522v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='DC]  13 Jan 2023  Techniques adopting this approach are referred to as Bayesian optimization (BO) methods and  have been an active area of research in ML for the last decade [5, 6, 7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Tuning the performance of ML models may benefit from the optimization of the  hyperparameters, defined as all those parameters that are not learned during the model training  procedure but encode some arbitrariness in the architecture of the model itself or in the procedure  to train it [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' In practice, hyperparameter optimization (HPO) studies require training the  model multiple times to explore the hyperparameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Since training ML models is  computationally expensive, HPO campaigns should focus as much as possible on those regions  of the hyperparameter space where the model performs better to reduce the time needed for  finding the best configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' On the other hand, the loss is often a noisy function of the  hyperparameters as multiple training procedures may result in different performance because of  the intrinsic randomness of the stochastic gradient-descent techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Exploring the hyperparameter space requires many independent trainings,  or trials,  that can run in parallel on different computing resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  In general, accessing more  resources enables the exploration of larger hyperparameter spaces, possibly resulting in better  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Opportunistic access to compute resources may provide valuable contribution to  HPO campagins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Unfortunately, coordinating studies on resources from different providers,  restrictions and regulations challenges the adoption of existing HPO services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  In this document, we propose Hopaas (Hyperparameter OPtimization As A Service),  implementing a set of RestAPIs to orchestrate HPO studies across multiple computing instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Computing nodes from multiple HPC centers can concur dynamically to the same optimization  study, requesting to the Hopaas server a set of hyperparameters to test and then sending  back the outcome of the training procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Several trials of one or more studies can be  tracked and monitored through the web interface provided by the Hopaas service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' A reference  implementation, with a server instance1 deployed on INFN Cloud resources and a simple client  package [10] wrapping the RestAPIs to Python functions, is also discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Hopaas API specification  We refer to a trial as a single training attempt with a specific set of hyperparameters to test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' A  study represents an optimization session and includes a collection of trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' In practice, a study  is unambiguously defined by the set of hyperparameters to optimize, the range of values where  searching the optimum, and the modality in which this search is carried out (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=', grid search,  Bayesian methods [5], or evolutionary algorithms [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  The core activity of the Hopaas service is to manage distributed optimization studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' A set  of RestAPIs is designed to create trials, finalize them, and update the service on intermediate  values of the objective function to enable early termination of the trial before the conclusion  of the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' The APIs named ask, tell and should prune implement these actions upon  POST HTTP requests, with user authentication based on an API token in the request path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  A computing node ready to test a set of hyperparameters will query the Hopaas server via  the ask API, including in the request body all the information needed to define an optimization  study unambiguously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' The Hopaas server will define a new trial, possibly assigning it to an  existing study, or creating a new one, and providing a unique identifier of the new trial and the  set of hyperparameters to evaluate as part of the HTTP response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Usually, the evaluation of the set of hyperparameters consists of training a model defined by  those hyperparameters aiming at the resulting value of the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' The evaluated  performance metric may correspond to the loss function computed during the training procedure  but, in general, it can be any numerical score obtained processing a given set of hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Once the evaluation is completed, the computing node will finalize the trial using the tell API,  1 Visit https://hopaas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='it for additional details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  POST REQUEST  ASK  POST REQUEST  TELL  Hopaas server  Processing Trial #1  Study A  Computing node  loss  trials  Computing node  Study B  Processing Trial #1  Study A  Study B  Trial #1  Trial #2  Trial #1  Trial #2  Trial #3  Computing node  Study B  Processing Trial #3  /  Computing node  Study B  Processing Trial #2  Processing Trial #2  Study A  Computing node  On-prem  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  A Hopaas server orchestrating multiple studies across multiple sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  whose body will include the unique identifier of the trial and the final evaluation of the objective  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  The Hopaas server may serve multiple ask requests from different sources, assigning them  to one or different studies, while updating the surrogate model each time a new evaluation is  made available by querying the tell API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' A schematic representation of the orchestration of  studies in multiple sites is reported in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Depending on the specific ML algorithm, intermediate evaluations of the objective function  can be accessed during the training procedure and used to abort non-promising trials (pruning)  without wasting computing power to take the training procedure to an end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Optionally, the  computing node may update the Hopaas server with intermediate evaluations of the objective  function by querying the should prune API for monitoring and pruning purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' The body of  a should prune request will contain the unique identifier of the trial, the intermediate value of  the loss function and an integer number encoding the progress of the training procedure, named  the step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' The HTTP response will indicate whether the study should be early terminated, or it  is sufficiently likely to result in an improvement over the previous tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  A reference Python front-end was developed aiming at a facilitated access to the Hopaas  service from Python applications [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' While Python is a primary choice for many scientific  applications, it should be noticed that the client simply wraps the RestAPIs into classes and  functions, as the Hopaas protocol is designed to be language-agnostic, relying on widely adopted  web communication standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' In addition, the Hopaas client is also framework-agnostic since  the evaluation of the objective function for a given set of hyperparameters can be implemented  with any framework and environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Implementation  The reference implementation for the Hopaas service running on INFN Cloud relies on  containerized applications orchestrated with docker-compose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' The web server implementing  the RestAPIs is a scalable set of Uvicorn instances running an application based on the FastAPI  framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' The BO algorithms are provided by integrating the back-end with Optuna, while  INPN  CLOUDCINECAawsCERNINPN  CLOUDhttp://hopaas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='it/api/ask/user_token  POST  http://hopaas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='infn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='it/api/tell/user_token  POST  200  empty HTTP response  Set-up of the optimization study  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' title, min/max, sampler, search space)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Retrieving the trial parameters  (both constant and optimizable parameters)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Objective function computation  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' closed formula, machine learning score)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Creation/loading of the optimization study  (same set-up allows to load existing optimization study)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Parameters sampled according to study set-up  (the prepared set of parameters defines an optimization trial)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Optimization study updated with trial results  (parallel tests enabled by gradientless optimization strategies)  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Ready for testing a new trial set  (both from the same or a new optimization study)  7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Ready for providing a new trial set  (score-driven suggestions enabled by default)  7b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Framework-agnostic optimization campaign  Computing instance  Hopaas server  +  Authorized users only!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  200  HTTP response with trial parameters  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Workflow of an optimization study with a client-server approach based on RestAPIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  future extensions to additional frameworks are planned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' The access to the Uvicorn instances  from the Internet is mediated by an NGINX reverse proxy accessed via the encrypted HTTPS  protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' A PostgreSQL instance is part of the docker-compose configuration to provide shared  persistency to the multiple instances of the web application back-end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  The workflow of the  interaction between the Hopaas server and computing nodes is depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  The same Hopaas server is designed to serve web-based user access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' A web application,  developed in HTML, CSS and JavaScript, is shipped to the client browser as defined by a set  of web-specific APIs in Uvicorn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' The web pages of the front-end provide dynamic visualizations  by fetching data from specialized APIs at regular intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Plots showing the evolution of the  loss reported by different studies and trials are obtained with the Chartist library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  The user authentication and authorization procedure of the web application is managed  relying on access tokens as defined by the OAuth2 standard, using the INFN GitLab instance  as identity provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Support for INDIGO IAM is also planned for the future [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Once  authenticated, users can generate multiple API tokens through the web application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Each API  token has a validity period defined at generation and can be revoked at any time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Tokens with  shorter validity are more appropriate for usage in public or untrusted contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Tuning the LHCb ultra-fast simulation models with Hopaas and Marconi 100  Machine Learning is an important research area in High Energy Physics (HEP), with first  applications dating back to the 1990s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Recent years have witnessed an explosion in the use of  ML techniques in HEP to face the computational challenges raised by the upcoming and future  runs of the Large Hadron Collider (LHC) [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' With an increasing number, complexity and  range of applications of the ML models, HPO is becoming popular in HEP [14], and specialized  frameworks targeting distributed computing are being developed [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  The reference implementation of the Hopaas service presented here has been successfully used  for HEP applications and, in particular, to optimize the parameterizations for Lamarr [15], a  novel LHCb ultra-fast simulation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Most of the parameterizations of Lamarr rely  scikitdmlc  XGBoostA  wwWINPN  CLOUDuvicornFastAPion Generative Adversarial Networks (GANs) [16], advanced algorithms taken from Computer  Vision that were demonstrated to be able to well reproduce the distributions obtained from  standard simulation techniques [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Adversarial models are particularly sensitive to the  choice of the hyperparameter configuration and require intensive optimization campaigns to  model accurately the target distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Several optimization studies have been orchestrated by the Hopaas service using diverse  computing instances, from scientific providers (like INFN, CERN and CINECA) and from  commercial cloud provider (like GCP or AWS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Most of the resources have been provided by  the CINECA supercomputer Marconi 100, with a custom network configuration to enable  the communication with the Hopaas server [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Hopaas was able to coordinate dozens of  optimization studies with hundreds of trials on each study from more than twenty concurrent  and diverse computing nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Conclusion and future work  Hyperparameter tuning and Bayesian methods for gradient-less optimization provide an effective  and simple mean of exploiting opportunistic compute resources to improve ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Unfortunately, environment variability and constraints set by different resource providers make  the application of existing HPO services challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  With Hopaas, we propose a solution  designed to require the addition of the thinnest possible layer in the model training application,  querying a central service via HTTPS and minimal RestAPIs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' A reference implementation with  a server instance running on INFN Cloud and a Python client was presented and tested in a real-  world application to coordinate hyperparameter optimization campaigns on multiple resource  providers including INFN, CERN, and CINECA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' In the future we will improve the quality of  the Web User Interface, for example enabling custom model documentation and sharing among  multiple users, and introduce support to multi-objective optimizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='  Acknowledgments  We would like to thank Doina Cristina Duma and the rest of the INFN Cloud group for the  technical support in the deployment and test of Hopaas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' We acknowledge enlightening and  motivating discussions with Diego Ciangottini, Stefano Dal Pra, Piergiulio Lenzi and Daniele  Spiga, especially on future applications and developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
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+page_content=' 2019 KDD’19 pp 2623–2631 (Preprint 1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='10902)  [9] Head T et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
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+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='5565057  [10] Barbetti M and Anderlini L 2023 URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
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+page_content=' 2020 EPJ Web Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' 245 07020  [13] Albertsson K et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' 2018 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' : Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
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+page_content='01112)  [15] Anderlini L et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
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+page_content=' 2014 NeurIPS’14 pp 2672–2680 (Preprint 1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='2661)  [17] Anderlini L et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content=' (LHCb) 2022 (Preprint 2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
+page_content='09947)  [18] Ratnikov F et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE5T4oBgHgl3EQfRg6t/content/2301.05522v1.pdf'}
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+Total energy-shaping control for mechanical systems via
+Control-by-Interconnection
+Joel Ferguson1
+Abstract— Application of IDA-PBC to mechanical systems
+has received much attention in recent decades, but its ap-
+plication is still limited by the solvability of the so-called
+matching conditions. In this work, it is shown that total energy-
+shaping control of under-actuated mechanical systems has a
+control-by-interconnection interpretation. Using this interpreta-
+tion, alternate matching conditions are formulated that defines
+constraints on the added energy, rather then the total closed-
+loop energy. It is additionally shown that, for systems that are
+under-actuated degree one with the mass matrix depending
+on a single coordinate, the kinetic energy matching conditions
+resolve to ODEs which can be evaluated numerically. Using this
+approach controllers are proposed for the benchmark cart-pole
+and acrobot systems.
+I. INTRODUCTION
+Energy-based methods for controlling nonlinear physical
+systems have been shown to be effective in a variety of
+physical domains [1]. Such methods consider the energy
+and structure of the system to be controlled to derive
+control strategies that exploit the natural system behaviours.
+Interconnection and Damping Assignment, Passivity-Based
+Control (IDA-PBC) is one such control methodology where
+the control input is designed such that the closed-loop can be
+interpreted as an alternate physical system with a different
+energy, interconnection and damping structure [2].
+While IDA-PBC has been applied to a broad range of
+systems, particular attention has been given to mechanical
+systems which exhibit a rich canonical structure [3], [4], [5].
+In the case of fully-actuated systems, IDA-PBC allows a
+user to arbitrarily modify the potential and kinetic energy
+of the closed-loop system [6], a process known as total
+energy shaping [3]. For under-actuated systems, however,
+application of IDA-PBC is limited by solutions that satisfy
+a set of PDEs, the so-called matching conditions. Much re-
+search effort has been committed to solving these equations,
+with solutions posed in several special cases [4], [5], [6].
+This design methodology has been applied to a number of
+benchmark examples such as the cart-pole, acrobot, spider
+crane, amongst others.
+Control-by-Interconnection (CbI) describes a sub-class of
+energy-based control methods that falls under the umbrella
+of IDA-PBC [7], [8]. Under this scheme, the controller is
+assumed to be a passive system that is interconnected with
+a passive plant to be controlled via the passive input-output
+pair. Casimirs, conserved quantities between the control sub-
+system and the plant, can be constructed to help shape the
+1Joel
+Ferguson
+is
+with
+the
+School
+of
+Engineer-
+ing,
+The
+University
+of
+Newcastle,
+Australia
+Email:
+joel.ferguson@newcastle.edu.au
+energy of the closed-loop system. It is known that potential
+energy shaping of fully-actuated mechanical systems falls
+into the class of CbI [7], [9]. Control of underactuated
+mechanical systems has been explored in the context of CbI
+by applying nonlinear PID controllers to both the standard
+and alternate passive outputs [10], [11], [12]. The idea of
+using PID for stabilisation of passive systems was formalised
+in [13] and a general characterisation of all passive outputs
+from a given system was characterised.
+In this work, the connection between IDA-PBC and CbI
+for under-actuated mechanical systems is explored. Using
+the bond graph formalism (see [14] for introduction), a
+control sub-system is proposed that allows for shaping of
+the kinetic and potential energies of the closed-loop system.
+By representing the controller as a passive interconnection,
+the requisite matching conditions are reformulated in terms
+of the added mass and added potential energy. Equivalence
+between the CbI and IDA-PBC is then established in the
+case of mechanical systems by identifying Casimirs relating
+the controller states to those of the plant. Finally, using the
+reformulated matching conditions, it is shown that in the case
+that the mass matrix depends on only one coordinate that the
+kinetic energy matching conditions can be formulated as an
+ODE that can be evaluated numerically for implementation.
+Notation. Function arguments are declared upon definition
+and are omitted for subsequent use. 0n×m denotes a n × m
+zeros matrix whereas In denotes a n×n identity matrix. For
+mappings H : Rn → R, we denote the transposed gradient as
+∇H :=
+� ∂H
+∂x
+�⊤. For P = P ⊤ ∈ Rn×n, λmin [P] , λmax [P]
+denotes the minimum and maximum (real) eigenvalues of P,
+respectively.
+II. BACKGROUND AND PROBLEM
+FORMULATION
+In this section a number of key concepts necessary for the
+subsequent developments are briefly revised.
+A. Control-by-interconnection
+In
+this
+work
+we
+consider
+input-state-output
+port-
+Hamiltonian systems (ISO-PHS) of the form
+˙xp = Fp(xp)∇xpHp(xp) + Gp(xp)up
+yp = G⊤
+p (xp)∇xpHp(xp)
+(1)
+where xp ∈ Rp is the state of the plant, Fp(xp) ∈ Rp×p is
+the combined interconnection and damping matrix satisfying
+Fp(xp) + F ⊤
+p (xp) ≤ 0, Hp(xp) ∈ R is the Hamiltonian,
+up ∈ Rm is the input, Gp(xp) ∈ Rp×m is the input
+arXiv:2301.03746v1  [eess.SY]  10 Jan 2023
+
+mapping matrix and yp ∈ Rm is the natural passive output
+corresponding to the input up.
+CbI assumes that the controller is a passive system that is
+interconnected with the plant (1) via a passive interconnec-
+tion. In this work, we consider a controller subsystem with
+two input-output ports, described by the ISO-PHS
+�
+�
+˙xc
+−yc1
+−yc2
+�
+� =
+�
+�
+K11(xc)
+K12(xc)
+K13(xc)
+K21(xc)
+K22(xc)
+K23(xc)
+K31(xc)
+K32(xc)
+K33(xc)
+�
+�
+�
+��
+�
+:=K(xc)
+�
+�
+∇xcHc
+uc1
+uc2
+�
+�
+(2)
+where xc ∈ Rc is the state of the controller, Hc(xc) ∈
+R is the controller Hamiltonian, uc1, yc1
+∈
+Rm and
+uc2, yc2 ∈ Rr are passive input-output pairs and K(xc) ∈
+R(p+m+r)×(p+m+r) satisfies K(xc) + K⊤(xc) ≤ 0 [15].
+The controller system (2) can be interconnected with the
+plant (1) via the passive interconnection
+up = −yc1
+uc1 = yp,
+(3)
+resulting in the closed-loop dynamics
+�
+�
+˙xp
+˙xc
+−yc2
+�
+� =
+�
+�
+Fp + GpK22G⊤
+p
+GpK21
+GpK23
+K12G⊤
+p
+K11
+K13
+K32G⊤
+p
+K31
+K33
+�
+�
+�
+��
+�
+Fcl
+�
+�
+∇xpHp
+∇xcHc
+uc2
+�
+�
+(4)
+where uc2, yc2 is a passive input-output pair to the inter-
+connected system. Noting that K + K⊤ ≤ 0, the closed-
+loop interconnection and damping structure Fcl satisfies
+Fcl + F ⊤
+cl ≤ 0 also.
+In the case of stabilisation, the objective is to construct
+the plant functions Hc(xc), K(xc) to ensure the existence
+of Casimirs which statically relate the controller states to
+functions of the plant states
+xc = fc(xp),
+(5)
+for fc(x) ∈ Rc. The Casimir functions and controller initial
+conditions are then designed to assign a desirable minimum
+to the total energy function
+W(xp) = H(xp) + Hc(xc)|xc=fc(xp).
+(6)
+It is noted that the Lyapunov candidate W(xp) can be gen-
+eralised to a function of H, Hc and the Casimirs xc −fc(xp)
+[15]. Methods to ensure the existence of and constructing
+Casimirs have been reported in [7] and the references therein.
+B. Underactuated mechanical systems
+The primary objective of this work is to apply CbI to the
+class of underactuated mechanical systems, described by the
+dynamics � ˙q
+˙p
+�
+=
+�0n×n
+In
+−In
+0n×n
+� �∇qH
+∇pH
+�
++
+�0n×m
+G
+�
+u
+H(q, p) = 1
+2p⊤M −1(q)p
+�
+��
+�
+:=T (q,p)
++V (q)
+y = G⊤∇pH,
+(7)
+where q ∈ Rn, p ∈ Rn are the configuration and momen-
+tum vectors, respectively, u ∈ Rm in the input, M(q) =
+M ⊤(q) > 0 is the inertia matrix and y is the natural passive
+output corresponding the the input u. The input mapping
+matrix G is assumed to be constant and have the structure
+G =
+�
+Im
+0(n−m)×m
+�
+,
+(8)
+where n − m < n is the degree of underactuation of the
+system1. The Hamiltonian H(q, p) is the sum of the kinetic
+energy T(q, p) and the potential energy V (q), which allows
+the gradient of H with respect to q to be written as
+∇qH(q, p) = ∇qT(q, p) + ∇qV (q).
+(9)
+A full-rank left-annihilator for the input mapping matrix (8)
+is defined as
+G⊥ =
+�0(n−m)×m
+I(n−m)
+�
+(10)
+which satisfies G⊥G = 0(n−m)×m.
+In the subsequent development, we will require an alter-
+nate representation of the gradient of the kinetic energy with
+respect to configuration ∇qT(q, p). Noting that the kinetic
+energy is quadratic in p, the gradient ∇qT(q, p) can always
+be factored into the form
+∇qT(q, p) = E(q, p)M −1(q)p,
+(11)
+for some matrix E(q, p) ∈ Rn×n. This has been previously
+noted in [16] using the Christoffel symbols. Note, however,
+that the matrix E(q, p) is non-unique and in this work we
+will use the representation given by
+∇qT(q, p) = 1
+2
+∂⊤
+∂q
+�
+M −1(q)p
+�
+p
+= 1
+2
+∂⊤
+∂q
+�
+M −1(q)p
+�
+M(q)
+�
+��
+�
+:=E(q,p)
+M −1(q)p.
+(12)
+In constructing a CbI interpretation to total energy shaping
+it is useful to define a virtual input-output pair for the system
+(7) by defining the input
+uv = Gu,
+(13)
+which allows the system to written similarly to a fully-
+actuated system as
+� ˙q
+˙p
+�
+=
+�0n×n
+In
+−In
+0n×n
+� �∇qH
+∇pH
+�
++
+�0n×n
+In
+�
+uv
+yv = ∇pH,
+(14)
+where uv, yv ∈ Rn. From the definition (13), it is clear that
+any input uv must satisfy
+G⊥uv = 0m×1,
+(15)
+1The assumed structure of G requires that the first m configuration
+coordinates are chosen to be collocated with the actuators. This class of
+dynamics falls into the broader class of ISO-PHS (1). For a more general
+input mapping matrix ¯G(q) ∈ Rn×m, there exists a change of coordinates
+recovering the structure (8) if the columns of ¯G(q) are involute.
+
+which will be ensured in subsequent control design. Assum-
+ing that (15) holds, the input u can be described as a function
+of uv by
+u = G⊤uv.
+(16)
+The advantage of constructing the virtual input-output pair is
+that the virtual output now describes the full velocity vector
+yv = M −1(q)p = ˙q,
+(17)
+a property that will be exploited when shaping the potential
+energy.
+C. IDA-PBC for underactuated mechanical systems
+IDA-PBC is a control design methodology whereby the
+control signal is designed such that the closed-loop dynamics
+have a port-Hamiltonian (pH) structure. When applied to
+underactuated mechanical systems, the target closed-loop
+dynamics have the structure
+� ˙q
+˙p
+�
+=
+�
+0n×n
+M −1(q)Md(q)
+−Md(q)M −1(q)
+J2(q, p) − GKdG⊤
+� �∇qHd
+∇pHd
+�
+Hd(q, p) = 1
+2p⊤M −1
+d (q)p + Vd(q),
+(18)
+where Md(q) = M ⊤
+d (q) > 0, Vd(q) are the desired closed-
+loop inertia matrix and potential energies, respectively,
+J2(q, p) = −J⊤
+2 (q, p) is skew-symmetric and Kd = K⊤
+d ≥ 0
+is a tuning parameter used for damping injection. If Vd
+is minimised at the target configuration, Hd qualifies as a
+Lyapunov function for the closed-loop system [4].
+The complexity of applying IDA-PBC to underactuated
+system is satisfying the so-called matching conditions. This
+conditions requires that the dynamics of the open-loop and
+closed-loop systems must agree on the spaces perpendicular
+to the control signal. The structure chosen fo the closed-loop
+system in (18) ensures that the dynamics of q agree with (7).
+Comparing the dynamics of p results in the condition
+G⊥ �
+∇qH − Md(q)M −1(q)∇qHd − J2(q, p)∇pHd
+�
+= 0(n−m)×1,
+(19)
+which
+defines
+a
+PDE
+that
+should
+be
+solved
+for
+Md(q), Vd(q), J2(q, p).
+Noting
+the
+structure
+of
+the
+Hamiltonians, this PDE can be separated into the components
+involving p, and those that do not by
+G⊥ �
+∇qT − Md(q)M −1(q)∇qTd − J2(q, p)M −1
+d (q)p
+�
+= 0(n−m)×1
+G⊥ �
+∇qV − Md(q)M −1(q)∇qVd
+�
+= 0(n−m)×1.
+(20)
+These expressions are known as the kinetic energy and
+potential energy matching equations, respectively.
+D. Contributions
+The objective of this work is to construct a CbI interpreta-
+tion of IDA-PBC when applied to underactuated mechanical
+systems. The contributions of this work are threefold:
+C.1 ISO-PHS with Casimirs that statically relate states are
+considered and a closed-form solution to remove the
+Casimirs by reducing the dimension of the state vector
+is proposed. This solution can be applied to the closed-
+loop dynamics of CbI implementations of the form (4)
+to describe the resulting dynamics as a function of xp
+only.
+C.2 A CbI controller for underactuated mechanical system
+of the form (2) is proposed and the resulting closed-
+loop is shown to be equivalent to the well-known dy-
+namics (18). The CbI interpretation generates alternate
+matching conditions to the expressions (20), describing
+constraints on the added mass and added potential
+energy.
+C.3 Using the alternate matching conditions, it is shown that
+the kinetic energy matching equations reduce to ODEs
+in the special case of underactuation degree one where
+the mass matrix is a function of only one configuration
+coordinate. It is demonstrated that numerical methods
+can be utilised in such cases to avoid solving these
+expressions analytically.
+E. Related works
+Significant attention has been given to solving the match-
+ing equations (20) in recent decades. In [3], [17] it was
+shown that is the system is under-actuated degree one and
+the mass matrix depends on a single un-actuated coordinate,
+the kinetic energy matching condition can be simplified to an
+ODE. Using a novel parametrisation of J2(q, p), a general so-
+lution for under-actuated degree one system was proposed in
+[4] under the assumption that the mass matrix depends only
+on the actuated coordinates. This approach was extended
+in [18] using a momentum transformation to simplify the
+matching equations. More recently, solutions to the potential
+energy matching equations were considered in [19] under
+the assumption that the mass matrix and potential energy
+functions were dependent on only one variable. Finally, a
+general solution to the special case of 2 degree-of-freedom
+system was proposed in [6]. The studies [20], [21] considered
+the effects of friction on the closed-loop stability using IDA-
+PBC.
+In recent works, alternate approaches to constructing so-
+lutions to the matching equations have been explored. The
+existence of conservative forces that cannot be factorised
+into a skew-symmetric matrix J2(q, p) was investigated in
+[22], which resulted in alternate matching equations. Im-
+plicit system representations were used in [23] to construct
+solutions in an over-parameterised space where the closed-
+loop dynamics were subject to constraints. By working in
+the larger dimension, a solution to a under-actuated degree
+2 crane system was proposed. Pfaffian differential equations
+were utilised in [24] which resulted in the kinetic energy
+
+PDEs being converted to an alternate form which admits
+simpler solutions.
+Some authors have investigated the possibility of avoiding
+the matching equations altogether by considering the control
+signal to be a CbI. The work [10] relied on a Lagrangian
+structure and several technical assumptions to verify the
+existence of a second passive output corresponding to the
+input u. Using this second output, a stabilising control was
+designed that ensured stability without requiring a solution
+to the matching PDEs. A similar approach was proposed in
+[12], [11] where a second passive output was utilised and
+the control assumed to have a PID structure.
+III. CASIMIR REDUCTION
+In this section, a method for reducing the state dimension
+of ISO-PHS with Casimirs is derived. The reduction method
+applies to general ISO-PHS with Casimirs and can be directly
+applied to the resulting closed-loop dynamics of CbI schemes
+of the form (4) to describe the system as a function of xp
+only. In the sequel, the reduction method will be used to show
+equivalence between the CbI controller for underactuated
+mechanical systems and the IDA-PBC dynamics (18). Before
+introducing the state reduction solution, a useful lemma is
+required.
+Lemma 1: Consider a square block matrix of arbitrary
+dimension
+A =
+�A11
+A12
+A21
+A22
+�
+(21)
+and assume that A22 is invertible. If the symmetric com-
+ponent of A is negative semi-definite, A + A⊤ ≤ 0, the
+symmetric component of the Schur complement
+A11 − A12A−1
+22 A21
+(22)
+is negative semi-definite also.
+Proof: First note that the Schur complement of X can
+be computed by
+�
+I
+−A⊤
+21A−⊤
+22
+�
+A
+�
+I
+−A−1
+22 A21
+�
+= A11 − A12A−1
+22 A21.
+(23)
+The symmetric component of this expression is negative
+semi-definite as A + A⊤ ≤ 0.
+The solution for reducing the dimension of ISO-PHS
+which exhibit Casimirs is now introduced. This development
+applies to systems of the form
+�
+�
+˙x1
+˙x2
+−y
+�
+� =
+�
+�
+F11(x)
+F12(x)
+F13(x)
+F21(x)
+F22(x)
+F23(x)
+F31(x)
+F32(x)
+F33(x)
+�
+�
+�
+��
+�
+F (x)
+�
+�
+∇x1H
+∇x2H
+u
+�
+�
+(24)
+where x ∈ Rp+c is the state of the system which has been
+partitioned into x1 ∈ Rp, x2 ∈ Rc, H(x1, x2) ∈ R is the
+Hamiltonian, F(x) ∈ R(p+c)×(p+c) is the full-rank intercon-
+nection and damping matrix satisfying F(x) + F ⊤(x) ≤ 0,
+u ∈ Rm is the input and y ∈ Rm is the corresponding passive
+output. It is assumed that the system contains a Casimir and
+the states have been partitioned such that the Casimir can be
+written as
+x2 = fc(x1),
+(25)
+where fc(x1) ∈ Rc is differentiable.
+The first step in constructing a minimal system represen-
+tation is defining a new set of coordinates given by
+w = x2 − fc(x1) = 0c×1,
+(26)
+which is identically equal to zero by construction. The
+system (24) can be described in the coordinates (x1, w) by
+�
+�
+˙x1
+−y
+˙w
+�
+� =
+�
+�
+Ip
+0p×c
+0p×m
+0m×p
+0m×c
+Im
+− ∂fc
+∂x1
+Ic
+0c×m
+�
+�
+�
+�
+˙x1
+˙x2
+−y
+�
+�
+=
+�
+�
+Ip
+0p×c
+0p×m
+0m×p
+0m×c
+Im
+− ∂fc
+∂x1
+Ic
+0c×m
+�
+� F
+×
+�
+�
+Ip
+0p×m
+− ∂⊤fc
+∂x1
+0c×p
+0c×m
+Ic
+0m×p
+Im
+0m×c
+�
+�
+�
+�
+∇x1Hr
+u
+∇wHr
+�
+�
+=
+�
+�
+¯F11
+¯F12
+¯F13
+¯F21
+¯F22
+¯F23
+¯F31
+¯F32
+¯F33
+�
+�
+�
+��
+�
+¯
+F
+�
+�
+∇x1Hr
+u
+∇wHr
+�
+� ,
+(27)
+where
+Hr(x1, w) :=H(x1, w + fc(x1))
+¯F11(x1) =F11(x)|x2=fc(x1)
+¯F12(x1) =F13(x)|x2=fc(x1)
+¯F13(x1) =F12(x) − F11(x)∂⊤fc
+∂x1
+����
+x2=fc(x1)
+¯F21(x1) =F31(x)|x2=fc(x1)
+¯F22(x1) =F33(x)|x2=fc(x1)
+¯F23(x1) =F32(x) − F31(x)∂⊤fc
+∂x1
+����
+x2=fc(x1)
+¯F31(x1) =F21(x) − ∂fc
+∂x1
+F11(x)
+����
+x2=fc(x1)
+¯F32(x1) =F23(x) − ∂fc
+∂x1
+F13(x)
+����
+x2=fc(x1)
+¯F33(x1) =F22(x) − F21(x)∂⊤fc
+∂x1
+− ∂fc
+∂x1
+F12(x)
++ ∂fc
+∂x1
+F11(x)∂⊤fc
+∂x1
+����
+x2=fc(x1)
+.
+(28)
+Recalling that w is identically equal to zero, ˙w is also
+equal to zero. Consequently, the final row of (27) is a
+constraint that needs to be resolved to construct a minimal
+system representation. There are two methods of reduction
+that will be considered. Firstly, in the transformed coordi-
+nates (27) it can occur that one or more columns of ¯F⋆3 is
+identically zero. Without loss of generality, it is assumed that
+
+the first d columns of ¯F⋆3 are equal to zero. To remove the
+zero rows, the full-rank matrix B is defined as
+B =
+�0d×(c−d)
+I(c−d)
+�
+,
+(29)
+which acts to select the non-zero columns of ¯F⋆3. The zero
+columns and corresponding rows are removed from (27) by
+�
+�
+˙x1
+−y
+B⊤ ˙w
+�
+� =
+�
+�
+¯F11
+¯F12
+¯F13B
+¯F21
+¯F22
+¯F23B
+B⊤ ¯F31
+B⊤ ¯F32
+B⊤ ¯F33B
+�
+�
+�
+��
+�
+:= ¯
+FB
+�
+�
+∇x1Hr
+u
+B⊤∇wHr
+�
+� ,
+(30)
+which does not modify the system dynamics. Note that as
+¯F + ¯F ⊤ ≤ 0, ¯FB + ¯F ⊤
+B ≤ 0 also. Using this representation,
+a method for resolving the remaining constraint equations is
+presented under the assumption that B⊤ ¯F33B is full rank.
+Proposition 1: Consider the pH system (24) with Casimir
+(25). If the matrix B⊤ ¯F33B is full-rank for all x1 the system
+can be described by a reduced-order model
+� ˙x1
+−y
+�
+= Fr(x1)
+�∇x1Hr
+u
+�
+,
+(31)
+where
+Hr(x1) =H (x1, fc(x1))
+Fr(x1) =
+� ¯F11
+¯F12
+¯F21
+¯F22
+�
+−
+� ¯F13B
+¯F23B
+�
+(B⊤ ¯F33B)−1 �
+B⊤ ¯F31
+B⊤ ¯F32
+�
+(32)
+and ¯F(x1) satisfies Fr(x1) + F ⊤
+r (x1) ≤ 0.
+Proof: The expression B⊤ ˙w, defined in (30), is iden-
+tically equal to 0(c−d)×1 by construction. Note, however,
+that the gradient B⊤∇wHr is not necessarily equal to
+zero. Assuming that B⊤ ¯F33B is full rank, the expression
+B⊤∇wHr can be described as
+B⊤∇wHr = − (B⊤ ¯F33B)−1 �
+B⊤ ¯F31
+B⊤ ¯F32
+� �
+∇x1Hr
+u
+�
+(33)
+Substituting this expression into the dynamics (30) resolves
+to the reduced dynamics (31).
+To verify that Fr + F ⊤
+r
+≤ 0, note that Fr is the Schur
+complement of ¯FB which satisfies ¯FB + ¯F ⊤
+B ≤ 0. It follows
+that Fr + F ⊤
+r ≤ 0 by application of Lemma 1.
+Proposition 1 showed that an ISO-PHS that exhibits a
+Casimir function can be described in a reduced state-space.
+The class of dynamics that are derived from application of
+CbI (4) that result in a Casimir of the form (5) falls into the
+class of systems (24). The following Corollary tailors the
+Casimir reduction for this important sub-class of dynamics.
+Corollary 1: If the closed-loop dynamics of a CbI scheme
+(4) exhibit a Casimir of the form (5), the system can be
+equivalently expressed in the form (31) where
+x1 =xp
+Hr(xp) =Hp(xp) + Hc (fc(xp))
+¯F11 =Fp + GpK22G⊤
+p
+¯F12 =GpK23
+¯F13 =GpK21 −
+�
+Fp + GpK22G⊤
+p
+� ∂⊤fc
+∂xp
+¯F21 =K32G⊤
+p
+¯F22 =K33
+¯F23 =K31 − K32G⊤
+p
+∂⊤fc
+∂xp
+¯F31 =K12G⊤
+p − ∂fc
+∂xp
+�
+Fp + GpK22G⊤
+p
+�
+¯F32 =K13 − ∂fc
+∂xp
+GpK23
+¯F33 =K11 − K12G⊤
+p
+∂⊤fc
+∂xp
+− ∂fc
+∂xp
+GpK21
++ ∂fc
+∂xp
+�
+Fp + GpK22G⊤
+p
+� ∂⊤fc
+∂xp
+(34)
+and B is suitably chosen as per (29) using the expressions
+¯F⋆3. The arguments have been dropped from the definitions
+of ¯F⋆⋆(xp) for the sake of readability.
+Proof:
+The result follows from direct application of
+Proposition 1 to the dynamics (4).
+IV. CONTROL-BY-INTERCONNECTION FOR
+MECHANICAL SYSTEMS
+In this section, a control-by-interconnection scheme for
+under-actuated mechanical systems is presented. A dynamic
+2-port control system is introduced with the intention that it
+will be interconnected to the plant (14) via one of the ports.
+The controller states are constructed to be statically related to
+the plant states after interconnection, resulting in Casimirs.
+By applying Proposition 1, the closed-loop dynamic are
+defined in a reduced space in which the dynamics coincide
+with standard total energy-shaping control (18).
+The proposed CbI scheme is shown in Figure 1. The
+intention of this control subsystem is to interconnect with the
+plant (14) via the uc1, yc1 power port. The second input uc2
+is available for subsequent control design, such as damping
+injection. The terms M(qa2), E(qa2, pa) are the plant mass
+matrix (7) and factorisation of the kinetic energy gradient
+(12) evaluated at the controller states whereas J(qa2, pa) =
+−J(qa2, pa)⊤ ∈ Rn×n is a skew-symmetric matrix to be
+chosen. The three-port storage element Ha(qa1, qa2, pa) has
+states qa1, qa2, pa ∈ Rn and energy function similar to
+mechanical systems,
+Ha(qa1, qa2, pa) = 1
+2p⊤
+a M −1
+a (qa2)pa
+�
+��
+�
+:=Ta(qa2,pa)
++ Vd(qa2) − V (qa1)
+�
+��
+�
+:=Va(qa1,qa2)
+,
+(35)
+
+Fig. 1: Mass shaping as CbI for under-actuated mechanical
+systems.
+where M −1
+a (qa2) is the inverse added mass, Ta(qa2, pa) is
+the added kinetic energy, Vd(qa2) is the desired closed-loop
+potential energy, V (qa1) is the plant potential energy function
+(7) evaluated at the plant state qa1 and Va(qa1, qa2) is the
+total added potential energy. It is important to note that,
+although M −1
+a (qa2) is represented as a matrix inverse, it
+need not be invertible nor positive. Indeed, it will be shown
+in subsequent developments that the key requirement is that
+M −1
+d (q) := M −1(q) + M −1
+a (q)
+(36)
+should be positive definite.
+In subsequent analysis it will be shown that the intercon-
+nection of the control system with the plant (14) via the
+interconnection
+uv = −yc1
+uc1 = yv,
+(37)
+yields Casimirs
+�
+�
+qa1
+qa2
+pa
+�
+� =
+�
+�
+In
+0n×n
+In
+0n×n
+0n×n
+In
+�
+�
+�
+q
+p
+�
+�
+��
+�
+fc(xp)
+.
+(38)
+Assuming the Casimirs exist, some intuition regarding the
+construction of the control system in Figure 1 can be
+provided. Both qa1, qa2 were constructed to be equal to q.
+Firstly ˙qa1 is equal to ˙q by interconnection with the plant
+virtual output yv via a 1-junction. To verify a similar relation
+for qa2, assume that qa2 = q, pa = p holds which results
+in ∇paHa = M −1
+a (q)p and uc1 + ∇paHa = M −1
+d p. With
+this in mind, the transformer can be seen to reconstruct the
+velocity ˙q = M −1(q)p for the bottom 1-junction, resulting
+in ˙qa1 = ˙q.
+To construct a Casimir pa = p, first note from (7) and
+(12) that the plant momentum dynamics can be expressed as
+˙p = −∇qV − E(q, p)M −1(q)p + uv.
+(39)
+The control structure acts to remove these forces from the
+plant via the right side of the control structure and re-
+introduce them via the top 0-junction where they are shared
+with the dynamics of pa. The ˙qa1 bond acts to cancel the
+gravity term from the plant −∇qV . Recalling that the bottom
+1-junction has flow equal to M −1(q)p, the right-side gyrator
+cancels the term −E(q, p)M −1(q)p from the plant. The left-
+side gyrator then re-introduces the force −E(q, p)M −1(q)p
+via the top 0-junction where it is shared between ˙p and ˙pa,
+establishing the desired Casimir.
+The claimed Casimir (38) is now formalised in the follow-
+ing Proposition. For this development, note that the gradients
+of the added energy Ha(·) satisfy
+∇qa1Ha = −∇qa1V
+∇qa2Ha = ∇qa2Ta + ∇qa2Vd
+∇qa2Ta = 1
+2
+∂⊤
+∂qa2
+�
+M −1
+a (qa2)pa
+�
+pa
+∇paHa = M −1
+a (qa2)pa
+(40)
+and the expressions A(·), B(·), C(·) in Figure 1 can be
+evaluated as
+A(qa2, pa, uc1) =M −1(qa2)Md(qa2) [uc1 + ∇paHa]
+B(qa2, pa, uc1) =∇qa2Ha − E⊤(qa2, pa)∇paHa
+− M(qa2)J(qa2, pa)Md(qa2)
+× [uc1 + ∇paHa]
+C(qa2, pa, uc1) =Md(qa2)M −1(qa2)∇qa2Ha
+− Md(qa2)M −1(qa2)E⊤(qa2, pa)∇paHa
+− Md(qa2)J(qa2, pa)Md(qa2)
+× [uc1 + ∇paHa] ,
+(41)
+with Md(·) defined in (36). To ensure that the Casimir exists,
+a number of requirements are imposed on the selection of
+the added inverse mass M −1
+a (q) and closed-loop potential
+energy Vd(q) which are equivalent of the standard matching
+conditions used in IDA-PBC (20).
+Proposition 2: Consider the control system in Figure 1
+and assume that it is interconnected to the plant (14) via the
+interconnection (37). If M −1
+a (qa), Vd(qa) are chosen such
+that
+G⊥C(qa2, pa, uc1)|qa2=q,pa=p = G⊥∇qV
+(42)
+and the controller states are initialised as qa1(0) = qa2(0) =
+q(0), pa(0) = p(0), the Casimir (38) holds for all time.
+Proof: Consider that at some time instant T (38) holds,
+implying that
+qa1(T) = qa2(T) = q(T), pa(T) = p(T).
+(43)
+It is shown that if (42) is satisfied, then the derivatives of
+the states also agree
+˙qa1(T) = ˙qa2(T) = ˙q(T), ˙pa(T) = ˙p(T),
+(44)
+
+establishing the existence of a Casimir for all future time.
+We proceed by first establishing the relationship for the
+configuration vector. From (37) and (14) uc = M −1(q)p
+which establishes ˙qa1(T) = ˙q(T). The input uc = M −1(q)p
+is substituted into A(·) (41) to find
+A|t=T = M −1(qa2)Md(qa2)
+�
+M −1(q)p + M −1
+a (qa2)pa
+�
+|t=T
+= M −1(q)p|t=T
+= ˙q|t=T ,
+(45)
+confirming that ˙qa2(T) = ˙q(T).
+Next we consider the behaviour of the momentum states.
+First note that, from the bond graph in Figure 1 and the
+definition (16), the plant input uv is given by
+u = G⊤uv
+= G⊤ [Guc2 − C(qa2, pa, uc1) + ∇qa1V (qa1)]
+= uc2 − G⊤ [C(qa2, pa, uc1) − ∇qa1V (qa1)] .
+(46)
+Using the control definition (46) and the condition (42), the
+plant dynamics (7) can be expanded as
+˙p = − ∇qT(q, p) −
+�G⊤∇qV (q)
+G⊥∇qV (q)
+�
++ Gu
+= − ∇qT(q, p) −
+�
+G⊤∇qV (q)
+G⊥∇qV (q)
+�
++ G
+�
+uc2 − G⊤ [C(qa2, pa, uc1) − ∇qa1V (qa1)]
+�
+= − ∇qT(q, p) + Guc2
+−
+�G⊤ {C(qa2, pa, uc1) + ∇qV (q) − ∇qa1V (qa1)}
+G⊥∇qV (q)
+�
+(47)
+Note that at time T, ∇qV (q)|t=T
+= ∇qa1V (qa1)|t=T .
+Additionally recall the assumption (42) which allows the
+simplification
+˙p|t=T = − ∇qT(q, p) − C(qa2, pa, uc1) + Guc2.
+(48)
+Recalling the identity (45), the dynamics of pa at time T can
+be expanded to
+˙pa = Gv − E(qa2, pa)A(·)|t=T − C(qa2, pa, uc1)|t=T
+= Gv − ∇qT(q, p)|t=T − C(qa2, pa, uc1)|t=T ,
+(49)
+which agrees with (48). As (48) and (49) agree at time T,
+(44) is verified for the momentum states. If at the initial time
+t = 0 we have qa(0) = q(0), pa(0) = p(0), it follows that
+qa(t) = q(t) and pa(t) = p(t) for all time via integration,
+completing the proof.
+Proposition 2 has established that the Casimir (38) holds
+under some technical assumptions that will be verified in
+subsequent design. Before proceeding, we note that the
+control subsystem in Figure 1 can be written in the form
+(2) with
+xc =
+�
+q⊤
+a1
+q⊤
+a2
+p⊤
+a
+�⊤
+Hc(qa1, qa2, pa) = Ha(qa1, qa2, pa)
+K11(qa2, pa) =
+�
+�
+0n×n
+0n×n
+0n×n
+0n×n
+0n×n
+M −1Md
+0n×n
+−MdM −1
+D − D⊤ + MdJMd
+�
+�
+K12(qa2, pa) =
+�
+�
+In
+M −1Md
+MdJMd − D⊤
+�
+�
+K13 =
+�
+�
+0n×m
+0n×m
+G
+�
+�
+K21(qa2, pa) =
+�
+−In
+−MdM −1
+D + MdJMd
+�
+K31 =
+�
+0m×n
+0m×n
+−G⊤�
+K22(qa2, pa) = MdJMd
+K23 = G
+K32 = −G⊤
+K33 = 0m×m
+D(qa2, pa) = MdM −1E⊤.
+(50)
+In the subsequent developments it is assumed that the
+requisite (42) of Proposition 2 holds, implying qa1(t) =
+qa2(t) = q(t), pa(t) = p(t). Condition (42) will be verified
+by choice of M −1
+a
+and Vd. Assuming the Casimir holds, the
+expressions for A(·), B(·), C(·) in (41) can be simplified to
+A(q, p) =M −1(q)p
+B(q, p) =∇qTa(q, p) + ∇qVd(q, p) − E⊤(q, p)M −1
+a (q)p
+− M(q)J(q, p)p
+C(q, p) =Md(q)
+�
+M −1(q)∇qTa(q, p) + M −1(q)∇qVd(q, p)
+−M −1(q)E⊤(q, p)M −1
+a (q)p − M −1(q)J(q, p)
+�
+(51)
+Recalling the definition of ∇qaTa in (40), it is noted that
+C(·) contains some terms which are quadratic in p and some
+that are functions of q only. The function C(·) is divided into
+C(q, p) =CKE(q, p) + CP E(q)
+CKE(q, p) =
+�
+M −1
+a (q) + M −1(q)
+�−1
+�
+��
+�
+Md(q)
+{Y (q, p) − J(q, p)} p
+CP E(q) =
+�
+M −1
+a (q) + M −1(q)
+�−1 M −1(q)∇qVd,
+(52)
+where KE represents kinetic energy, PE represents poten-
+tial energy and Y is defined as
+Y (q, p) =1
+2M −1(q)∂⊤
+∂q
+�
+M −1
+a (q)p
+�
+− 1
+2
+∂
+∂q
+�
+M −1(q)p
+�
+M −1
+a (q).
+(53)
+As Y is linear in p it can be written as
+Y (q, p) =
+n
+�
+i=1
+piY i(q),
+(54)
+
+where
+Y i(q) =1
+2M −1(q)∂⊤
+∂q
+�
+M −1
+a (q)ei
+�
+− 1
+2
+∂
+∂q
+�
+M −1(q)ei
+�
+M −1
+a (q).
+(55)
+The
+key
+constraint
+for
+control
+design
+is
+choosing
+M −1
+a (q), Vd(q) satisfying the matching condition (42). From
+the definition of C(·) in (51), the constraint equation is
+a function of both M −1
+a (q) and
+�
+M −1
+a (q) + M −1(q)
+�−1,
+making direct design of this matrix difficult. To simplify
+the design process, an alternate characterisation of (42) is
+introduced.
+In the following proposition, the inverse mass matrix,
+inverse added mass matrix, interconnection matrix and Y (·)
+are partitioned as
+�
+m11(q)
+m⊤
+21(q)
+m21(q)
+m22(q)
+�
+=
+�G⊤
+G⊥
+�
+M −1(q)
+�
+G
+G⊥⊤�
+�
+ma11(q)
+m⊤
+a21(q)
+ma21(q)
+ma22(q)
+�
+=
+�G⊤
+G⊥
+�
+M −1
+a (q)
+�
+G
+G⊥⊤�
+�
+J11(q, p)
+−J⊤
+21(q, p)
+J21(q, p)
+J22(q, p)
+�
+=
+�G⊤
+G⊥
+�
+J(q, p)
+�
+G
+G⊥⊤�
+�Y11(q, p)
+Y12(q, p)
+Y21(q, p)
+Y22(q, p)
+�
+=
+�G⊤
+G⊥
+�
+Y (q, p)
+�
+G
+G⊥⊤�
+.
+(56)
+Using the above definitions, an alternate characterisation of
+(42) is presented.
+Proposition 3: The matching condition (42) is satisfied if:
+• The added mass matrix M −1
+a (q) is chosen such that
+D(q)
+�
+Y i(q) + Y i⊤(q)
+�
+D⊤(q) = 0(n−m)×(n−m)
+(57)
+for all i ∈ {1, . . . , n} where
+D(q) =
+�(m21 + ma21)(m11 + ma11)−1
+−In−m
+�
+.
+(58)
+• The desired potential energy Vd(q) satisfies
+s1(q)G⊥∇qV = −s2(q)G⊤∇qVd − s3(q)G⊥∇qVd
+= −D(q)M −1(q)∇qVd,
+(59)
+where
+s1(q) = (m22 + ma22)
+− (m21 + ma21)(m11 + ma11)−1(m⊤
+21 + m⊤
+a21)
+(60)
+is the Schur complement of M −1(q) + M −1
+a (q) and
+s2(q) = (m21 + ma21)(m11 + ma11)−1m11 − m21
+s3(q) = (m21 + ma21)(m11 + ma11)−1m⊤
+21 − m22.
+(61)
+Proof:
+From the graph in Figure 1 and the intercon-
+nection (37), the virtual input uv is given by
+uv = Gu =Gv − C + ∇qV.
+(62)
+Recalling the definition of G, (62) is equivalent to (42).
+Collecting the terms u, v and left multiplying by M −1
+a +M −1
+results in
+�
+M −1
+a
++ M −1�
+G(u − v)
+=
+�
+M −1
+a
++ M −1�
+{−CKE − CP E + ∇qV }
+(63)
+Due to the structure of G in (8), (63) has the left annihilator
+D(·), defined in (58).
+Left multiplying (63) by D(·) and separating the compo-
+nents into those relating to the kinetic and potential energies
+result in
+0(n−m)×1 = −D
+�
+M −1
+a
++ M −1�
+CKE
+(64)
+0(n−m)×1 = −D
+�
+M −1
+a
++ M −1�
+{CP E − ∇qV } .
+(65)
+Using (52), (65) is expanded to
+0(n−m)×1 =D
+�
+M −1
+a
++ M −1�
+∇qV
+− DM −1∇qVa,
+(66)
+which can be seen to agree with (59) after expanding.
+Now considering the constraint on the kinetic energy
+expression (64), the definition (52) is substituted to find
+0(n−m)×n =D {−Y + J} .
+(67)
+Using the relevant definitions, the first component of (67)
+can be solved for J21(q, p) as
+J21(q, p) =(m21 + ma21)(m11 + ma11)−1(−Y11 + J11)
++ Y21.
+(68)
+Substituting this expression back into the second component
+of (67) reveals the constraint
+0(n−m)×(n−m)
+=(m21 + ma21)(m11 + ma11)−1(−Y12 − J⊤
+21)
+− (−Y22 + J22)
+=(m21 + ma21)(m11 + ma11)−1 �
+−Y12 − Y ⊤
+21
+�
+− (m21 + ma21)(m11 + ma11)−1(−Y11 + J11)⊤
+× (m11 + ma11)−1(m⊤
+21 + m⊤
+a21) − (−Y22 + J22)
+= − D
+�
+−Y11 + J⊤
+11
+−Y12 − Y ⊤
+21
+0(n−m)×m
+−Y22 + J22
+�
+D⊤.
+(69)
+The term J22 is taken as below to solve the skew-symmetric
+part of this expression,
+J22
+= −1
+2D
+�
+−Y11 + Y ⊤
+11 + J⊤
+11 − J11
+−Y12 − Y ⊤
+21
+Y ⊤
+12 + Y21
+−Y22 + Y ⊤
+22
+�
+D⊤,
+(70)
+where J11 ∈ Rm×m is a free skew-symmetric term. The
+symmetric part of (69) must also be equal to zero, implying
+that
+D
+�
+Y + Y ⊤�
+D⊤ = 0(n−m)×(n−m).
+(71)
+Finally, noting that this must be true for each pi, the
+condition (57) follows.
+
+Remark 1: The expression (57) implicitly defines a set of
+PDEs that must be satisfied by any choice of M −1
+a (q). From
+the definition of Y i in (55), the first m equations are describe
+partial differential equations involving the partial derivatives
+of ma11, ma21. The remaining n − m equations describe
+partial differential equations involving the partial derivatives
+of ma21, ma22. This structure can be useful for resolving the
+equations into a standard representation for solving.
+Corollary 2: In the special case of under-actuation degree
+1, if M −1, M −1
+a
+is a function of only 1 configuration variable
+qi, the kinetic energy matching equations (57) can be reduced
+to a set of ODEs.
+d
+dqi
+�
+m⊤
+a21
+ma22
+�
+= g
+�
+ma11, d
+dqi
+ma11, M −1, d
+dqi
+M −1
+�
+, (72)
+where g(·) ∈ Rn is a function implicitly defined by the
+matching conditions (57) and ma11(qi) can be chosen freely.
+Proof: Assuming that the mass matrix M −1 is function
+only of a single configuration variable qi, we will also impose
+that the added mass M −1
+a
+is a function only of the same
+variable. As a consequence, the matching expression (57) is
+now only a function in the single variable qi. Notably, all
+partial derivatives of M −1
+a
+with respect to qk, where k ̸= i,
+are equal to zero.
+Noting Remark 1, the first n − 1 expressions of (57) pro-
+duce differential equations involving the partial derivatives
+of ma11, ma21. The dimension of ma21 is 1 × (n − 1), so
+the first n − 1 equations can be solved simultaneously to
+find an expression for
+d
+dqi m⊤
+a21. The nth expressions of (57)
+can then be resolved for an expression for
+d
+dqi ma22, which
+has dimension 1. Combining these expressions, the matching
+equations (57) can be resolved into an ODE of the form (72).
+Remark 2: Corollary 2 describes situations in which the
+kinetic energy matching equations can be reduced to an
+ODE. The solution, however, will depend on the choice of
+ma11(qi) and may not be globally defined. This poses the
+question of how should the function ma11(qi) be chosen to
+ensure an appropriate solution M −1
+a (qi)—a nonlinear control
+problem!
+The results of Corollary 1 describe the degrees of freedom
+that exist when constructing a solution to the added inverse
+mass matrix. Similar degrees of freedom exist in the defini-
+tion of the closed-loop potential energy that can be exploited
+to ensure positivity of the chosen function. The following
+Corollary defines a free function that can be utilised to this
+effect.
+Corollary 3: Suppose that there exists a full rank matrix-
+valued function K(q) ∈ Rm×m such that the integral
+Γ(q) =
+�
+K(q)G⊤ �
+M −1
+a (q) + M −1(q)
+�
+M(q) dq,
+(73)
+exists. The desired closed-loop potential energy can be
+chosen as
+Vd(q) = Vm(q) + Vf(Γ(q)),
+(74)
+where Vm(·) must be chosen to satisfy the potential energy
+matching conditions (59) and Vf(·) is a free function that
+does not impact the matching equations. Consequently, the
+matching equation (59) can be equivalently written as
+s1(q)G⊥∇qV = −s2(q)G⊤∇qVm − s3(q)G⊥∇qVm
+= −D(q)M −1(q)∇qVm.
+(75)
+Proof: Computing the gradient of Vd results in
+∇qVd = ∇qVm + ∂⊤Γ
+∂q ∇ΓVf
+= ∇qVm + M
+�
+M −1
+a
++ M −1�
+GK⊤∇ΓVf.
+(76)
+From the definition of D(q) in (58), we have the identity
+D(q)M −1M(q)
+�
+M −1
+a (q) + M −1(q)
+�
+G =
+�Im
+⋆�
+G
+= 0(n−m)×m
+(77)
+Substituting the expression (76) into (59) and noting the
+above expression results in the simplified matching equation
+(75).
+Remark 3: Vf(·) is a free function precisely because Γ is
+an integral of the passive output yc2. The potential energy
+Vf could be alternatively constructed as a capacitor element
+added to the input uc2 in Figure 1.
+Now we arrive at one of the key results of this work, the
+equivalence of the proposed CbI scheme and total energy-
+shaping control of underactuated mechanical systems. As-
+suming that the CbI scheme has been constructed to satisfy
+the required matching conditions to ensure the existence of a
+Casimir of the form qa = q, pa = p, Proposition 1 is applied
+to reconstruct the reduced closed-loop structure (18).
+Proposition 4: Consider the underactuated mechanical
+system with virtual input (14) and assume that Ma(q), Vd(q)
+are chosen such that the conditions of Proposition 3 are sat-
+isfied in some neighbourhood of a point (q, p) = (q⋆, 0n×1).
+If the control signal is chosen as
+u(q, p) =v − G⊤ �
+Md(q)M −1(q)
+�
+−E⊤(q, p)M −1
+a (q)p
++∇qaHa(qa, pa) − M(q)J(q, p)p] − ∇qV }
+(78)
+where
+Md(q) =
+�
+M −1
+a (q) + M −1(q)
+�−1 ,
+(79)
+the following hold:
+• The closed-loop dynamics have the form
+�
+˙q
+˙p
+�
+=
+�
+0n×n
+M −1(q)Md(q)
+−Md(q)M −1(q)
+J2(q, p)
+� �∇qHd
+∇pHd
+�
++
+�0n×m
+G
+�
+v
+Hd(q, p) = 1
+2p⊤M −1
+d (q)p + Vd(q)
+y = G⊤∇pHd,
+(80)
+where
+J2(q, p) =Md(q)
+�
+J(q, p) + M −1(q)
+�
+E(q, p) − E⊤(q, p)
+�
+×M −1(q)
+�
+Md(q) + Md(q)M −1(q)E⊤(q, p)
+− E(q, p)M −1(q)Md(q)
+(81)
+
+• If Md(q), Vd(q) satisfy
+Md(q) > 0,
+Vd(q) > 0
+(82)
+in some neighbourhood of (q, p)
+=
+(q⋆, 0n×1),
+(q⋆, 0n×1) is a stable equilibrium of the closed-loop
+system for v = 0m×1.
+• If the input signal v is used for damping injection
+v = −KdG⊤y
+(83)
+for some positive Kd ∈ Rm×m and the equilibrium
+(q, p) = (q⋆, 0n×1), (q⋆, 0n×1) is locally detectable
+from the output y, the point (q⋆, 0n×1) is asymptotically
+stable.
+Proof: Interconnection of the mechanical system with
+the control subsystem results in a closed-loop of the form
+(4), where xc, Hc and K⋆⋆ are defined in (50) and
+xp =
+�q
+p
+�
+Fp =
+�
+0n×n
+In
+−In
+0n×n
+�
+Gp =
+�0n×n
+In
+�
+.
+(84)
+From (38), we have that
+∂fc
+∂xp
+=
+�
+�
+In
+0n×n
+In
+0n×n
+0n×n
+In
+�
+� .
+(85)
+To verify the claim, Corollary 1 is applied which requires
+a suitable definition of B. Expanding the definitions of ¯F⋆3
+from (34) reveals
+¯F13 =
+�0n×n
+0n×n
+−In
+0n×n
+In − MdM −1
+D
+�
+¯F23 =
+�0n×n
+0n×n
+0n×n
+�
+¯F33 =
+�
+�
+0n×n
+0n×n
+0n×n
+0n×n
+0n×n
+In
+0n×n
+−In
+0n×n
+�
+� ,
+(86)
+resulting in the choice
+B =
+�
+�
+0n×n
+0n×n
+In
+0n×n
+0n×n
+In
+�
+� .
+(87)
+Expanding the expression B⊤ ¯F33B results in
+B⊤ ¯F33B =
+�0n×n
+−In
+In
+0n×n
+�
+(88)
+which is invertible, ensuring that Corollary 1 can be applied.
+Expanding the definitions of Fr in (32) results in the reduced
+dynamics
+�
+�
+˙q
+˙p
+−y
+�
+� =
+�
+�
+0n×n
+M −1Md
+0n×n
+−MdM −1
+¯J2
+G
+0n×n
+−G⊤
+0n×n
+�
+�
+�
+��
+�
+Fr
+�
+�
+∇qHd
+∇pHd
+v
+�
+�
+¯J2 = MdJMd + D − D⊤ + MdM −1D⊤ − DM −1Md,
+(89)
+which agrees with (80) when substituting in the definition
+for D in (50). Stability and asymptotic stability of the point
+(q⋆, 0n×1) follows from Proposition 1 of [17].
+Remark 4: From Proposition 4 it is clear that M −1
+a (q)
+does not need to be a positive matrix. Rather, the closed-
+loop mass M −1
+d
+must be positive to ensure stability fo the
+system. In cases that M −1
+a
+is positive, the control sub-system
+in Figure 1 is passive.
+V. EXAMPLE APPLICATIONS
+In this section the matching conditions derived in Propo-
+sition 3 are used to construct stabilising control laws for
+the cart-pole and acrobot systems. In both cases, the mass
+matrix depends on only one configuration variable, so the
+kinetic energy matching conditions can be reduced to ODEs
+as detailed in Corollary 2. This enables the solutions to be
+constructed numerically, removing the need to analytically
+solve the equations.
+Both
+examples
+were
+prepared
+in
+Matlab
+2022a
+and
+the
+source
+code
+is
+available
+via
+https://github.com/JoelFerguson/Underactuated Mechanical CbI.
+A. Cart-pole example
+The cart-pole system, shown in Figure 2, attempts to
+balance the pole of length ℓ and mass mp in the upright
+position by applying a force F to the cart with mass mc.
+The state q1 describes the horizontal displacement of the cart
+whereas q2 describes the angle of the pole from vertical in
+the clockwise direction. The cart-pole system can be written
+Fig. 2: The cart-pole system attempts to balance the pole in
+the upright position by regulating the force F.
+as a pH system of the form (7) with
+q =
+�q1
+q2
+�
+M(q) =
+� mc + mp
+mpl cos q2
+mpl cos q2
+mpl2
+�
+V (q) = mpgl cos q2
+G =
+�1
+0
+�
+.
+(90)
+In the subsequent control design, the parameters mc = mp =
+l = 1, g = 9.8 have been used.
+
+dw
+mcThe mass matrix of the cart-pole system depends only on
+q2, the unactuated coordinate. The added inverse mass is
+assumed to also be a function of q2 also, allowing it to be
+written as
+M −1
+a (q2) =
+�
+ma11(q2)
+m⊤
+a21(q2)
+ma21(q2)
+ma22(q2)
+�
+.
+(91)
+As noted in Corollary 2, the kinetic energy matching equa-
+tions (57) can be reduced to an ODE as both M −1, M −1
+a
+are a function of only one variable. The associate ODE is of
+the form (72) for qi = q2 where ma11(q2) is a free function
+to be chosen. The ODE can be evaluated using numerical
+solvers.
+Before solving the ODE associated with the kinetic energy
+matching equations, consideration should be given to how
+the resulting mass matrix impacts the closed-loop potential
+energy Vd. Recalling (74), the closed-loop potential energy
+is composed of a free term Γ(·) and a term Vm(q) which
+must satisfy (75), where s1, s2, s3 are defined in (60), (61).
+As the potential V, M −1, M −1
+a
+are all functions of only q2,
+Vm is also assumed to be a function of q2 only, reducing
+(75) to the ODE
+∇q2Vm = −s1(q2)
+s3(q2)∇q2V,
+(92)
+which can be evaluated numerically once a solution for
+M −1
+a (q2), and hence s1(·), s3(·), are found. noting that the
+vector field ∇q2V is divergent from the point q2 = 0, the
+closed-loop vector field ∇q2Vm should reverse the direction
+locally. This is ensured if the ratio s1(q)
+s3(q) is positive in some
+neighbourhood of the origin. Recalling that s1(q) is the Schur
+complement of M −1 + M −1
+a , which is necessarily positive,
+it is required that s3(q) be positive in some neighbourhood
+of q2 = 0. The values
+ma11(0) = 0
+ma21(0) = −2
+ma22(0) = 8,
+(93)
+where chosen which result in s1(0) = 1, s3(0) = 1 and
+λmin
+�
+M −1(0) + M −1
+a (0)
+�
+= 0.917 > 0.
+The added inverse mass matrix can now be found by
+numerically evaluating the ODE (72). The term ma11(q2) is
+a free function that was chosen to be constant ma11(q2) =
+0,
+∂
+∂q2 ma11 = 0 for this example. The resulting functions for
+ma21(q2), ma22(q2) were found to exist on the interval q2 ∈
+[−0.48, 0.48] and are shown in Figure (3). From Proposition
+4, M −1(q2)+M −1
+a (q2) should be positive to ensure stability,
+so the minimum eigenvalue of this expression is shown in
+the same figure.
+The closed-loop potential energy Vm(q2) can now be
+obtained by numerically by evaluating the ODE (92). The
+terms s1(·), s3(·) are evaluated using the solutions to M −1
+a
+shown in Figure (3). The resulting function Vm(q2) is shown
+in Figure (4). As expected, the function is positive in some
+neighbourhood of q2 = 0 due to the choice of the added
+mass at q2 = 0 in (93).
+-0.5
+0
+0.5
+q2
+-2
+0
+2
+4
+6
+8
+ma11
+ma12
+ma22
+-0.5
+0
+0.5
+q2
+0.1
+0.15
+0.2
+min[M-1+Ma
+-1]
+Fig. 3: A solution for the inverse added mass M −1
+a
+was found
+to exist for the cart-pole system on the domain q2 between
+±0.48 radians.
+-0.5
+0
+0.5
+q2
+0
+1
+2
+3
+4
+5
+Vm
+Fig. 4: Solution for the added potential energy Vm(q2) for
+the cart-pole system.
+The proposed functions of M −1
+a , Vm can be used to
+construct a controller to stabilise the pendulum in the upright
+position. To ensure stability of q1 = 0 also, the free term
+Vf(Γ(q)), defined in (74), is constructed. The function Γ(·)
+defined by the integral (73), where K(q) is a free function
+chosen to ensure solvability. Noting that M −1, M −1
+a
+are
+functions of q2 only, the parametrisation
+�β1(q2)
+β2(q2)�
+= G⊤ �
+M −1
+a (q2) + M −1(q2)
+�
+M(q2)
+(94)
+is introduced. The free function is chosen as K(q2) =
+1
+β1(q2),
+resulting in
+Γ(q) =
+� �
+1
+β2(q2)
+β1(q2)
+�
+dq
+= q1 +
+� β2(q2)
+β1(q2) dq2,
+(95)
+which can be solved numerically from the initial condition
+Γ(02×1) = 0. The function Vf(·) was taken as Vf(Γ(q)) =
+1
+2κΓ(q)2 with κ = 5 for simulation. A contour plot of the
+resulting closed-loop potential energy is shown in Figure (5).
+Note that a minimum has been assigned to q = 02×1. As a
+final control design stage, damping is injected via the new
+passive input/output pair with
+v = −5G⊤(M −1
+a
++ M −1)p.
+(96)
+The complete control signal is defined by the expression (46).
+The cart-pole system was simulated for 5 seconds from ini-
+tial conditions q(0) = (0, 0.3), p(0) = (0, 0). The resulting
+state evolution and closed-loop energy Hd is shown in Figure
+
+-0.4
+-0.3
+-0.2
+-0.1
+0
+0.1
+0.2
+0.3
+0.4
+q2
+-0.5
+0
+0.5
+q1
+log(Vd)
+Fig. 5: Contour plot of the closed-loop potential energy
+Vd(q) = Vm(q2) + Vf(Γ(q)) for the cart-pole system on
+log scale.
+6. As expected, the proposed controller stabilises the origin
+and the closed-loop energy Hd decreases monotonically.
+0
+1
+2
+3
+4
+5
+time (s)
+-2
+0
+2
+q1
+q2
+p1
+p2
+0
+1
+2
+3
+4
+5
+time (s)
+0
+1
+2
+Hd
+Fig. 6: Numerical simulation of cart-pole system in closed-
+loop with CbI scheme.
+B. Acrobot example
+The acrobot system, shown in Figure 7, consists of 2 links
+with an actuator supplying a input torque τ fixed between
+the base and second links. The base link has displacement of
+q2, measured from vertical, length ℓ2, mass m2, moment of
+inertia Jℓ1 and centre of mass ℓc2 from the base pivot point.
+The actuated link has displacement of q1 measured relative
+to the base link, length ℓ1, mass m1, moment of inertia Jℓ1
+and centre of mass ℓc1 from the actuated pivot point. The
+control objective of this system is to stabilise the upright
+equilibrium position (q1, q2) = (0, 0). The acrobot system
+Fig. 7: The acrobot system attempts to balance in the vertical
+position by manipulating the torque generated by an actuator
+between the two links.
+can be written as a pH system of the form (7) with
+M(q) =
+�
+c2
+c2 + c3 cos q1
+c2 + c3 cos q1
+c1 + c2 + 2c3 cos q1
+�
+V (q) = c4g cos q2 + c5g cos(q1 + q2)
+G =
+�1
+0
+�
+,
+(97)
+where
+c1 = m2ℓ2
+c2 + m1ℓ2
+2 + Jℓ2
+c2 = m1ℓ2
+c1 + Jℓ1
+c3 = m1ℓ2ℓc1
+c4 = m2ℓc2 + m1ℓ1
+c5 = m1ℓc1.
+(98)
+For the purposes of simulation, we take the values g = 9.8,
+c1 = 2.3333, c2 = 5.3333, c3 = 2, c4 = 3, c5 = 2 which
+were previously used in [5], [25].
+In this example, the total energy-shaping controller pro-
+posed in [5] is reconstructed as a CbI control scheme by
+solving the matching conditions of Proposition 3. In that
+work, the closed-loop mass matrix was chosen to be the
+constant matrix
+M −1
+d
+=
+� 0.3385
+−0.9997
+−0.9997
+5.9058
+�
+(99)
+which will be recovered in subsequent computations.
+The mass matrix of the acrobot system depends only on
+q1, the actuated coordinate. The added inverse mass matrix
+is assumed to be a function of only q1 also, resulting in the
+structure
+M −1
+a (q1) =
+�
+ma11(q1)
+m⊤
+a21(q1)
+ma21(q1)
+ma22(q1)
+�
+.
+(100)
+As the system is underactuated degree 1 and the mass matrix
+is a function of only one variable, the kinetic energy match-
+ing equations can be reduced to an ODE as per Corollary
+2. The resulting ODE has the form (72) with qi = q1 and
+where ma11(q1) is a free function.
+
+In order to recover the result (99), this free function
+ma11(q1) is chosen as
+ma11(q1) = G⊤ �
+M −1
+d
+− M −1(q1)
+�
+G
+= 0.3385 − c1 + c2 + 2c3 cos q1
+c1c2 − c2
+3 cos2(q1) .
+(101)
+The initial conditions ma12(0), ma22(0) are similarly defined
+as
+ma12(0) = G⊥ �
+M −1
+d
+− M −1(0)
+�
+G = −0.1313
+ma12(0) = G⊥ �
+M −1
+d
+− M −1(0)
+�
+G⊥⊤ = 5.2743.
+(102)
+The added inverse mass was evaluated numerically and the
+results are shown in Figure 8. As the previously reported
+solution (99) is globally defined, it is unsurprising that the
+inverse added mas is also globally defined. As expected, the
+minimum eigenvalue of M −1 + M −1
+a
+is constant also.
+-2
+0
+2
+q1
+-2
+0
+2
+4
+6
+8
+ma11
+ma12
+ma22
+-2
+0
+2
+q1
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+min[M-1+Ma
+-1]
+Fig. 8: A solution for the added mass M −1
+a
+for the acrobot
+was found to exist globally.
+Solving the potential energy PDE (75) is difficult due to
+the open-loop potential energy being a function of both q1
+and q2. This dependence implies that Vm cannot be resolved
+directly using an ODE solver. Considering the structure of
+V in (97), it is proposed that the closed-loop energy Vm has
+the structure
+Vm(q) = f1(q1) sin(q2) + f2(q1) cos(q2),
+(103)
+which has derivatives
+∇q1Va =∂f1
+∂q1
+sin(q2) + ∂f2
+∂q1
+cos(q2)
+∇q2Va =f1(q1) cos(q2) − f2(q1) sin(q2).
+(104)
+The open-loop potential energy has gradients
+∇q1V = − c5g sin(q1) cos(q2) − c5g cos(q1) sin(q2)
+∇q2V = − c4g sin(q2) − c5g sin(q1) cos(q2)
+− c5g cos(q1) sin(q2).
+(105)
+Substituting the expressions (104) and (105) into (59) and
+matching coefficients results in the system of equations
+� ∂f1
+∂q1
+∂f2
+∂q1
+�
+=
+1
+s2(q1)
+×
+�c4gs1(q1) + c5gs1(q1) cos(q1) + s3(q1)f2(q1)
+c5gs1(q1) sin(q1) − s3(q1)f1(q1)
+�
+,
+(106)
+which can be evaluated numerically. The values of f1, f2
+at the origin should be chosen to ensure that the origin is
+an equilibrium point and Vm is positive in q2. Considering
+the expressions (105), (106), the origin is an equilibrium for
+f1(0) = 0. The energy function (103) is locally positive
+with respect to q1 for f2(0) negative. For the purpose of
+simulation, f2(0) = −50 was used. The resulting function
+Vm is shown in Figure 9.
+Fig. 9: Solution for the added potential energy Vm(q2).
+Considering Figure 9, it is clear that Vm is not positive
+definite with respect to the origin. Note, however, that q2 = 0
+has been stabilised. To ensure stability of q1 = 0 also,
+the free term Vf(Γ(q)), defined in (74), is constructed. The
+function Γ(·) defined by the integral (73), where K(q) is
+a free function chosen to ensure solvability. Noting that
+M −1, M −1
+a
+are functions of q1 only, the parametrisation
+�
+β1(q1)
+β2(q1)
+�
+= G⊤ �
+M −1
+a (q1) + M −1(q1)
+�
+M(q1)
+(107)
+is introduced. The free function is chosen as K(q1) =
+1
+β2(q1),
+resulting in
+Γ(q) =
+� �
+β1(q1)
+β2(q1)
+1
+�
+dq
+=
+� β1(q1)
+β2(q1) dq1 + q2,
+(108)
+which can be solved numerically from the initial condition
+Γ(02×1) = 0. The function Vf(·) was taken as Vf(Γ(q)) =
+1
+2κΓ(q)2 with κ = 250 for simulation. A contour plot of
+the resulting closed-loop potential energy on a log scale is
+shown in Figure 10. Note that a minimum has been assigned
+to q = 02×1. As a final control design stage, damping is
+injected via the new passive input/output pair with
+v = −5G⊤(M −1
+a
++ M −1)p.
+(109)
+The complete control signal is defined by the expression (46).
+The acrobot system was simulated for 20 seconds from ini-
+tial conditions q(0) = (0, 0.5), p(0) = (0, 0). The resulting
+
+100
+m
+0
+-100
+-200
+4
+2
+0
+-2
+a2
+44
+2
+0
+-2
+q1-3
+-2
+-1
+0
+1
+2
+3
+q2
+-3
+-2
+-1
+0
+1
+2
+3
+q1
+log(Vd)
+Fig. 10: Contour plot of the closed-loop potential energy
+Vd(q) = Vm(q) + Vf(Γ(q)) on log scale for the acrobot
+system.
+state evolution and closed-loop energy Hd is shown in Figure
+11. As expected, the proposed controller stabilises the origin
+and the closed-loop energy Hd decreases monotonically.
+0
+5
+10
+15
+20
+time (s)
+-1
+0
+1
+2
+q1
+q2
+0
+5
+10
+15
+20
+time (s)
+-10
+0
+10
+p1
+p2
+0
+5
+10
+15
+20
+time (s)
+0
+20
+40
+Hd
+Fig. 11: Numerical simulation of acrobot system in closed-
+loop with CbI scheme.
+VI. CONCLUSIONS AND FUTURE WORKS
+In this work total energy shaping has been shown to
+have a CbI interpretation which results in alternate matching
+equations related to the added inverse mass. These equations
+were utilised to construct controllers for the cart-pole and
+acrobot systems, both of which have the property that the
+mass matrix depends on only one variable, using numerical
+methods. While the proposed approach is effective, a number
+of technical aspects of this approach require further investi-
+gation. In particular:
+• As detailed in Corollary 2, The kinetic energy matching
+equations can be posed as ODEs in the special case
+that the mass matrix depends on only one configuration
+variable. This property allows the matching equations
+to be evaluated numerical using ODE solvers. Further
+investigation into solving the matching equations in the
+case that the mass matrix is a function of multiple
+configuration variables is required. In some cases it
+may be possible to decouple the dependence on each
+coordinate, recovering equivalent ODEs. Alternatively,
+the numerical evaluation of the matching PDEs should
+be investigated.
+• When evaluating the kinetic energy matching equations
+in (72), the term ma11(qi) is a free function that can
+be used to control the resulting added inverse mass.
+As seen in the cart-pole example of Section V-A, poor
+choice of this function results in the solution only being
+defined on a small domain. Conversely, in the acrobot
+example of Section V-A this term was chosen to ensure
+a global solution to the matching equations. Choice of
+this function defines a nonlinear control problem that
+should be investigated to ensure desirable behaviour of
+the result.
+• In both examples of Section V the controllers were
+designed to stabilise the origin of the respective sys-
+tems. While this was achieved and verified numerically,
+asymptotic stability was not established. Asymptotic
+stability requires that the passive output of the closed-
+loop system is zero-state detectable, a task that is non-
+trivial for underactuated systems. Further investigation
+into methods for injecting damping into the unactuated
+momentum channels of the closed-loop system is re-
+quired. It is hoped that the CbI interpretation of the
+controller shown in Figure 1 may provide new insight
+into how this might be achieved.
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+

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@@ -0,0 +1,3435 @@
+arXiv:2301.11576v1  [math.PR]  27 Jan 2023
+EMPIRICAL PROCESS
+SAMPLED ALONG A STATIONARY PROCESS
+GUY COHEN AND JEAN-PIERRE CONZE
+Abstract. Let (Xℓ)ℓ∈Zd be a real random field (r.f.)
+indexed by Zd with common
+probability distribution function F. Let (zk)∞
+k=0 be a sequence in Zd. The empirical
+process obtained by sampling the random field along (zk) is �n−1
+k=0[1Xzk ≤s − F(s)].
+We give conditions on (zk) implying the Glivenko-Cantelli theorem for the empirical
+process sampled along (zk) in different cases (independent, associated or weakly corre-
+lated random variables). We consider also the functional central limit theorem when the
+Xℓ’s are i.i.d.
+These conditions are examined when (zk) is provided by an auxiliary stationary pro-
+cess in the framework of “random ergodic theorems”.
+Contents
+Introduction
+2
+1.
+General results on the empirical process along a sub-sequence
+3
+1.1.
+Preliminaries
+3
+1.2.
+A Glivenko-Cantelli type theorem
+10
+1.3.
+A sufficient condition for a FCLT for the sampled empirical process
+12
+2.
+Local times for ergodic sums
+15
+2.1.
+Auxiliary general results
+15
+2.2.
+Non centered cocycles
+22
+2.3.
+Counterexamples
+23
+3.
+Examples
+27
+3.1.
+Random walks
+27
+3.2.
+Extensions of the r.w. case
+32
+3.3.
+Step functions over rotations
+34
+Date: January 30, 2023.
+2010 Mathematics Subject Classification. Primary: 60F05, 28D05, 22D40, 60G50; Secondary: 47B15,
+37A25, 37A30.
+Key words and phrases. Empirical process, sampling along a stationary process, local times, Glivenko-
+Cantelli theorem, functional central limit theorem, random walks.
+1
+
+2
+GUY COHEN AND JEAN-PIERRE CONZE
+4.
+About limit theorems along ergodic sums
+37
+4.1.
+Glivenko-Cantelli theorem along ergodic sums
+37
+4.2.
+Discussion: universal sequences
+39
+References
+40
+Introduction
+For a sequence (Xk) of real i.i.d. random variables with common probability distribution
+function F, the empirical process is defined by �n−1
+k=0 [1Xk≤s − F(s)]. Recall two classical
+results.
+(A) the Glivenko-Cantelli theorem:
+a.s. the sequence of empirical distribution functions Fn(s) :=
+1
+n
+�n−1
+k=0 1Xk≤s converges
+uniformly to F, i.e. sups |Fn(s) − F(s)| → 0;
+(B) a functional central limit theorem (FCLT): if the r.v.s Xk have a common distribution
+F over [0, 1], then
+the process
+1
+√n
+�n−1
+k=0 [1Xk≤s − F(s)] converges weakly to a Brownian bridge in the space
+of cadlag functions on [0, 1].
+In this paper we study the extension of these results when the process is sampled along a
+subsequence, analogously to what is done for limit theorems in random scenery.
+In the sequel, for d ≥ 1, (Xℓ)ℓ∈Zd will be a real random field (r.f.) indexed by Zd defined
+on a probability space (Ω, F, P) with common probability distribution function F. The
+expectation on (Ω, P) is denoted by E. We consider in particular the case of a r.f. of i.i.d.
+r.v.’s or of stationary associated r.v.’s.
+Let (zk)∞
+k=0 be a sequence in Zd. The process obtained by sampling the random field along
+(zk) is Wn(s) := �n−1
+k=0[1Xzk≤s − F(s)].
+We will call Wn(s) “empirical process sampled along (zk)”, or simply “sampled empirical
+process”. A general question is whether the above results (A), (B) extend to the sampled
+empirical process Wn(s), in particular when (zk) is given by another stationary process
+with values in Zd.
+In Section 1, we give conditions on (zk) implying that (A) and (B) are still valid for
+an empirical process sampled along (zk) in different cases: independent, associated or
+weakly correlated random variables. The conditions are expressed in terms of the following
+quantities associated to the sequence (zk) in Zd: local time, maximal local time and
+number of self-intersections (up to time n) defined, for n ≥ 1, by
+Nn(ℓ) := #{0 ≤ k ≤ n − 1 : zk = ℓ},
+Mn := max
+ℓ
+Nn(ℓ), Vn := #{0 ≤ j, k ≤ n − 1 : zj = zk}.
+(1)
+
+EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS
+3
+They satisfy �
+ℓ Nn(ℓ) = n and n ≤ Vn = �
+ℓ N2
+n(ℓ) ≤ nMn ≤ n2.
+In the other sections, (zk) is given by a stationary process (or equivalently by the sequence
+(Skf(x))k≥1 of ergodic sums of a function f over a dynamical system).
+The conditions found in Section 1 lead to study the local times, maximum number of visits,
+number of self-intersections for the sequence (Skf(x)). General remarks are presented in
+Section 2. Then in Section 3, we consider two families of examples: random walks and
+some ergodic sums over a rotation.
+The Glivenko-Cantelli theorem along ergodic sums (extension of (A)) is strongly related
+to random ergodic theorems, in particular to results in [23] and [25]. This is discussed in
+the last Section 4.
+Finally let us mention the quenched FCLT for the 2-parameters process
+Wn(s, t) :=
+[nt]−1
+�
+k=0
+[1XZk(x)≤s − F(s)], (s, t) ∈ [0, 1]2.
+When (Xℓ) is a r.f. of i.i.d. r.v.’s indexed by Z2 and when the sampling is provided by a
+2-dimension centered random walk (Zk) with a moment of order 2, the weak convergence
+for a.e. x toward a Kiefer-M¨uller process can be shown. This will be the content of a
+forthcoming paper.
+Acknowledgements. Part of this research was done during visits of the first author to
+the IRMAR at the University of Rennes 1 and of the second author to the Center for
+Advanced Studies in Mathematics at Ben Gurion University. The authors are grateful to
+their hosts for their support.
+1. General results on the empirical process along a sub-sequence
+1.1. Preliminaries. In this subsection, results on the empirical process along a sub-
+sequence are shown for independent variables, as well for some of them for wider classes
+(associated, PDQ and weakly correlated random variables). We start by recalling some
+notions and auxiliary results.
+1) Associated variables
+Definition (cf. [17]): A finite set of real random variables T = (T1, T2, . . . , Tn) is said to
+be associated if Cov[f(T), g(T)] ≥ 0, for every coordinate-wise non-decreasing functions
+f = f(x1, ..., xn) and g = g(x1, ..., xn) for which E[f(T)], E[g(T)], E[f(T) g(T)] exist. An
+infinite set of random variables is associated if any finite subset of it is associated.
+Association of random variables is preserved under taking subsets and forming unions of
+independent sets (of associated random variables). In particular a family of independent
+variables is associated.
+
+4
+GUY COHEN AND JEAN-PIERRE CONZE
+Clearly, orthogonal associated random variables are independent. Examples of (non inde-
+pendent) stationary associated processes with absolutely summable series of correlations
+are provided by some Ising models. References to such examples of stationary Zd random
+fields which satisfies the FKG inequalities and with absolutely summable correlations
+can be found in Newman’s paper [26]. Notice that the FKG inequalities expresses the
+association property of the r.v.’s.
+2) PQD variables
+Two r.v.’s X, Y are called (cf. [24]) positively quadrant dependent (PQD) if,
+P(X > x, Y > y) ≥ P(X > x) P(Y > y), ∀x, y ∈ R.
+The property is preserved by centering. Any pairwise associated r.v.’s are pairwise PQD.
+Pairwise independent random variables are pairwise PQD associated.
+Two random variables X and Y are PQD if and only if for every non-decreasing functions
+f and g, Cov(f(X), g(Y )) ≥ 0 (whenever the covariance exists) ([17, Theorem 4.4]).
+3) We will use the following results:
+a) Maximal inequality of Newman and Wright [27, Inequality (12)]:
+If (Wi) is a sequence of centered associated, square integrable random variables, it holds:
+P( max
+1≤j≤n |
+j
+�
+i=1
+Wi| ≥ λ ∥
+n
+�
+i=1
+Wi∥2) ≤ 2P(|
+n
+�
+i=1
+Wi| ≥ (λ −
+√
+2) ∥
+n
+�
+i=1
+Wi∥2), ∀λ ≥ 0.
+(2)
+b) Hoeffding’s identity (see [2, Theorem 3.1])
+Let X, Y be random variables with finite second moments. For any absolutely continuous
+functions f, g on R, such that E[f 2(X) + g2(Y )] < ∞, it holds
+Cov(f(X), g(Y )) =
+� ∞
+−∞
+� ∞
+−∞
+f ′(x)g′(y)[P(X > x, Y > y) − P(X > x)P(Y > y)]dxdy.
+In particular, if X, Y are PDQ random variables, if |f ′|, |g′| ≤ M a.e., we have
+|Cov(f(X), g(Y ))| ≤ M2Cov(X, Y ).
+4) Uniformity in the analogues of Glivenko-Cantelli theorem will follow from the lemma:
+Lemma 1.1. [7, Lemma, p 140] Let Fn, F be a family of right continuous distributions on
+R. Assume that, for each point x in a dense countable set Q ⊂ R, we have Fn(x) → F(x).
+Let J be the set of jumps of F and assume that Fn(x)−Fn(x−) → F(x)−F(x−) for every
+x ∈ J. Then Fn(x) → F(x) uniformly in R.
+A strong law of large numbers
+First we state a law of large numbers for bounded r.v.’s valid under weak hypotheses.
+
+EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS
+5
+Let (Uℓ)ℓ∈Zd be a r.f. indexed by Zd of square integrable r.v’s on a probability space
+(Ω, F, P). Let (zk)k≥0 be a sequence in Zd, d ≥ 1, with numbers of self-intersections
+Vn, n ≥ 1. The partial sums along (zk) are denoted by Sn :=
+n−1
+�
+k=0
+Uzk.
+By the Cauchy-Schwarz inequality, if �
+ℓ supr |⟨Ur+ℓ, Ur⟩| < +∞, if holds for a finite
+constant C0:
+∥
+n−1
+�
+i=0
+Uzi∥2
+2 =
+�
+ℓ
+�
+r
+Nn(r + ℓ)Nn(r)⟨Ur+ℓ, Ur⟩ ≤ Vn
+�
+ℓ
+sup
+r |⟨Ur+ℓ, Ur⟩| = C0Vn.
+(3)
+In particular if the r.f. is stationary and the series of correlations is absolutely summable
+(i.e., �
+ℓ∈Zd |⟨X0, Xℓ⟩| < +∞), then the spectral density of the r.f.
+exists and is the
+continuous non-negative function ρ on Td with Fourier coefficients
+�
+Td e2πi⟨ℓ,t⟩ ρ(t) dt =
+⟨X0, Xℓ⟩ and it holds:
+∥Sn∥2
+2 = ∥
+n−1
+�
+i=0
+Uzi∥2
+2 ≤ Vn
+�
+ℓ
+|⟨Uℓ, U0⟩|.
+(4)
+Proposition 1.2. Suppose the r.v.’s Uℓ on (Ω, P) centered and uniformly bounded by the
+same constant K, ∥Uℓ∥∞ ≤ K, ∀ℓ. Assume that (zk) is such that
+Vn ≤ C1
+n2
+(log n)β , for constants C1, β.
+(5)
+1) Then, if β > 1 and
+�
+ℓ∈Zd
+sup
+r∈Zd |⟨Ur+ℓ, Ur⟩| < +∞,
+2) or if β > ζ for some ζ ∈ [1, 2] and the r.f. (Uℓ) is stationary with
+�
+ℓ∈Zd
+|⟨Uℓ, U0⟩|ζ < ∞,
+the (strong) LLN holds: Sn(ω)
+n
+→ 0, for P-a.e ω.
+Proof. 1) For convenience, if t is in R+, we define St as S[t]. From (3) it follows
+�
+(|Sn|
+n )2 dP ≤ C0
+Vn
+n2 ≤ C0C1
+1
+(log n)β .
+Therefore, putting β = 1 + η and α = 1 − η/2 (which implies αβ > 1) we have
+�
+k
+�
+(|S2kα|
+2kα )2 dP ≤ C0C1
+�
+k
+1
+(log 2kα)β = C′ �
+k
+1
+kαβ < +∞;
+hence:
+lim
+k→+∞
+S2kα
+2kα = 0, a.e.
+For n ≥ 1, let kn be such that 2(kn)α ≤ n < 2(kn+1)α (that is: kn = [(log2 n)1/α]).
+We put qn := 2(kn+1)α − 2(kn)α and pn = n − 2(kn)α ≤ qn.
+
+6
+GUY COHEN AND JEAN-PIERRE CONZE
+For qn, the following estimate holds: qn = 2(kn)α(2(kn+1)α−(kn)α − 1) ∼ C′′ 2(kn)α
+(kn)1−α.
+Using the uniform boundedness of the r.v.’s, we can write:
+|Sn
+n − S2(kn)α
+2(kn)α | = |S2(kn)α + �2(kn)α+pn
+i=2(kn)α
+Uzi
+2(kn)α + pn
+− S2(kn)α
+2(kn)α | = |2(kn)α �2(kn)α+pn
+i=2(kn)α
+Uzi − pnS2(kn)α
+2(kn)α(2(kn)α + pn)
+|
+≤ 2(kn)α �2(kn)α+pn
+i=2(kn)α
+|Uzi| + pn|S2(kn)α|
+2(kn)α(2(kn)α + pn)
+≤ 2(kn)α �2(kn)α+qn
+i=2(kn)α |Uzi| + qn|S2(kn)α|
+2(kn)α(2(kn)α)
+≤ qnK2(kn)α + qn|S2(kn)α|
+2(kn)α(2(kn)α)
+=
+qn
+2(kn)α (K + |S2(kn)α|
+2(kn)α ). ≤
+C
+2(kn)1−α(K + |S2(kn)α|
+2(kn)α ) → 0.
+2) We consider now the stationary case. Since ζ = 1 is special case of 1), we assume
+ζ ∈]1, 2]. We put β = ζ + η, where η is > 0 in view of the hypothesis.
+First, suppose that ζ = 2. Then under the hypothesis, the r.f. has a spectral measure νϕ
+absolutely continuous with respect to the Lebesgue measure λ on the torus with a density
+ρ ∈ L2(dt) given by the Fourier series ρ(t) = �
+ℓ∈Zd⟨Uℓ, U0⟩ e2iπ⟨ℓ,t⟩.
+Using the inequality λ{ρ > Mn} ≤ M−2
+n ∥ρ∥2
+2, we can write:
+∥Sn∥2
+2
+n2
+= 1
+n2
+�
+Td |
+n−1
+�
+j=0
+e2πi⟨zj,t⟩|2 dνϕ(t) ≤ Mn
+n2
+�
+Td |
+n−1
+�
+j=0
+e2πi⟨zj,t⟩|2 dt +
+�
+ρ>Mn
+ρ dt
+≤
+Mn
+Vn
+n2 + (λ{ρ > Mn})
+1
+2∥ρ∥2 ≤ Mn
+Vn
+n2 + M−1
+n ∥ρ∥2
+2.
+Taking Mn = (log n)1+ 1
+2 η, we obtain the bound
+1
+n2 ∥
+n−1
+�
+j=0
+Uzj∥2
+2 ≤
+C
+(log n)1+ 1
+2 η
+and then we finish the proof as in 1).
+Now, suppose that
+�
+ℓ∈Zd
+|⟨Uℓ, U0⟩|ζ < ∞ with 1 < ζ < 2. The spectral density ρ exists
+and is in L2(λ), since
+�
+ℓ∈Zd
+|⟨Uℓ, U0⟩|2 < ∞. Moreover it belongs to Lζ′(λ) where ζ, ζ′ are
+conjugate exponents (see: [31], p. 102, or [21] Th. 31.22), and it satisfies:
+∥ρ∥ζ′ ≤ (
+�
+ℓ∈Zd
+|⟨Uℓ, U0⟩|ζ)1/ζ.
+H¨older’s inequality implies:
+�
+ρ>Mn ρ dt ≤ (λ{ρ > Mn})1/ζ∥ρ∥ζ′. As
+λ{ρ > Mn} ≤ M−ζ′
+n
+�
+ρζ′ dt = M−ζ′
+n
+∥ρ∥ζ′
+ζ′,
+
+EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS
+7
+it follows:
+�
+ρ>Mn
+ρ dt ≤ M−ζ′/ζ
+n
+∥ρ∥1+ζ′/ζ
+ζ′
+.
+Therefore, we obtain
+1
+n2
+�
+Td |
+n−1
+�
+j=0
+e2πi⟨zj,t⟩|2 dνϕ(t) ≤ Mn
+Vn
+n2 +
+�
+ρ>Mn
+ρ dt ≤ Mn
+Vn
+n2 + M−ζ′/ζ
+n
+∥ρ∥1+ζ′/ζ
+ζ′
+.
+Now we take Mn such that : Mn/(log n)β = M−ζ′/ζ
+n
+, i.e. Mn = (log n)β/ζ′. We get
+1
+n2 ∥
+n−1
+�
+j=0
+Uzj∥2
+2 ≤
+C
+(log n)β(1−1/ζ′) =
+C
+(log n)β/ζ =
+C
+(log n)1+η/ζ with η > 0,
+and the end of the proof is as above.
+□
+Remarks 1.3. 1) Let us give an example of a non stationary r.f. (Uℓ) which satisfies
+Condition 1) of the previous proposition.
+We take (Uℓ = Vℓ Wℓ, ℓ ∈ Zd), where (Vℓ) and (Wℓ) are two r.f.’s independent from
+each other, with (Vℓ) centered stationary and such that �
+ℓ∈Zd |⟨Vℓ, V0⟩| < ∞, and (Wℓ)
+satisfying supℓ,p |⟨Wℓ+p, Wℓ⟩| < ∞.
+The r.f. (Wℓ) can be viewed as a (multiplicative) noise (which can be non stationary)
+independent from the r.f. (Uℓ). Clearly the condition in 1) is satisfied.
+2) For a stationary r.f. (Uℓ) with a bounded spectral density (but with a series of corre-
+lations which may be not absolutely summable), then like in 1) the condition β > 1 is
+sufficient for the conclusion of the theorem.
+Now, we give a pointwise bound for the sampled sums, first for i.i.d. r.v.’s, then for a
+stationary random field (Uℓ)ℓ∈Zd of associated r.v.’s.
+Proposition 1.4. 1) Suppose that the r.v.’s Uℓ, ℓ ∈ Zd, are i.i.d., centered, uniformly
+bounded by a constant K, ∥U0∥∞ ≤ K, and that E|U0|2 = 1. Then it holds
+lim sup
+n
+|Sn|
+√Vn (2 log log n)
+1
+2 ≤ K, P-a.e.
+(6)
+If Vn = o(n2 (log log n)−1), then lim
+n
+Sn
+n = 0, P-a.e.
+2) Suppose the random field stationary and the r.v.’s Uℓ centered associated.
+a) For all ε > 0, it holds, with σn := ∥ �n−1
+i=0 Uzi∥2:
+lim sup
+n
+|Sn|
+σn (log σn)
+1
+2+ε ≤ 1, P-a.e.
+(7)
+b) If moreover the r.f. has a summable series of correlations, then, for all ε > 0,
+|Sn| = O(
+�
+Vn (log n)
+1
+2+ε), P-a.e.
+(8)
+
+8
+GUY COHEN AND JEAN-PIERRE CONZE
+If Vn ≤ Cn2 (log n)−(1+η)) for some constants C, η > 0, then lim
+n
+Sn
+n = 0, P-a.e.
+Proof.
+A) Recall that σn = ∥ �n−1
+i=0 Uzi∥2. In case 2) we may assume ∥U0∥2 = 1, and
+then in all cases σn ≤ n and by association σn ≥ n
+1
+2. We have in case 1) σn = √Vn and
+in case 2b), for associated variables, by (4): σn ≤ (�
+p⟨Up, U0⟩)
+1
+2 √Vn. By association, σn
+is non-decreasing and tends to infinity.
+For ρ > 1, let nk = nk(ρ) be a strictly increasing sequence of integers such that ρk <
+σnk ≤ ρk+1. Since 1 ≤ σ2
+k+1 − σ2
+k ≤ 1 + 2k, such a sequence exists after a certain rank. By
+the choice of (nk) we have
+ρk < σnk ≤ ρk+1 < σnk+1 ≤ ρk+2.
+(9)
+Moreover, we have σnk+1/σnk ≤ ρ2 and, since σn ≤ n, nk ≥ ρk.
+Let (λn) be a non
+decreasing sequence of positive numbers such that
+λnk >
+√
+2, lim supk λnk+1/λnk ≤ 1,
+�
+k P
+��� �nk−1
+i=0 Uzi
+�� ≥ (λnk −
+√
+2) ∥ �nk−1
+i=0 Uzi∥2
+�
+< ∞.
+(10)
+By the previous inequalities and by Newman-Wright’s inequality (2) for the sequence of
+centered associated random variables 1 (Wi) = (Uzi), we have
+�
+k
+P(
+max
+0≤j≤nk−1
+��
+j
+�
+i=0
+Uzi
+�� ≥ λnk ∥
+nk−1
+�
+i=0
+Uzi∥2) ≤ 2
+�
+k
+P(|
+nk−1
+�
+j=0
+Uzj| ≥ (λnk−
+√
+2)∥
+nk−1
+�
+j=0
+Uzj∥2) < +∞.
+By the Borel-Cantelli lemma, it follows:
+lim sup
+k
+max0≤j≤nk+1−1
+�� �j
+i=0 Uzi
+��
+λnk+1 σnk+1
+≤ 1, P-a.e.
+Hence P-a.e.
+lim sup
+k
+max0≤j<nk+1−1 | �j
+i=0 Uzi|
+λnkσnk
+≤ lim sup
+k
+�λnk+1
+λnk
+σnk+1
+σnk
+�
+≤ ρ2.
+(11)
+Observe that, if |Si| > ρ2λiσi, for some i ∈ [nk, nk+1[, then max0≤j<nk+1 |Sj| > ρ2λnkσnk.
+This shows:
+{|Sn| > ρ2λnσn, i.o.} ⊂ { max
+0≤j<nk+1 |Sj| > ρ2λnkσnk, i.o.}.
+By this inclusion and (11) it follows: lim sup
+n
+| �n−1
+i=0 Uzi|
+λnσn
+≤ ρ2, P-a.e.
+Taking ρ = ρn with ρn ↓ 1, we obtain
+lim sup
+n
+| �n−1
+i=0 Uzi|
+λnσn
+≤ 1, P-a.e.
+(12)
+B) Choice of a sequence (λk) such that (10) is satisfied.
+1as it is a subset of a set of associated r.v.’s
+
+EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS
+9
+Case 1)
+Suppose that the Uk’s are i.i.d. r.v.’s. Recall that if (Wj, j ≥ 1) are centered bounded
+sequence of independent random variables on (Ω, P), for any finite sum of the Wj’s it
+holds by Hoeffding’s inequality for differences of martingale ([20]), for every ε > 0:
+P(|
+�
+j
+Wj| > ε) ≤ 2 exp(−1
+2
+ε2
+�
+j ∥Wj∥2∞
+).
+(13)
+We apply it to the family (Nn(ℓ)Uℓ, ℓ ∈ Zd). From the hypotheses, we have:
+�
+ℓ
+∥Nn(ℓ)Uℓ∥2
+∞ ≤ K2 �
+ℓ
+N2
+n(ℓ) = K2Vn.
+With ε = (λ −
+√
+2)√Vn, (13) implies:
+P
+��� �
+ℓ
+Nn(ℓ) Uℓ
+�� ≥ (λ −
+√
+2)
+�
+Vn
+�
+≤ 2 exp
+�
+− 1
+2(λ −
+√
+2)2 Vn
+K2Vn
+�
+= 2 exp
+�
+−
+1
+2K2(λ −
+√
+2)2�
+.
+Let c, δ be such that c > δ > K2. In the previous inequality, we take
+λ = λn = (2c log log n)
+1
+2.
+Let k(c, δ) be such that λnk >
+√
+2 and c(1−
+2
+√c log log nk ) ≥ δ > 1, for k ≥ k(c, δ). We have:
+∞
+�
+k=k(c,δ)
+P
+���
+nk−1
+�
+i=1
+Uzi
+�� ≥ (λnk −
+√
+2) ∥
+nk−1
+�
+i=1
+Uzi∥2
+�
+≤ 2
+∞
+�
+k=k(c,δ)
+exp
+�
+−
+1
+2K2(λnk −
+√
+2)2�
+≤
+2
+exp K2
+∞
+�
+k=k(c,δ)
+exp
+�
+− c
+K2 log log nk)(1 −
+2
+√c log log nk
+)
+�
+≤
+2
+exp K2
+∞
+�
+k=k(c,δ)
+1
+(k log ρ)
+δ
+K2 < ∞.
+Now we can apply (12). It follows:
+lim sup
+n
+| �n−1
+i=0 Uzi|
+�
+2c(log log n)Vn
+≤ 1, P-a.e.
+Taking c = cn with cn ↓ K2, we get (6).
+Case 2)
+For general associated r.v.’s, we use simply that P
+��� �n−1
+i=0 Uzi
+�� ≥ λ ∥ �n−1
+i=0 Uzi∥2
+�
+≤
+1
+λ2.
+We take λn = (log σn)
+1
+2 +ε, with ε > 0. By (9) we have λnk ≥ (k log ρ)
+1
+2+ε, and therefore,
+for a constant C1: �
+k
+1
+λ2nk ≤ C1
+�
+k k−(1+2ε) < +∞; hence condition (10).
+
+10
+GUY COHEN AND JEAN-PIERRE CONZE
+Moreover we have k log ρ ≤ log σnk ≤ log σnk+1 ≤ (k + 2) log ρ; hence
+λnk+1
+λnk
+=
+�log σnk+1
+log σnk)
+� 1
+2+ε ≤ (1 + 2
+k)
+1
+2+ε → 1.
+By (12), this proves (7) in 2a)
+For case 2b) we have σ2
+n ≤ Vn
+�
+p⟨Up, U0⟩ and σn ≤ n, hence it yields (8). The last
+conclusion in case 2b) is now clear. Remark that it follows also from Proposition 1.2.
+□
+1.2. A Glivenko-Cantelli type theorem.
+Empirical process
+Let us consider a random field of r.v.’s (Xℓ, ℓ ∈ Zd) on (Ω, F, P) with common distribution
+function F. Let (zk) ⊂ Zd be a sequence with self-intersections (Vn).
+Notation. We say that (Xℓ, ℓ ∈ Zd) satisfies a Glivenko-Cantelli theorem along a sequence
+(zk) in Zd if
+lim
+n sup
+s | 1
+n
+n
+�
+k=1
+1(−∞,s](Xzk(ω)) − F(s)| = 0, for P-a.e.ω.
+We show now a Glivenko-Cantelli theorem along a sequence (zk) under various hypotheses
+on (zk) and on (Xℓ) (mixing, i.i.d., associated or PQD).
+Le (Xℓ, ℓ ��� Zd) be a r.f. Denoting by σ(Xℓ) the σ-algebra generated by the random
+variable Xℓ, we define a coefficient of mixing by
+γ(ℓ) := sup
+r∈Zd
+sup
+A∈σ(Xr), B∈σ(Xℓ+r)
+|P(A ∩ B) − P(A)P(B)|.
+(14)
+Observe that for every s, t ∈ R it holds:
+sup
+r∈Zd |⟨1Xr≤s − P(Xr ≤ s), 1Xℓ+r≤t − P(Xℓ+r ≤ t)⟩| ≤ γ(ℓ), ∀ℓ ∈ Zd.
+(15)
+By (15) and Proposition 1.2, we get:
+Theorem 1.5. Let (zk) be such that Vn ≤ C1
+n2
+(log n)β , for constants C1 > 0, β.
+If �
+ℓ∈Zd γ(ℓ) < +∞ and β > 1,
+or if the r.f. is stationary and �
+ℓ∈Zd γ(ℓ)ζ < +∞, for some ζ ∈ [1, 2] and β > ζ,
+then (Xℓ, ℓ ∈ Zd) satisfies a Glivenko-Cantelli theorem along (zk).
+Using Proposition 1.4, we consider now the i.i.d. and associated cases.
+Theorem 1.6. a) If (Xℓ)ℓ∈Zd is a r.f. of i.i.d. r.v.’s, then under the condition Vn =
+o(n2 (log log n)−1) it satisfies a Glivenko-Cantelli theorem along (zk).
+b) If (Xℓ)ℓ∈Zd is a strictly stationary r.f. of associated r.v.’s such that �
+ℓ⟨Xℓ, X0⟩ con-
+verges, then, under the condition Vn = O(n2 log−(1+η) n) for some η > 0, for a.e. ω, we
+
+EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS
+11
+have for each continuity point s of F:
+lim
+n
+1
+n
+n−1
+�
+k=0
+1(−∞,s](Xzk(ω)) = F(s).
+(16)
+If F is continuous, the convergence is uniform in s.
+Proof.
+a) Denote by Fn(s)(ω) the means
+1
+n
+�n−1
+k=0 1(−∞,s](Xzk(ω)). Let Q be a dense
+countable set of continuity points of F.
+For every s ∈ Q, by the assumption on Vn and Proposition 1.4, there is a null set N(s)
+such that, for a sequence εn tending to 0, for every ω ̸∈ N(s),
+|Fn(s)(ω) − F(s)| ≤ εn(Vn log log n)− 1
+2|
+n−1
+�
+k=0
+�
+1(−∞,s](Xzk) − F(s)
+�
+| → 0.
+Then Fn(s)(ω) → F(s) for every ω outside the null set N := ∪s∈QN(s) and for s ∈ Q.
+Similarly by Proposition 1.4, for every s in the set J of jumps of F, we have Fn(s)(ω) −
+Fn(s−)(ω) → F(s) − F(s−) a.e. As J is countable, this convergence holds for every s ∈ J
+and ω ̸∈ ˜N, where ˜N is a null set.
+Outside the null set N ∪ ˜N, Lemma 1.1 applied with Q and J implies the result.
+b) We consider now the case of a strictly stationary random field of associated r.v.’s. Let
+s be a continuity point of the common distribution F. For every ǫ > 0 there exists δ > 0,
+such that F(s + δ) − F(s − δ) ≤ ǫ. As in [30], for δ > 0 and s, define the approximated
+step function hδ,s by hδ,s(x) = 0, if x ≤ s − δ and hδ,s(x) = 1 + x−s
+δ
+if s − δ ≤ x ≤ s,
+otherwise, hδ,s(x) = 1. It is a non decreasing continuous function with h′
+δ,s(x) = 1/δ for
+s−δ < x < s. It follows from the above Hoeffding’s identity applied to this approximated
+step function (see [2]):
+Cov(hδ,s(Xℓ), hδ,s(X0)) ≤ δ−2⟨Xℓ, X0⟩,
+Cov(hδ,s+δ(Xℓ), hδ,s+δ(X0)) ≤ δ−2⟨Xℓ, X0⟩.
+By association and non decreasing,
+�
+hδ,s(Xℓ)
+�
+as well as
+�
+hδ,s+δ(Xℓ)
+�
+are stationary r.f.s
+of associated r.v.’s, and we may apply Proposition 1.4 to their centered versions (also
+associated). The condition simply reads, for τ = s, s + δ:
+�
+ℓ
+Cov(hδ,τ(Xℓ), hδ,τ(X0)) ≤ δ−2 �
+ℓ
+⟨Xℓ, X0⟩ < ∞.
+We put Sn = �n−1
+k=0 hδ,s(Xzk) and Sn = �n−1
+k=0 hδ,s+δ(Xzk).
+By hδ,s+δ(x) ≤ 1{x>s} ≤
+hδ,s(x), it holds Sn ≤ �n−1
+k=0 1(s,∞)(Xzk) ≤ Sn. Hence by Proposition 1.4, we have almost
+everywhere 1
+nSn → E[hδ,s+δ(X0)] and 1
+nSn → E[hδ,s(X0)]. Since
+E[hδ,s(X0)] ≤ F(s) − F(s − δ) + 1 − F(s) ≤ ǫ + 1 − F(s),
+E[hδ,s+δ(X0)] ≥ 1 − F(s + δ) = 1 − F(s) − (F(s + δ) − F(s)) ≥ 1 − F(s) − ǫ,
+
+12
+GUY COHEN AND JEAN-PIERRE CONZE
+we conclude
+1 − F(s) − ǫ ≤ lim inf
+n
+1
+n
+n−1
+�
+k=0
+1(s,∞)(Xzk) ≤ lim sup
+n
+1
+n
+n−1
+�
+k=0
+1(s,∞)(Xzk) ≤ 1 − F(s) + ǫ.
+Subtracting the 1’s and taking ǫ → 0, we get (16).
+□
+PQD variables.
+The result shown for associated variables can be extended to the class of PDQ variables,
+but with a stronger condition on the local times of the sequence (zk).
+Proposition 1.7. Let (Uℓ) be a centered stationary random field of pairwise PQD r.v.’s
+such that �
+ℓ⟨Uℓ, U0⟩ converges. Let (zk) be a sequence of points with maximal local times
+sequence (Mn). If �
+n≥1
+Mn
+n2 < +∞, then 1
+n(Uz0 + · · · + Uzn−1) converges a.e. to 0.
+Proof. We apply the following result of [4]: let (Yj : j ≥ 1) be a sequence of pairwise
+centered PQD r.v.’s.
+with finite variance. If �
+j≥1 j−2Cov(Yj, �j
+i=1 Yi) converges and
+supj E|Yj| < ∞, then n−1 �n
+i=1 Yi → 0 a.e.
+Taking for Yj the (still) pairwise PQD r.v.’s Uzj, we get the result, since Cov(Uzj, Uz1 +
+· · · + Uzj) ≤ Mj
+�
+ℓ⟨U0, Uℓ⟩.
+□
+Now, we consider the empirical distribution.
+Theorem 1.8. Let (Xℓ) be a centered strictly stationary random field of pairwise PQD
+r.v.’s with distribution function F such that �
+ℓ⟨Xℓ, X0⟩ converges. Let (zk) be a sequence
+of points with maximal local times sequence (Mn). If �
+n≥1
+Mn
+n2 < +∞, then for each
+continuity point s of F, we have for a.e. ω: limn
+1
+n
+�n−1
+k=0 1(−∞,s](Xzk(ω)) = F(s).
+In particular, if F is continuous, the above convergence is uniform over s.
+Proof. The r.f.s hδ,s(Xℓ) and hδ,s+δ(Xℓ) are still stationary pairwise PQD. The proof is
+analogous to the proof of Theorem 1.6. For the last statement, we use Lemma 1.1.
+□
+Remark. If Mn = O(n (log n)−(1+η)), then Vn = O(n2 (log n)−(1+η)).
+If Vn ≤ C
+n2
+(log n)β , with β > 2, then
+�
+n≥1
+Mn
+n2 < +∞.
+As shown in Section 3, �
+n≥1
+Mn
+n2 converges when the sampling is done along random
+walks, but diverges in some examples of sampling along “deterministic” random walks.
+1.3. A sufficient condition for a FCLT for the sampled empirical process.
+After a Glivenko-Cantelli theorem for sampled empirical processes, we consider now the
+Functional Central Limit Theorem (FCLT). Let (zk) be in Zd, d ≥ 1, with the associated
+quantities Nn(ℓ), Mn and Vn defined by (1).
+Before restricting to a r.f. of i.i.d. r.v.’s, first we examine the variance in the more general
+situation where the series of correlations is absolutely summable.
+
+EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS
+13
+Kernel associated to a sequence (zk) and variance.
+Let Kn be the kernel (which is a real even function on Td depending on n ≥ 0) defined
+by the equivalent formulas:
+Kn(t)
+=
+|
+n−1
+�
+k=0
+e2πi⟨zk,t⟩|2 = n + 2
+n−1
+�
+k=1
+n−k−1
+�
+j=0
+cos(2π⟨zk+j − zj, t⟩) = |
+�
+ℓ∈Zd
+Nn(ℓ) e2πi⟨ℓ,t⟩|2
+=
+n + 2
+�
+ℓ
+�n−1
+�
+k=1
+n−k−1
+�
+j=0
+1zk+j−zj=ℓ
+�
+cos(2π⟨ℓ, t⟩).
+(17)
+If (Xℓ, ℓ ∈ Zd) is a stationary r.f. such that �
+ℓ∈Zd |⟨Xℓ, X0⟩| < +∞, the spectral density
+ρ is continuous and we have:
+�
+|
+n−1
+�
+k=0
+Xzk|2dP =
+�
+Td Kn(t) ρ(t) dt ≤ ∥ρ∥∞Vn ≤ (
+�
+ℓ∈Zd
+|⟨Xℓ, X0⟩|)Vn.
+One can ask if there is an asymptotic variance, i.e., a limit for the normalised quantity
+V −1
+n
+�
+|
+n−1
+�
+k=0
+Xzk|2dP which is bounded if the series of correlations is absolutely summable.
+The existence of asymptotic variance is shown in [11] in the case of summation along a
+random walk. We will come back to the question of its positivity for transient random
+walks in Subsection 3.1.
+Functional Central limit Theorem in the i.i.d. case
+The following result gives a sufficient condition for a Functional Central limit Theorem
+(FCLT) along a sequence (zk) in the i.i.d. case.
+The standard Brownian bridge process W 0(s) is the centered Gaussian process W 0(s) :=
+W(s)−sW(1) in C(0, 1), where W(s) is the Wiener process. It has the properties W 0(0) =
+W 0(1) = 0 and E[W 0(s1)W 0(s2)] = s1 ∧ s2 − s1s2.
+Let (Xk)k∈Zd be i.i.d. random variables with common probability distribution F in [0, 1].
+We put W 0
+F = W 0 ◦ F. Let Yn(s) be the random element in D[0, 1] defined by
+Yn(s) =
+1
+√Vn
+n−1
+�
+k=0
+[1Xzk≤s − F(s)] =
+1
+√Vn
+�
+ℓ∈Zd
+Nn(ℓ) [1Xℓ≤s − F(s)].
+Theorem 1.9. Yn(s) →D[0,1] W 0
+F(s), if (zk) satisfies the condition
+lim
+n
+M2
+n
+Vn
+= 0,
+(18)
+Proof.
+The result follows from the Cram´er-Wold device, if we prove convergence of the
+finite dimensional distributions and tightness. The variance is
+E[Yn(s)]2 = 1
+Vn
+�
+ℓ
+N2
+n(ℓ) E[1Xℓ≤s − F(s)]2 = σ2(s) = F(s)(1 − F(s)).
+(19)
+
+14
+GUY COHEN AND JEAN-PIERRE CONZE
+1) Finite dimensional distributions. The convergence follows from Lindeberg’s theorem
+for triangular arrays of independent random variables as in [3, thm 7.2]. The Lindeberg’s
+condition for the triangular array of independent r.v.’s
+�Nn(ℓ)[1Xℓ≤s − F(s)]
+√Vn
+�
+ℓ,n follows
+from
+1
+Vn
+�
+ℓ
+�
+{Nn(ℓ)|1Xℓ≤s−F (s)|≥ ε√Vn}
+N2
+n(ℓ) |1Xℓ≤s − F(s)|2dP
+≤
+1
+Vn
+�
+ℓ
+N2
+n(ℓ)
+�
+{supℓ Nn(ℓ)|1X0≤s−F (s)|≥ε√Vn}
+|1X0≤s − F(s)|2dP → 0,
+for every ε > 0, since Vn = �
+ℓ N2
+n(ℓ) and supℓ Nn(ℓ)
+√Vn
+→ 0, by assumption.
+For the correlation of the process taken at s1 and s2, it holds by independence:
+E[Yn(s1)Yn(s2)]
+=
+1
+Vn
+�
+ℓ1,ℓ2
+Nn(ℓ1)Nn(ℓ2)E[(1Xℓ1≤s1 − F(s1))(1Xℓ2≤s2 − F(s2))]
+=
+1
+Vn
+�
+ℓ
+N2
+n(ℓ)(F(s1 ∧ s2) − F(s1)F(s2)) = F(s1 ∧ s2) − F(s1)F(s2).
+This proves the convergence in distribution: Yn(s) → W 0
+F(s) for every s.
+Let us show now the convergence of the finite dimensional distributions. Starting with
+the asymptotic distribution of aYn(s1) + bYn(s2), by the above computation, we have
+E[(aYn(s1) + bYn(s2))2] =
+a2F(s1)(1 − F(s1)) + b2F(s2)(1 − F(s2)) + 2ab(F(s1 ∧ s2) − F(s1)F(s2)).
+(20)
+As above, it is easily seen that Lindeberg’s condition is satisfied for the triangular array
+defined by aYn(s1)+bYn(s2). It means that the asymptotic distribution of aYn(s1)+bYn(s2)
+is centered Gaussian with variance as computed above.
+Note that E[(aW 0(s1) + bW 0(s2))2] is also given by (20) above.
+Similarly, for every s1 ≤ · · · ≤ sr, it holds
+(Yn(s1), · · · , Yn(sr)) →dist (W 0
+F(s1), . . . , W 0
+F(sr)).
+Tightness. First we suppose F continuous. Following the method of [3], it is enough to
+show that for s ≤ t ≤ r, uniformly in n,
+E[(Yn(t) − Yn(s))2(Yn(r) − Yn(t))2] ≤ C(F(r) − F(s))2.
+Putting F(u, v) := F(v) − F(u), f(ℓ, u, v) := 1u<Xℓ≤v − F(u, v), for u < v, we compute
+E[(Yn(t) − Yn(s))2(Yn(r) − Yn(t))2] = 1
+V 2
+n
+E
+�� �
+ℓ
+Nn(ℓ)f(ℓ, s, t)
+�2� �
+ℓ
+Nn(ℓ)f(ℓ, t, r)
+�2�
+.
+
+EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS
+15
+By expansion and independence, the above expression is sum of three types of terms:
+1
+V 2
+n
+�
+ℓ
+N4
+n(ℓ) [A], 1
+V 2
+n
+�
+ℓ1,ℓ2
+N2
+n(ℓ1)N2
+n(ℓ2) [B], 1
+V 2
+n
+�
+ℓ1̸=ℓ2
+N2
+n(ℓ1)N2
+n(ℓ2) [C],
+with A = E[f 2(ℓ, s, t)f 2(ℓ, t, r)], B = E[f 2(ℓ1, s, t)] E[f 2(ℓ2, t, r)],
+C = E[f(ℓ1, s, t)f(ℓ1, t, r))] E[f(ℓ2, s, t)f(ℓ2, t, r))].
+By stationarity and since the intervals do not overlap, we have
+A
+= F(s, t)F 2(t, r) + F 2(s, t)F(t, r) − 3F 2(s, t)F 2(t, r),
+B
+= F(s, t)(1 − F(s, t)) · F(t, r)(1 − F(t, r)), C = F 2(s, t)F 2(t, r).
+Since 0 ≤ F(s, t), F(t, r), F(s, r) ≤ 1 and F(s, t), F(t, r) ≤ F(s, r), it follows
+A ≤ 2F 3(s, r) ≤ 2F 2(s, r), B ≤ F 2(s, r), C ≤ F 4(s, r) ≤ F 2(s, r).
+Recall that Vn = �
+ℓ N2
+n(ℓ). Using ∥ · ∥ℓ4 ≤ ∥ · ∥ℓ2 for the bound of the first term, we have
+for some fixed constant C > 0:
+E[(Yn(t) − Yn(s))2(Yn(r) − Yn(t))2] ≤ C(F(r) − F(s))2, ∀n.
+Hence by [3, Theorem 15.6], for non decreasing continuous F, the sequence of processes
+(Yn(s)) is tight in D(0, 1). This proves that, if F is continuous, then Yn →D(0,1) (W 0 ◦ F).
+Now, for a general F a classical method is to use a generalized inverse. Let us describe it
+briefly. We consider first the uniform empirical process. Let (ζk) be uniformly distributed
+i.i.d. r.v.’s. Denote the empirical process along (zk) with respect to (ζk) by Un(s). By
+applying what we have just proved for a continuous distribution, Un(s) →D(0,1) W 0(s).
+Now let F −1(t) := inf{s : t ≤ F(s)}. We get P(F −1(ζ0) ≤ s) = P(X0 ≤ s) = F(s), so
+Yn(s) =dist. Un(F(s)). We may then proceed as in Billingsley ([3, Theorem 5.1]) to deduce
+the FCLT for Yn(s) with W 0(F(s)) as limit.
+□
+2. Local times for ergodic sums
+In the previous section about limit theorems for the empirical process sampled along (zk),
+we have found sufficient conditions on the quantities Vn and Mn associated to (zk). When
+(zk) is given by a “cocycle”, zk = Skf(x), one can ask if these conditions are satisfied.
+We start with some general facts and construct counterexamples for which condition (18)
+is not satisfied. In the next section, we will discuss two very different examples of cocycles:
+first the case of random walks, then “stationary random walks” generated by a rotation.
+2.1. Auxiliary general results.
+First we introduce some notation and make general remarks.
+
+16
+GUY COHEN AND JEAN-PIERRE CONZE
+Notation 2.1. Let (X, B, µ) be a probability space and T a measure preserving trans-
+formation acting on X such that the dynamical system (X, B, µ, T) is ergodic.
+Let f be a measurable function on X with values in Zd, d ≥ 1. Its ergodic sums generated
+by the iteration of T, denoted by fk (or Skf), are
+fk(x) :=
+k−1
+�
+j=0
+f(T jx), k ≥ 1, f0(x) = 0.
+The sequence (fk(x), k ≥ 1) can be viewed as a “stationary random walk” defined on
+(X, B, µ). It will be called a “cocycle” and denoted by (µ, T, f) or simply (T, f).
+For x ∈ X, we put (cf. (1)) N0(x, ℓ) = 0 and, for n ≥ 1,
+Nn(T, f, x, ℓ)
+:=
+#{1 ≤ k ≤ n : fk(x) = ℓ}, ℓ ∈ Zd,
+Mn(T, f, x)
+:=
+max
+ℓ∈Zd Nn(T, f, x, ℓ),
+Vn(T, f, x)
+:=
+#{1 ≤ j, k ≤ n : fj(x) = fk(x)} =
+�
+ℓ∈Zd
+N2
+n(x, ℓ).
+Most of the time, we will omit T and f in the notation and write simply Nn(x, ℓ), Mn(x),
+Vn(x). We have �
+ℓ Nn(x, ℓ) = n and n ≤ Vn(x) ≤ n Mn(x).
+A question is to know if the following conditions hold for a.e. x:
+Vn(x) = o(n2 (log log n)−1) or Vn(x) ≤ C1
+n2
+(log n)β , with β > 1,
+(21)
+lim
+n
+M2
+n(x)
+Vn(x) = 0.
+(22)
+For a random walk this is related to a question studied in [16] and later in [15]: How
+many times does the walk revisit the most frequently visited site in the first n steps?
+Cylinder map.
+We denote by ˜Tf the map (sometimes called cylinder map) (x, ℓ) →
+(Tx, ℓ + f(x)) acting on X × Zd, endowed with the infinite invariant measure ˜µ defined
+as the product of µ by the counting measure on Zd.
+For ϕ : X × Zd → R the ergodic sums for the cylinder map are
+˜Snϕ(x, ℓ) :=
+n−1
+�
+k=0
+ϕ( ˜T k
+f (x, ℓ)) =
+n−1
+�
+k=0
+ϕ(T kx, ℓ + fk(x)).
+With ϕ0 := 1X×{0}, it holds
+˜Snϕ0(x, −ℓ) =
+n−1
+�
+k=0
+1X×{0}(T kx, −ℓ + fk(x)) = #{0 ≤ k ≤ n − 1 : fk(x) = ℓ}.
+Therefore, ˜Snϕ0(x, −ℓ) = Nn(ℓ)(x).
+
+EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS
+17
+Recurrence/transience. It can be shown that a cocycle (µ, T, f) (over an ergodic dynamical
+system) is either recurrent or transient. For f with values in Zd, it means that either
+Skf(x) = 0 infinitely often for a.e. x, or Skf(x) = 0 finitely often for a.e. x. In the latter
+case, we have limk |Skf(x)| = +∞, µ-a.e.
+Let Rn(x) = {ℓ ∈ Zd : fk(x) = ℓ for some k ≤ n} be the “range” of the cocycle, i.e., the
+set of points visited by fk(x) up to time n.
+In [14] the following result is shown (for the general case of a cocycle with values in a
+countable group): let G be a countable group and f : X → G. If the cocycle (T, f) is
+recurrent, then Card(Rn(x)) = o(n) for a.e. x. If it is transient, there exists c > 0 such
+that Card(Rn(x)) ∼ c n for a.e. x.
+Using the lemma below, this implies for a.e. x:
+lim inf
+n
+Vn(x)
+n
+> 0 in the transient case, Vn(x)
+n
+→ +∞ in the recurrent case.
+(23)
+To show (23) we use the following inequality valid for a general sequence (zk):
+Lemma 2.2. If A is a non empty subset in Zd, we have:
+Vn ≥
+��n−1
+k=0 1zk ∈A
+�2
+Card(A)
+.
+(24)
+Proof. Cauchy-Schwarz inequality implies:
+n−1
+�
+k=0
+1zk ∈A =
+�
+ℓ∈A
+n−1
+�
+k=0
+1zk =ℓ ≤ (
+�
+ℓ∈A
+(
+n−1
+�
+k=0
+1zk =ℓ)2)
+1
+2 (Card(A))
+1
+2 ≤ V
+1
+2
+n (Card(A))
+1
+2.
+□
+If zk = Skf(x), this show (23). Indeed by taking A = Rn(x) we get
+Vn(x) ≥
+n2
+Card(Rn(x)).
+(25)
+Lemma 2.3. The following formulas hold for quantities defined in 2.1.
+Vn(x)
+=
+n + 2
+n−1
+�
+k=1
+n−k−1
+�
+j=0
+(1fk(T jx)=0),
+(26)
+=
+2[Nn−1(Tx, 0) + Nn−2(T 2x, 0) + ... + N1(T n−1x, 0)] + n, n ≥ 2,
+(27)
+Mn(x)
+=
+max[Nn(x, 0), 1 +
+max
+1≤k≤n−1 Nn−k(T kx, 0)] ≤ 1 +
+max
+0≤k≤n−1 Nn(T kx, 0),
+(28)
+=
+Mn−1(Tx) + 1ℓ(n−1,Tx)=0 ≤ Mn−1(Tx) + 1.
+(29)
+Proof. a) From fk(x) = f(x) + fk−1(Tx), k ≥ 1, it follows
+Nn(x, ℓ) = Nn−1(Tx, ℓ − f(x)) + 1f(x)=ℓ, n ≥ 1.
+(30)
+
+18
+GUY COHEN AND JEAN-PIERRE CONZE
+Therefore we have:
+�
+ℓ∈Zd
+N2
+n(x, ℓ) =
+�
+ℓ∈Zd
+[Nn−1(Tx, ℓ − f(x)) + 1f(x)=ℓ]2 =
+�
+ℓ∈Zd
+[Nn−1(Tx, ℓ) + 1ℓ=0]2
+=
+�
+ℓ̸=0
+[Nn−1(Tx, ℓ)]2 + [Nn−1(Tx, 0) + 1]2 =
+�
+ℓ
+[Nn−1(Tx, ℓ)]2 + 2Nn−1(Tx, 0) + 1.
+Hence the relation
+Vn(x) = Vn−1(Tx) + 2Nn−1(Tx, 0) + 1.
+(31)
+We have V1(x) = 1 and by the previous relation we get by induction (26) and (27).
+b) For x ∈ X, let ℓ(n, x) (a most visited site by Sk(x) up to time n) be defined by
+ℓ(n, x)
+:=
+0, if Nn(x, 0) ≥ Nn(x, ℓ), for all ℓ ̸= 0,
+else
+:=
+ℓ1, if ℓ1 is such that Mn(x) = Nn(x, ℓ1) > Nn(x, 0).
+Let pn(x) ∈ [1, n] be the first visit of Sk(x) to ℓ(n, x) for k = 1, ..., n.
+By definition
+Mn(x) = Nn(x, ℓ(n, x)).
+We have Mn(x) = Nn(x, 0) if ℓ(n, x) = 0, else Mn(x) = Nn−pn(x)(T pn(x)x, 0) + 1, by the
+cocycle relation Spn(x)+k(x) − Spn(x)(x) = Sk(T pn(x)x). This implies:
+Mn(x) ≤ max[Nn(x, 0), Nn−pn(x)(T pn(x)x, 0) + 1] ≤ max[Nn(x, 0), max
+1≤k≤n Nn−k(T kx, 0) + 1].
+It follows (noticing that N0(x, 0) = 0):
+Mn(x) ≤ 1 +
+max
+0≤k≤n−1 Nn−k(T kx, 0) ≤ 1 +
+max
+0≤k≤n−1 Nn(T kx, 0).
+(32)
+This shows (28).
+c) Observe also the following relation: by (30) we have:
+Mn(x)
+=
+sup
+ℓ
+[Nn−1(Tx, ℓ − f(x)) + 1ℓ−f(x)=0] = sup
+ℓ
+[Nn−1(Tx, ℓ) + 1ℓ=0]
+=
+max [sup
+ℓ̸=0
+Nn−1(Tx, ℓ), Nn−1(Tx, 0) + 1].
+If ℓ(n − 1, Tx) = 0, then Nn−1(Tx, 0) ≥ supℓ̸=0 Nn−1(Tx, ℓ). If ℓ(n − 1, Tx) ̸= 0, then
+Nn−1(Tx, 0) < supℓ̸=0 Nn−1(Tx, ℓ). This shows (29).
+Remark 2.4. By (28), if Kn is a uniform bound over x of Nn(x, 0), then Mn(x) ≤ Kn.
+Likewise, if Nn(x, 0) ≤ Kn, for a.e. x, then Mn(x) ≤ Kn, for a.e. x.
+Now we show that the set of x ∈ X such that limn
+M2
+n(x)
+Vn(x) = 0 has measure 0 or 1.
+Lemma 2.5. It holds: lim
+n [M2
+n(x)
+Vn(x) − M2
+n(Tx)
+Vn(Tx) ] = 0.
+If T is ergodic, there is a constant γ ∈ [0, 1] such that lim sup
+n
+M2
+n(x)
+Vn(x) = γ for a.e. x.
+
+EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS
+19
+Proof. We use (31) and (29). Putting ε = 1ℓ(n−1,Tx)=0, we have:
+|M2
+n(x)
+Vn(x) − M2
+n−1(Tx)
+Vn−1(Tx) | = |M2
+n−1(Tx) + ε(2Mn−1(Tx) + 1)
+Vn−1(Tx) + 2Nn−1(Tx, 0) + 1 − M2
+n−1(Tx)
+Vn−1(Tx) |
+= |ε(2Mn−1(Tx) + 1)
+Vn(x)
+− (2Nn−1(Tx, 0) + 1)
+Vn(x)
+M2
+n−1(Tx)
+Vn−1(Tx) |
+≤ 2Mn−1(Tx) + 1
+Vn(x)
++ 2Nn−1(Tx, 0) + 1
+Vn(x)
+≤ 4Mn−1(Tx)
+Vn(x)
++
+2
+Vn(x) ≤
+4
+√n +
+2
+Vn(x).
+For the last inequality we use that either Mn(x) ≥ √n, hence Mn(x)
+Vn(x) ≤
+1
+Mn(x) ≤
+1
+√n, or
+Mn(x) < √n, hence Mn(x)
+Vn(x) ≤
+√n
+n =
+1
+√n. Therefore,
+|M2
+n(x)
+Vn(x) − M2
+n−1(Tx)
+Vn−1(Tx) | → 0.
+(33)
+Observe now that
+Mn(x) = Mn−1(x) + εn, where εn = 0 or = 1,
+and εn = 1 if and only if there is ℓn such that
+Mn−1(x) = #{1 ≤ k ≤ n − 1 : fk(x) = ℓn} and fn(x) = ℓn.
+We have
+M2
+n(x) = M2
+n−1(x) + cn, with cn = εn(1 + 2Mn−1(x))
+and Nn(x, ℓ) = Nn−1(x, ℓ) + ε′
+n(ℓ), with ε′
+n(ℓ) = 1fn(x)=ℓ and �
+ℓ∈Zd ε′
+n(ℓ) = 1.
+Therefore,
+Vn(x) =
+�
+ℓ∈Zd
+(Nn−1(x, ℓ) + ε′
+n(ℓ))2 = Vn−1(x) + 2
+�
+ℓ∈Zd
+ε′
+n(ℓ)Nn(x, ℓ)) + 1,
+0 ≤ Vn(x) = Vn−1(x) + dn, with dn ≤ 2Mn(x) + 1.
+|M2
+n(x)
+Vn(x) − M2
+n−1(x)
+Vn−1(x) | = |M2
+n−1(x) + cn
+Vn−1(x) + dn
+− M2
+n−1(x)
+Vn−1(x) | ≤
+cn
+Vn(x) +
+dn
+Vn(x)
+M2
+n−1(x)
+Vn−1(x)
+≤ εn(1 + 2Mn−1(x)
+Vn(x)
++ 2Mn(x) + 1
+Vn(x)
+M2
+n−1(x)
+Vn−1(x) ≤
+2
+Vn(x) + 4Mn(x)
+Vn(x)
+≤
+2
+Vn(x) + 4
+√n → 0.
+From (33) and the convergence above, it follows limn [ M2
+n(x)
+Vn(x) − M2
+n(Tx)
+Vn(Tx) ] = 0. By ergodicity
+of T, this shows the lemma
+□
+Case of a coboundary
+The case when f is coboundary degenerates. Indeed, the following holds:
+Proposition 2.6. If f is a coboundary we have:
+a) lim infn
+Mn(x)
+n
+> 0, for a.e. x;
+b) there is a constant β > 0 such that
+1
+n2Vn(x) → β, for a.e. x;
+c) for a.e. x, lim infn
+M2
+n(x)
+Vn(x) > 0.
+
+20
+GUY COHEN AND JEAN-PIERRE CONZE
+Proof. Suppose that f is coboundary, f = TΦ − Φ. Since f has values in Zd and T is
+ergodic, for all component Φj of Φ, e2πiΦj is a constant. It follows that Φ has also its
+values in Zd up to an additive constant and we can assume that Φ has values in Zd.
+a) We have lim infn
+Mn(x)
+n
+≥ lim infn
+1
+nNn(x, 0) > 0, for a.e. x. The positivity results the
+following simple argument:
+For R ≥ 1, let AR denote the set ∪ℓ:∥ℓ∥≤R(Φ = ℓ). Since, for each ℓ, by Birkhoff’s theorem,
+limn
+1
+n
+�
+0≤k≤n−1 1Φ(T kx)=ℓ = µ(Φ = ℓ), it holds
+1
+nNn(x, 0) ≥
+�
+ℓ∈AR
+1(Φ=ℓ)(x) 1
+n
+n−1
+�
+k=0
+1(Φ=ℓ)(T kx) →
+�
+ℓ∈AR
+1(Φ=ℓ)(x) µ(Φ = ℓ).
+Therefore, for every R ≥ 1, lim infn
+Mn(x)
+n
+≥ lim infn
+Nn(x,0)
+n
+≥ �
+ℓ∈AR 1(Φ=ℓ)(x) µ(Φ = ℓ),
+and the limit when R → ∞ at right is > 0, for a.e. x.
+b) For Vn, we have:
+Vn(f, x)
+=
+�
+ℓ∈Zd
+N2
+n(x, ℓ) =
+�
+ℓ∈Zd
+#{0 ≤ k ≤ n − 1 : Φ(T kx) − Φ(x) = ℓ}2
+=
+�
+ℓ∈Zd
+#{0 ≤ k ≤ n − 1 : Φ(T kx) = ℓ}2 =
+�
+ℓ∈Zd
+�
+�
+0≤k≤n−1
+1Φ(T kx)=ℓ
+�2,
+hence: 1
+n2
+�
+ℓ∈AR
+�
+�
+0≤k≤n−1
+1Φ(T kx)=ℓ
+�2 =
+�
+ℓ∈AR
+�1
+n
+�
+0≤k≤n−1
+1Φ(T kx)=ℓ
+�2 →
+�
+ℓ∈AR
+(µ(Φ = ℓ))2.
+This implies, for every R ≥ 1,
+lim inf
+n
+1
+n2
+�
+ℓ∈Zd
+�
+�
+0≤k≤n−1
+1Φ(T kx)=ℓ
+�2 ≥ lim
+n
+1
+n2
+�
+ℓ∈AR
+�
+�
+0≤k≤n−1
+1Φ(T kx)=ℓ
+�2 =
+�
+ℓ∈AR
+(µ(Φ = ℓ))2.
+It follows: lim inf
+n
+1
+n2
+�
+ℓ∈Zd
+�
+�
+0≤k≤n−1
+1Φ(T kx)=ℓ
+�2 ≥
+�
+ℓ∈Zd
+µ(Φ = ℓ)2. For the complementary
+of AR, it holds:
+�
+ℓ:∥ℓ∥>R
+� �
+0≤k<n
+1Φ(T kx)=ℓ
+�2 =
+�
+0≤j,k<n
+�
+ℓ:∥ℓ∥>R
+1Φ(T jx)=ℓ 1Φ(T kx)=ℓ
+≤
+�
+0≤j,k<n
+(
+�
+ℓ:∥ℓ∥>R
+1Φ(T jx)=ℓ) (
+�
+ℓ:∥ℓ∥>R
+1Φ(T kx)=ℓ) ≤
+�
+0≤j,k<n
+1Ac
+R(T jx)1Ac
+R(T kx) =
+� �
+0≤k<n
+1Ac
+R(T kx)
+�2.
+It follows for the upper bound:
+lim sup
+n
+1
+n2
+�
+ℓ∈Zd
+�
+�
+0≤k≤n−1
+1Φ(T kx)=ℓ
+�2
+≤ lim
+n
+�
+ℓ∈AR
+�1
+n
+�
+0≤k≤n−1
+1Φ(T kx)=ℓ)2 + lim
+n
+�1
+n
+�
+0≤k<n
+1Ac
+R(T kx)
+�2
+=
+�
+ℓ∈AR
+(µ(Φ = ℓ))2 + µ(Ac
+R)2
+→
+R→∞
+�
+ℓ∈Zd
+µ(Φ = ℓ)2.
+
+EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS
+21
+This shows b) with β = �
+ℓ∈Zd µ(Φ = ℓ)2 > 0.
+c) Follows from a) and b).
+□
+Proposition 2.7. There is a constant β ≥ 0 such that, for a.e. x, limn
+Vn(x)
+n2
+= β. We
+have β > 0 if and only if the cocycle (T, f) is a coboundary.
+Proof. The case of a coboundary follows from Proposition 2.6.
+Suppose now that the cocycle is not a coboundary. From (26), we can write
+Vn(x)
+n2
+= 1
+n + 2
+n
+n−1
+�
+k=1
+1
+n
+n−k−1
+�
+j=0
+(1fk(T jx)=0)
+≤ 1
+n + 2
+n
+n−1
+�
+k=1
+1
+n
+n−1
+�
+j=0
+(1fk(T jx)=0) = 1
+n + 2
+n
+n−1
+�
+j=1
+Nn(T jx, 0)
+n
+.
+We will show that 1
+n
+n−1
+�
+j=0
+Nn(T jx, 0)
+n
+tend to 0 a.e.
+By the ergodic theorem of Dunford and Schwarz (in the space of infinite measure X × Z)
+applied to ˜Tf and φ0 = 1X×{0}, which is bounded and in Lp(X × Z), for every p ≥ 1, we
+get a function ˜φ0(x) which is ˜Tf-invariant and in L1(X × Z) and
+lim
+n
+Nn(x, 0)
+n
+= ˜φ0(x), a.s.
+As f is not a coboundary, ˜φ0 is zero a.e. (cf. for instance [12].)
+Observe that ∥ supn≥L
+Nn(x,0)
+n
+∥2 → 0, as L goes to +∞. Indeed, for every 0 < ε ≤ 1,
+letting Aε,L := {x : supn≥L
+Nn(x,0)
+n
+> ε}, we have µ(Aε,L) → 0, when L → +∞. Since
+Nn(x,0)
+n
+≤ 1, it follows, for L big enough:
+� �
+sup
+n≥L
+(Nn(x, 0)
+n
+)
+�2 dµ ≤ ε2 + µ(Aε,L) ≤ 2ε.
+We put Λn(x) := sup
+s≥n
+Ns(x, 0)
+s
+. By the previous observation, we have limn ∥Λn∥2 = 0.
+Let us consider the following maximal function for the action of T:
+˜Λn(x) = sup
+1≤r<∞
+1
+r
+r−1
+�
+j=0
+Λn(T jx) = sup
+1≤r<∞
+1
+r
+r−1
+�
+j=0
+sup
+s≥n
+Ns(T jx, 0)
+s
+.
+(34)
+From a classical maximal inequality, we have ∥˜Λn∥2 ≤ 2∥Λn∥2 → 0.
+Observe also that, from the definition of ˜Λn in (34), the following inequalities hold:
+˜Λn(x) ≥ sup
+r,s≥n
+1
+r
+r−1
+�
+j=0
+Ns(T jx, 0)
+s
+≥ 1
+n
+n−1
+�
+j=0
+Nn(T jx, 0)
+n
+.
+
+22
+GUY COHEN AND JEAN-PIERRE CONZE
+The sequence sup
+r,s≥n
+1
+r
+r−1
+�
+j=0
+Ns(T jx, 0)
+s
+is non negative and decreasing. Since ∥˜Λn∥2 → 0,
+the L2-norm of its limit in (X, µ) is zero. The result follows.
+□
+Remark 2.8. (see also section 4 and [25])
+Let (Uℓ)ℓ∈Zd be a r.f. of square integrable r.v.’s on a probability space (Ω, F, P) stationary
+in the weak sense and such that �
+ℓ |⟨Uℓ, U0⟩| < +∞. By (4) and Proposition 2.7, if f is
+not a coboundary, it holds
+1
+n2 ∥
+n−1
+�
+k=0
+Ufk(x)∥2
+2 ≤ C Vn(x)
+n2
+→ 0, for µ-a.e. x.
+Another result of norm convergence whose proof is like the proof of Proposition 1.2 is the
+following. Suppose that the r.f. is stationary. Let ϕ be an observable on the dynamical
+system (Ω, P, θ) with a spectral measure νϕ. We have:
+�
+Ω
+|
+n−1
+�
+j=0
+ϕ ◦ θzj|2 dP =
+�
+T1 |
+n−1
+�
+j=0
+e2πizjt|2 dνϕ(t).
+Assume that νϕ is absolutely continuous with respect to the Lebesgue measure on the
+torus, and let ρ ∈ L1(dt) such that dνϕ(t) = ρ(t)dt. For ε > 0 there is M such that
+�
+ρ>M ρ dt < ε. We have
+1
+n2
+�
+Td |
+n−1
+�
+j=0
+e2πi⟨zj,t⟩|2 dνϕ(t)
+≤
+M
+n2
+�
+Td |
+n−1
+�
+j=0
+e2πi⟨zj,t⟩|2 dt +
+�
+ρ>M
+ρ dt ≤ M Vn
+n2 + ε.
+This shows that Vn
+n2 → 0 implies 1
+n2
+�
+Ω
+|
+n−1
+�
+j=0
+ϕ ◦ θzj|2 dP → 0. This is satisfied by every
+ϕ ∈ L2(P), if the dynamical system has a Lebesgue spectrum.
+In particular, taking zk = fk(x), by Proposition 2.7, if f is not a coboundary, it holds
+1
+n2
+�
+Ω
+|
+n−1
+�
+j=0
+ϕ(θfj(x)ω)|2 dP(ω) → 0, for a.e. x.
+When the spectral density is square integrable, as we have seen in Proposition 1.2, the
+pointwise convergence holds under quantitative hypothesis on the sequence (zk).
+2.2. Non centered cocycles.
+In an ergodic dynamical system (X, µ, T), if f : X → R is an integrable function with
+µ(f) > 0, by the ergodic theorem for the ergodic sums ST
+n f(x) = �n−1
+k=0 f(T kx), it holds
+for a.e. x: limn
+1
+nSnf(x) > 0 and therefore limn ST
+n f(x) = +∞. If f has values in Z, as
+
+EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS
+23
+the process ST
+n f(x) visits finitely often each site, one can think there is a chance that the
+following condition is satisfied:
+lim
+n
+M2
+n(T, f, x)
+Vn(T, f, x) = 0.
+(35)
+A case where (35) is satisfied is the following: let X be a topological compact space,
+T : X → X a continuous map, which is uniquely ergodic with µ as unique invariant
+measure. Let f : X → Z be an integrable function such that µ(f) ̸= 0. Assume f to be
+Riemann-integrable (i.e. such that, for every ε > 0, there are two continuous functions
+ψ0, ψ1 with ψ0 ≤ f ≤ ψ1 and µ(ψ1 − ψ0) ≤ ε).
+Then, the ergodic means of f converge uniformly, and this implies the existence of N
+such that 1
+n|ST
+n f(x)| ≥ 1
+2|µ(f)| > 0 for n ≥ N and every x. It follows that the number of
+visits of ST
+n f(x) to 0 is ≤ N, for every x. By remark 2.4, Mn(x) ≤ N, for every x, and a
+fortiori (35) is satisfied.
+Nevertheless, we will see that (35) may fail in non uniform cases: there are dynamical
+systems and sets B of positive measure such that, for f = 1B,
+lim sup
+n
+M2
+n(T, f, x)
+Vn(T, f, x) = 1.
+(36)
+2.3. Counterexamples.
+In this subsection, we construct a transient counterexample, and also a recurrent coun-
+terexample with a function f of null integral such that (36) is satisfied.
+To construct these counterexamples, we start by considering a general ergodic dynamical
+system (X, µ, T) and a measurable set B ⊂ X of positive measure. Let TB be the induced
+map on B, R(x) = RB(x) = inf{k ≥ 1 : T kx ∈ B} the first return time of x in B and
+Rn(x) = RB
+n (x) := �n−1
+k=0 R(T k
+Bx) the n-th return time of x in B.
+We take x ∈ B. If f is a function such that f = 0 outside B, the position of the sums
+up to time Rn−1(x) are the positions of the ergodic sums STB
+n f for the induced map up
+to time n, that is:
+{f(x), f(x) + f(TBx), ..., f(x) + f(TBx) + ... + f(T n−1
+B
+x)}.
+For a site ℓ, the number of visits up to time Rn−1(x) of the ergodic sums for T is
+NRn−1(x)(x, ℓ) =
+n−1
+�
+k=0
+RB(T k
+Bx) 1STB
+k
+f(x)=ℓ
+and therefore
+VRn−1(x)(T, x) =
+�
+ℓ
+[
+n−1
+�
+k=0
+RB(T k
+Bx) 1S
+TB
+k
+f(x)=ℓ]2.
+(37)
+
+24
+GUY COHEN AND JEAN-PIERRE CONZE
+Case f = 1B. Clearly �n−1
+k=0 f(T k
+Bx) = n. For the map T, the ergodic sums of f are
+incremented by 1 when and only when the iterates T jx visit the set B. Otherwise, they
+stay fixed. The times of visits in B, for x ∈ B, are 0, R(x), R(x) + R(TBx), .... We have:
+for x ∈ B,
+Rn−1(x)+t
+�
+j=0
+f(T jx) = n, for t = 0, ..., Rn(x) − Rn−1(x) − 1.
+For Nn(T, x, ℓ) = Nn(T, f, x, ℓ), it holds:
+Nn(T, x, ℓ)
+=
+0, if n < Rℓ(x),
+=
+t, if n = Rℓ(x) + t, with 0 ≤ t < Rℓ+1(x) − Rℓ(x),
+=
+Rℓ+1(x) − Rℓ(x) = R(T ℓ
+Bx), if n ≥ Rℓ+1(x).
+For L ≥ 1, we have for the time preceding the L-th return to the basis for f = 1B:
+MRL(x)−1(T, f, x) = max
+ℓ≤L R(T ℓ
+Bx), VRL(x)−1(T, f, x) =
+�
+ℓ≤L
+R2(T ℓ
+Bx).
+(38)
+In order to compute an explicit example, it is easier to start from a given map S and
+construct a special flow T over this map.
+Let ϕ : X → N be integrable and ≥ 1. The (discrete time) special map T = Tϕ is defined
+on ˜X := {(x, k), x ∈ X, k = 0, ..., ϕ(x) − 1} ⊂ X × R,
+by T(x, k) := (x, k + 1), if 0 ≤ k < ϕ(x) − 1, := (Sx, 0), if k = ϕ(x) − 1.
+Let ˜µ be the probability measure defined on ˜X by ˜µ(A × {k}) = µ(ϕ)−1 µ(A), for k ≥ 0
+and A ⊂ {x : k ≤ ϕ(x) − 1}. It is Tϕ-invariant. The space X can be identified with the
+subset B = {(x, 0), x ∈ X} of ˜X with normalized measure. The set B is the basis and
+ϕ − 1 the roof function of the special map Tϕ.
+As for the map S we will take an ergodic rotation, the special flow Tϕ will be also ergodic
+for the measure ˜µ on ˜X.
+Observe that the recurrence time R(x) = RB(x) for the special flow in the basis B is ϕ(x)
+and the n-th return time of x in B is Rn(x) = RB
+n (x) = �n−1
+k=0 ϕ(Skx).
+For S, let us take a rotation S = Sα on X = T/Z by α mod 1, where α is irrational. We
+denote by qn the denominators of α. We will construct the measure preserving transfor-
+mation which is the special flow (with discrete time) over Sα with a roof function ϕ such
+that, for cocycle generated by 1B in the system ( ˜X, ˜µ, T), lim inf
+n
+Vn(T, x)
+M2
+n(T, x) = 1.
+We will use the next lemma with p = pn, q = qn, the numerators and denominators of α.
+Lemma 2.9. Let p, q ≥ 1, (p, q) = 1, be such that |α − p/q| < 1/q2. For every x, there is
+a value 0 ≤ i < q such that x + iα mod 1 ∈ [0, 2/q].
+More generally, for every interval I of length 2/q, for every x, there is a value 0 ≤ i < q
+such that x + iα mod 1 ∈ I.
+
+EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS
+25
+Proof. It is well known that there is exactly one value of jα mod 1, for 0 ≤ j < q, in each
+interval [ ℓ
+q, ℓ+1
+q [, ℓ = 0, ..., q − 1. Let us recall a proof. For j = 0, jα ∈ [0, 1/q[. The map
+j → ℓj = jp mod q, which is injective, is a permutation of the set {1, ..., q − 1} onto itself.
+We have α = p/q + γ, with |γ| < 1/q2.
+Assuming γ > 0, it follows: jα mod 1 ∈ [ ℓj
+q , ℓj
+q + j
+q2] ⊂ [ ℓj
+q , ℓj+1
+q [, for j = 1, ..., q − 1. The
+case γ < 0 is treated the same way.
+Now let us prove the first point. Let x be in [0, 1[. There is i0 ∈ {0, ..., q − 1} such that
+x = i0
+q + θ, with 0 ≤ θ < 1/q. By the claim, there is i ∈ [0, q[ such that iα mod 1 ∈
+[ q−i0
+q , q−i0+1
+q
+]. Hence x + iα mod 1 ∈ [θ, 1
+q + θ] ⊂ [0, 2
+q].
+□
+Let (λn) be an increasing sequence of positive integers which will be subjected below to
+growth conditions. First we assume that it satisfies the condition:
+qλn+1 ≥ 3qλn, ∀n ≥ 1.
+(39)
+Denote by Jn the interval Jn = [
+3
+qλn+1
+, 3
+qλn
+]. For the roof function, we take, with εn =
+1
+n2,
+ϕ = 1 +
+�
+n≥1
+⌊εnqλn⌋1Jn.
+The function ϕ is integrable:
+�
+ϕdµ ≤ 1 + 3 �
+n εn. Observe also that, by (39), the length
+of Jn is > 2/qλn and that (εnqλn) is not decreasing for n ≥ 2 .
+Let x be in the basis. By construction, the orbit of x under the iteration of Tϕ is that
+of the rotation Sα until it enters the set Bc, complementary of B at some time. Then it
+stays in this set, until it reaches the roof and comes down to the basis. Then the dynamic
+is that of the rotation, until again Sj
+αx falls in the set ϕ > 1 and so on.
+Let Wn(x) be the first visit of Sjx in Jn. By lemma 2.9, we have Wn(x) ≤ qλn.
+Now we choose f to get a transient counterexample and a recurrent one.
+Transient counterexample.
+We take f = 1 on the basis and 0 outside.
+The sequence (λn) is taken such that
+qλn ≥ n (qλn−1)2, n ≥ 1.
+(40)
+By (38), we obtain (recall that now TB, the induced map in the basis B, is the rotation
+S = Sα and R(T j
+Bx) = ϕ(Sj
+αx)):
+MRWn(x)(x)−1(T, x)
+=
+max
+j≤Wn(x) ϕ(Sjx),
+(41)
+VRWn(x)(x)−1(T, x)
+=
+�
+j≤Wn(x)
+ϕ2(Sjx).
+(42)
+In the above formula, ϕ(Sjx) is either 1 or (for some k ≤ n−1) 1+⌊εkqλk⌋ ≤ 1+εn−1qλn−1,
+excepted for the last term which is 1 + ⌊εnqλn⌋.
+
+26
+GUY COHEN AND JEAN-PIERRE CONZE
+The maximum in (41) (given by the first visit to Jn) is 1 + ⌊εnqλn⌋ ≥ εnqλn. As we have
+seen, this first visit for the iterates Sjx occurs at a time ≤ qλn. It follows by (40):
+VRWn(x)(x)−1(T, x)
+M2
+RWn(x)(x)−1(T, x)
+≤
+qλn
+(εn−1 qλn−1)2
+(εn qλn − 1)2 + 1 ≤ (εn−1
+εn
+)2 (qλn−1)2
+qλn
+1
+(1 − (εn qλn)−1)2 + 1
+≤
+2 (
+n
+n − 1)2 (qλn−1)2
+qλn
++ 1 ≤ 4
+n + 1, for n big enough.
+This shows: lim sup
+n
+M2
+n(T, f, x)
+Vn(T, f, x) = 1. The result is proved for x in the basis B, but is
+satisfied for a.e. x ∈ ˜X, since lim sup
+n
+M2
+n(T, f, x)
+Vn(T, f, x) is a.e. constant by ergodicity of the
+special flow and Lemma 2.5.
+Remark that Skf(x) → +∞ for every point x.The sequence (Nn(x, 0)) is bounded for
+every x, but not uniformly in x.
+Recurrent counterexample.
+In order to obtain a recurrent counterexample, we now use a special cocycle over a rotation
+by α (with α bpq) studied later (see Subsection 3.3).
+Let f defined on the basis by f(x) = 1[0, 1
+2[(x) − 1[ 1
+2,1[(x) and 0 outside, and Skf(x) =
+�k−1
+i=0 f(x + iα mod 1). By (37), we have
+VRn−1(x)(T, f, x)
+=
+�
+ℓ
+[
+n−1
+�
+k=0
+ϕ(x + kα) 1Skf(x)=ℓ]2
+=
+�
+ℓ
+[
+n−1
+�
+k=0
+(1 +
+�
+j
+εjqλj1Jj(x + kα)) 1Skf(x)=ℓ]2.
+Observe that for a constant C, 1 + �
+j<n εjqλj1Jj(x + kα)) ≤ Cqλn−1. Using the bounds
+for the special function f and α bpq, this implies:
+VRWn−1(x)(T, f, x)
+≤
+�
+ℓ
+[
+qλn
+�
+k=0
+(1 +
+�
+j<n
+εjqλj1Jj(x + kα)) 1Skf(x)=ℓ]2
+≤
+C2 �
+ℓ
+[
+qλn
+�
+k=0
+qλn−1 1Skf(x)=ℓ]2
+≤
+C2q2
+λn−1
+�
+ℓ
+[
+qλn
+�
+k=0
+1Skf(x)=ℓ]2 ≤ C2q2
+λn−1 q2
+λn/
+�
+log qλn.
+Put Ln = SWn(x)f(x) for the site visited by the cocycle when Sjx enters Jn. We have
+MRWn(x)(T, f, x) ≥ NRWn(x)(T, f, x, Ln(x)) = εnqλn.
+
+EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS
+27
+Hence:
+0 ≤
+VRWn(x)(T, f, x)
+M2
+RWn(x)(T, f, x) − 1
+≤
+C2 q2
+λn−1 q2
+λn
+�
+log qλn
+1
+ε2nq2
+λn
+= C2 n4q2
+λn−1
+�
+log qλn
+.
+Now, we choose a growth condition on (λn) stronger than (40), such that the above bound
+tends to 0.
+This shows the result for x in the basis, hence on the whole space using again Lemma 2.5.
+3. Examples
+In general, for a dynamical system (X, µ, T) and a cocycle (T, f), it seems difficult to get
+a precise estimate of the quantities Nn(x, ℓ), Mn(x), Vn(x). In this section we present two
+types of cocycles for which this is possible, first in the case of strong stochastic properties,
+in particular for the classical case of random walks, then when they are generated by step
+functions over rotations.
+3.1. Random walks.
+1-dimensional cocycle satisfying the LIL.
+We start be a remark on the the law of iterated logarithm (LIL). Suppose that (T, f) is
+a 1-dimensional cocycle which satisfies the LIL. Then for a constant c1 > 0, for a.e. x,
+the inequality |fn(x)| > c1 (n ln ln n)
+1
+2 is satisfied only for finitely many values of n. This
+implies that, for a.e. x, there is N(x) such that |fn(x)| ≤ (c1 n ln ln n)
+1
+2, for n ≥ N(x);
+so that, for N(x) ≤ k < n, |fk(x)| ≤ (c1 k ln ln k)
+1
+2 ≤ (c1 n ln ln n)
+1
+2.
+Therefore we have Card(Rn(x)) ≤ c2(x) (n ln ln n)
+1
+2, with an a.e. finite constant c2(x).
+In dimension 1, by (25), we get that for a.e. x there is c(x) > 0 such that
+Vn(x) ≥ C(x) n
+3
+2 (ln ln n)− 1
+2.
+The case where a LIL is valid includes the case of a 1-dimensional r.w. centered with
+finite variance, but also the class of cocycles for which a martingale method can be used.
+Random walks.
+Now we consider sequences given by a random walk. For random walks in Zd, the quan-
+tities Vn(x), Mn(x) have been studied in many papers since the 50’s. Mn(x) is called
+“maximal multiplicity of points on a random walk” by Erd¨os and Taylor [16]. Below, we
+give a brief survey of several results for r.w.s. First we recall some definitions.
+Let (ζi)i≥0 be a sequence of i.i.d. random vectors on a probability space (X, µ) with
+values in Zd and common probability distribution ν. The associated random walk (r.w.)
+Z = (Zn) in Zd starting from 0 is defined by Z0 := 0,
+Zn := ζ0 + ... + ζn−1, n ≥ 1.
+
+28
+GUY COHEN AND JEAN-PIERRE CONZE
+A r.w. can be seen as a special case of cocycle. Indeed, the r.v.’s ζi can be viewed as the
+coordinate maps on (X, µ) obtained as (Zd)Z equipped with the product measure ν⊗Z
+and with the shift T acting on the coordinates. We have ζi = ζ0 ◦ T i and the cocycle
+relation Zn+n′ = Zn + Zn′ ◦ T n, ∀n, n′ ≥ 0.
+Let S := {ℓ ∈ Zd : P(ζ0 = ℓ) > 0} be the support of ν and L the sub-lattice of Zd
+generated by S. Let D be the sub-lattice of Zd generated by {ℓ − ℓ′, ℓ, ℓ′ ∈ S}.
+For simplicity (and without loss of generality) in what follows we will assume that the
+random walk Z is aperiodic (L = Zd). We exclude also the “deterministic” case (i.e.,
+when P(ζ0 = ℓ) = 1 for some ℓ ∈ Zd) in dimension 1 (the deterministic case in higher
+dimension is excluded by aperiodicity).
+Notice that all the pointwise limits or bounds mentioned now for random walks are
+a.s.
+statements.
+These bounds will show that conditions (22), (21) are satisfied by
+Vn(x), Mn(x) a.s. for on random walks under mild assumptions.
+Recurrence/transience.
+Recall that a r.w. Z = (Zn) is recurrent if �∞
+n=1 µ(Zn = 0) = +∞ and otherwise transient.
+Recurrence occurs if and only if µ(Zn = 0 infinitely often) = 1, and transience if and only
+if µ(Zn = 0 infinitely often) = 0 (cf. [8], [9]).
+For an aperiodic r.w. Z in dimension d with a moment of order 2 (for d = 1, a moment
+of order 1 suffices), for d = 1, 2, Z is recurrent if and only if it is centered. For d ≥ 3, it
+is always transient.
+Variance.
+Let (Xℓ, ℓ ∈ Zd) be a stationary centered r.f. with summable correlation and spectral
+density ρ. We have
+1
+n∥
+n−1
+�
+k=1
+XZk(x)∥2
+2 =
+�
+Td
+1
+n|
+n−1
+�
+k=0
+e2πi⟨Zk(x),t⟩|2 dt =
+�
+Td
+1
+nKn(x, t) ρ(t) dt,
+where, using (17) with zk = Zk(x) and Zk(x) − Zj(x) = Zk(T jx), 1
+nKn reads
+1
+nKn(x, t)
+= 1 + 2
+�
+ℓ
+�n−1
+�
+k=1
+1
+n
+n−k−1
+�
+j=0
+1Zk(T jx)=ℓ
+�
+e2πi⟨ℓ,t⟩.
+(43)
+As already recalled, the existence of the asymptotic variance limn Vn(x)−1 �
+| �n−1
+k=0 XZk(x)|2dµ
+has been shown in [11] and the positivity of the limit has been discussed.
+The asymptotic variance may be zero in case a coboundary condition is satisfied. An
+interesting situation is that of the sums along a transient (non deterministic) r.w., where
+the asymptotic variance is always > 0. Below we will recall briefly a proof.
+
+EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS
+29
+Transient case
+For a transient random walk we use the following general result (Lemma 3.14 in [11]):
+Lemma 3.1. If (X, µ, T) is an ergodic dynamical system and (ϕk)k≥1 a sequence of func-
+tions in L1(X, µ) such that �
+k≥1 ∥ϕk∥1 < ∞, then
+lim
+n
+1
+n
+n−1
+�
+k=1
+n−k−1
+�
+j=0
+ϕk(T jx) =
+∞
+�
+k=1
+�
+ϕk dµ, for a.e. x.
+(44)
+Therefore, for a transient random walk, we obtain for µ-a.e. x:
+lim
+n
+Vn(x)
+n
+= 1 + 2 lim
+n
+n−1
+�
+k=1
+1
+n
+n−k−1
+�
+j=0
+1Zk(T jx)=0 = 1 + 2
+∞
+�
+k=1
+µ(Zk = 0) < +∞
+and the normalisation for the variance is by n up to a finite constant factor.
+Variance in the non deterministic transient case.
+Now we recall the proof of the positivity of the asymptotic variance.
+Let Ψ(t) = E[e2πi⟨ζ0,t⟩], t ∈ Td. Observe that Ψ(t) ̸= 1 for t ̸= 0 in Td, when the r.w. is
+aperiodic and |Ψ(t)| < 1, for t ̸∈ Γ1, where Γ1 is the closed subgroup {t ∈ Td : e2πi⟨r,t⟩ =
+1, ∀r ∈ D}. We put, for t ∈ Td\{0} and 0 ≤ λ < 1,
+Φ(t) := 1 − |Ψ(t)|2
+|1 − Ψ(t)|2 = ℜe[1 + Ψ(t)
+1 − Ψ(t)],
+Φλ(t) := 1 − λ2|Ψ(t)|2
+|1 − λΨ(t)|2 = −1 + 2
+∞
+�
+k=0
+λkℜe(Ψ(t)k) = −1 + 2
+∞
+�
+k=0
+λkµ(Zk = ℓ) cos(2π⟨ℓ, t⟩),
+where the last relation follows from ℜe(Ψ(t)k) = ℜe(E[e2πi⟨Zk,t⟩]) = �
+ℓ µ(Zk = ℓ) cos(2π⟨ℓ, t⟩).
+We put Φ(0) = 0.The function Φ is even, non-negative and Φ(t) = 0 only on Γ1, which is
+̸= Td when the r.w. is non deterministic (if the r.w. is deterministic, µ(ζ0 = ℓ) = 1 for
+some ℓ ∈ Zd and this implies |Ψ(t)| ≡ 1, but this case is excluded). Therefore Φ is ̸= 0
+a.e. for the Lebesgue measure on Td.
+Proposition 3.2. (cf. [28]) Let Z = (Zn) be a transient aperiodic random walk in Zd.
+There is a non-negative constant M such that the Fourier coefficients of 1
+nKn converges
+to those of Φ + Mδ0 and lim
+n
+� 1
+nKn ρ dt > 0.
+Proof. We use that, if (Zn) is a transient, for all ℓ ∈ Zd, we have �∞
+k=1 µ(Zk = ℓ) < +∞.
+Therefore, the series I(ℓ) := −1ℓ=0 + �∞
+k=0 [µ(Zk = ℓ) + µ(Zk = −ℓ)] converges and by
+(43) and Lemma 3.1, the even functions 1
+nKn(x, .) satisfy:
+�
+Td
+1
+nKn(x, .) cos 2π⟨ℓ, .⟩ dt
+= −1ℓ=0 +
+n−1
+�
+k=0
+1
+n
+n−k−1
+�
+j=0
+[1Zk(T jx)=ℓ + 1Zk(T jx)=−ℓ] →
+n→∞ I(ℓ).
+
+30
+GUY COHEN AND JEAN-PIERRE CONZE
+Note that above the sum over k is written starting from 0. By letting n tend to infinity
+in the relation
+−1ℓ=0 +
+∞
+�
+k=0
+λk[µ(Zk = ℓ) + µ(Zk = −ℓ)]
+=
+�
+Td cos 2π⟨ℓ, .⟩ [−1 + 2ℜe(
+1
+1 − λΨ(.))] dt =
+�
+Td cos 2π⟨ℓ, t⟩ Φλ(.) dt,
+we get since the left sum tends to I(ℓ):
+I(ℓ) = lim
+λ↑1
+�
+Td cos 2π⟨ℓ, t⟩ Φλ(t) dt.
+Taking ℓ = 0 in the previous formula, it follows from Fatou’s lemma:
+I(0) = 1 + 2
+∞
+�
+k=1
+µ(Zk = 0) = lim
+λ↑1
+�
+Td Φλ(t) dt ≥
+�
+Td lim
+λ↑1 Φλ(t) dt =
+�
+Td Φ(t) dt.
+This shows the integrability of Φ on Td and we can write with a constant M ≥ 0
+I(0) = lim
+λ↑1
+�
+Td Φλ(t) dt =
+�
+Td lim
+λ↑1 Φλ(t) dt + M =
+�
+Td Φ(t) dt + M.
+Let Uη be the ball of radius η > 0 centered at 0. By aperiodicity of the r.w., Ψ(t) ̸= 1 for
+t in Uc
+η, the complementary in Td of Uη, This implies supt∈Ucη supλ<1 Φλ(t) < +∞.
+Therefore, we get: lim
+λ↑1
+�
+Ucη
+cos 2π⟨ℓ, t⟩ Φλ(t) dt =
+�
+Ucη
+cos 2π⟨ℓ, t⟩ Φ(t) dt, hence:
+I(ℓ) =
+�
+Ucη
+cos 2π⟨ℓ, t⟩ Φ(t) dt + lim
+λ↑1
+�
+Uη
+cos 2π⟨ℓ, t⟩ Φλ(t) dt, ∀η > 0,
+which can be be written:
+−
+�
+Uη
+cos 2π⟨ℓ, .⟩ Φ dt = I(ℓ) −
+�
+Td cos 2π⟨ℓ, .⟩ Φ dt − lim
+λ↑1
+�
+Uη
+cos 2π⟨ℓ, .⟩ Φλ dt.
+(45)
+Let ε > 0. By positivity of Φλ, we have, for η(ε) small enough:
+(1 − ε)
+�
+Uη(ε)
+Φλ dt ≤
+�
+Uη(ε)
+cos 2π⟨ℓ, .⟩ Φλ dt ≤ (1 + ε)
+�
+Uη(ε)
+Φλ dt;
+By subtracting
+�
+Uη(ε) cos 2π⟨ℓ, t⟩ Φ(t) dt in the previous inequalities and (45), we get:
+(1 − ε)
+�
+Uη(ε)
+Φλ dt −
+�
+Uη(ε)
+cos 2π⟨ℓ, .⟩ Φ dt
+≤ I(ℓ) −
+�
+Td cos 2π⟨ℓ, .⟩ Φ dt − lim
+λ↑1
+�
+Uη(ε)
+cos 2π⟨ℓ, .⟩ Φλ dt +
+�
+Uη(ε)
+cos 2π⟨ℓ, .⟩ Φλ dt
+≤ (1 + ε)
+�
+Uη(ε)
+Φλ dt −
+�
+Uη(ε)
+cos 2π⟨ℓ, .⟩ Φ dt;
+
+EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS
+31
+As we can chose λ such that
+| − lim
+λ↑1
+�
+Uη(ε)
+cos 2π⟨ℓ, .⟩ Φλ dt +
+�
+Uη(ε)
+cos 2π⟨ℓ, .⟩ Φλ dt| ≤ ε,
+we obtain:
+−ε + (1 − ε)
+�
+Uη(ε)
+Φλ dt −
+�
+Uη(ε)
+cos 2π⟨ℓ, .⟩ Φ dt
+≤ I(ℓ) −
+�
+Td cos 2π⟨ℓ, .⟩ Φ dt ≤ ε + (1 + ε)
+�
+Uη(ε)
+Φλ dt −
+�
+Uη(ε)
+cos 2π⟨ℓ, .⟩ Φ dt
+For ε small enough,
+�
+Uη(ε) cos 2π⟨ℓ, .⟩ Φ dt can be made arbitrary small, as well as ε supλ<1
+�
+Uη Φλ dt,
+since Φ is integrable and supλ<1
+�
+Td Φλ dt < ∞.
+This shows that I(ℓ) −
+�
+Td cos 2π⟨ℓ, .⟩ Φ dt −
+�
+Uη(ε) Φλ dt can be made arbitrarily small for
+ε > 0 small and λ close to 1. The same is true for ℓ = 0 and also for the difference
+[I(ℓ) −
+�
+Td cos 2π⟨ℓ, .⟩ Φ dt −
+�
+Uη(ε) Φλ dt] − [I(0) −
+�
+Td Φ dt −
+�
+Uη(ε) Φλ dt]
+= [I(ℓ) −
+�
+Td cos 2π⟨ℓ, .⟩ Φ dt] − [I(0) −
+�
+Td Φ dt] = [I(ℓ) −
+�
+Td cos 2π⟨ℓ, .⟩ Φ dt] − M].
+Therefore I(ℓ) =
+�
+Td cos 2π⟨ℓ, t⟩ Φ(t) dt + M for all ℓ and the Fourier coefficients of 1
+nKn
+converges to those of Φ + Mδ0.
+As the non-negative sequence ( 1
+nKn) is bounded in L1-norm and the density ρ is contin-
+uous, this proves
+� 1
+nKnρ dt →
+�
+Φρ dt + Mρ(0). Moreover, the limit is > 0 since both
+Φ and ρ are not 0 a.e.
+It is shown in [28] that M = 0 for d > 1.
+□
+Behaviour of Mn(x).
+In the transient case (d ≥ 3) (at least for a simple r.w.), Erd¨os and Taylor (1960) proved
+that for a constant γ > 0 depending on the dimension,
+lim
+n
+Mn(x)
+log n = γ.
+Recurrent case
+In dimension 1, H. Kesten has shown that lim sup
+n
+Mn
+√
+n ln ln n
+=
+√
+2/σ.
+Therefore in
+dimension 1, we have the following lower and upper bounds for Vn:
+C1(x) n
+3
+2 (ln ln n)− 1
+2 ≤ Vn(x) ≤ C2(x) n
+3
+2(ln ln n)
+1
+2.
+Dimension d = 2.
+There is a deterministic rate (law of large numbers): for a constant C0.
+�
+Vn dµ
+n log n → C0 and Vn(x)
+n log n → C0, for a.e. x.
+
+32
+GUY COHEN AND JEAN-PIERRE CONZE
+For a planar simple random walk, Erd¨os and Taylor [16] have shown:
+lim sup
+n
+Mn(x)
+(log n)2 ≤ 1
+π.
+(46)
+The result has been extended by Dembo, Peres, Rosen and Zeitouni [15], who proved for
+an aperiodic centered random walk on Z2 with moments of all orders:
+lim
+n
+Mn(x)
+(log n)2 =
+1
+2π det(Γ)
+1
+2 ,
+where Γ is the covariance matrix associated to the random walk.
+As shown in the proof in [15], it suffices to suppose that the 2-dimensional r.w. is aperiodic.
+Moreover, the proof for the upper bound is based on the local limit theorem which uses
+only the existence of the moment of order 2. Therefore, assuming the existence of the
+moment of order 2, the upper bound (46) holds.
+It follows in this case: there exist C(x) a.e finite such that:
+M2
+n(x)
+Vn(x) ≤ C(x)(log n)3
+n
+.
+3.2. Extensions of the r.w. case.
+1) Consequence of the Local Limit Theorem (LLT).
+The Local Limit Theorem, when it is satisfied by the cocycle (T, f), gives some pointwise
+information on Vn(x). For example, if d = 2, the following lemma holds:
+Lemma 3.3. Suppose that the LLT holds and d = 2. Then, for every ε > 0, there is an
+integrable function C (depending on ε), such that:
+Nn(x, 0) ≤ C(x)(ln n)2+ε, Vn(x) ≤ C(x) n (log n)2+ε.
+(47)
+Proof. By the LLT, it holds, for n ≥ 1,
+�
+Nn(., 0)dµ =
+n
+�
+k=1
+�
+1fk=0 dµ ≤ C
+n
+�
+k=1
+1
+k ≤ C ln n.
+Let ε be a positive constant. Putting Γ(x) = �∞
+n=1 n−(2+ε)N2n(x, 0), we have:
+�
+Γ(x) dµ(x) ≤ C
+∞
+�
+n=1
+n−(2+ε)n = C
+∞
+�
+n=1
+n−(1+ε) < +∞,
+so that N2n(x, 0) ≤ Γ(x) n2+ε, where Γ is integrable.
+If 2kn ≤ n < 2(kn+1), then with p = 2 + ε, we have for n big enough,
+Nn(x, 0)
+(log2 n)p ≤ N2(kn+1)(x, 0)
+kp
+n
+≤ (kn + 1)p
+kp
+n
+N2(kn+1)(x, 0)
+(kn + 1)p
+= (1 + 1/kn)p Γ(x) ≤ 2Γ(x).
+
+EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS
+33
+For Vn(x), by (27) we have:
+�
+Vn(x) dµ(x) = 2
+n−1
+�
+k=1
+�
+Nn−k(x, 0) dµ(x) + n = O(n log n).
+As above for Nn(x, 0), the pointwise bound (47) follows.
+□
+Among example of cocycles satisfying a LLT, there are the r.w.’s (but with more precise
+results as recalled above), but also cocycles generated by functions with values in Zd
+depending on a finite number of coordinates over a sub-shift of finite type endowed with
+a Gibbs measure ([19]), ([18]).
+2) Functions depending on a finite number of coordinates on a Bernoulli scheme.
+Now we try to bound Mn(x) in situation which extends slightly that of random walks.
+Suppose that (X, µ, T) is a Bernoulli scheme with X = IN, where I is a finite set. Let
+f : x → f(x1, ..., xr) be a centered function from X to Zd, d ≥ 1, depending on a finite
+number of coordinates.
+Let us consider the generalized random walk (Zn) defined by the sequence of ergodic sums
+Zn(x) = fn(x) = X0(x) + ... + Xn−1(x), where Xk(x) = f(T kx).
+Lemma 3.4. For all m ≥ 1 and for constants Cm, C′
+m independent of ℓ, we have:
+�
+Nm
+n (., ℓ) dµ =
+�
+[
+n
+�
+k=1
+1fk=ℓ]m dµ
+≤ Cmnm/2, for d = 1,
+≤ C′
+m(Logn)m, for d = 2.
+Proof. We bound the sum �
+1≤k1<k2<...<km≤n µ(fk1 = ℓ, fk2 = ℓ, ..., fkm = ℓ).
+For r ≤ k1 < k2 < ... < km ≤ n, writing Xk instead of T kf, we have:
+µ(X0 + ... + Xk1−1 = ℓ, Xk1 + ... + Xk2−1 = 0, ..., Xkm−1 + ... + Xkm−1 = 0)
+=
+�
+a1,1,...,ar,1,...,a1,m,...,ar,m∈S
+µ[
+X0 + ... + Xk1−r = ℓ − (a1,1 + ... + ar,1), Xk1−r+1 = a1,1,..., Xk1−1 = ar,1,
+Xk1 + ... + Xk2−r = −(a1,2 + ... + ar,2), Xk2−r+1 = a1,2, ..., Xk2−1 = ar,2, ...
+Xkm−1 + ... + Xkm−r = −(a1,m + ... + ar,m), Xkm−r+1 = a1,m, ..., Xkm−1 = ar,m];
+which is less than:
+�
+a1,1,...,ar,1,...,a1,m,...,ar,m∈S
+µ[X0 + ... + Xk1−r = ℓ − (a1,1 + ... + ar,1),
+Xk1 + ... + Xk2−r = −(a1,2 + ... + ar,2), ..., Xkm−1 + ... + Xkm−r = −(a1,m + ... + ar,m)].
+
+34
+GUY COHEN AND JEAN-PIERRE CONZE
+Now inside [.] the events are independent. By independence and stationarity, the preceding
+sum is
+�
+a1,1,...,ar,1,...,a1,m,...,ar,m∈S
+µ[X0 + ... + Xk1−r = ℓ − (a1,1 + ... + ar,1)]
+µ[X0 + ... + Xk2−k1−r = −(a1,2 + ... + ar,2)] ...
+µ[ X0 + ... + Xkm−km−1−r = −(a1,m + ... + ar,m)].
+With τn(k) = supℓ µ(X0 + ... + Xk−1 = ℓ) 1[0,n](k), we get the following bound
+�
+1≤k1<k2<...<km≤n
+µ(fk1 = ℓ, fk2 = ℓ, ..., fkm = ℓ)
+=
+�
+r≤k1<k2<...<km≤n
+µ(X1 + ... + Xk1−1 = ℓ, Xk1 + ... + Xk2−1 = 0, ..., Xkm−1 + ... + Xkm−1 = 0)
+≤ srm
+�
+r≤k1<k2<...<km≤n
+τn(k1 − r) τn(k2 − k1 − r) ... τn(km − km−1 − r)
+≤ Csrm �
+k
+(τn ∗ τn ∗ ... ∗ τn)(k) ≤ Csrm(
+�
+k
+τn(k))m.
+We have srm terms for the first sum, where the cardinal |S| is denoted by s,.
+Now we can use, as for the usual r.w., convolution and the local limit theorem.
+□
+From the lemma, it follows easily in the recurrent case that for a.e. x, for all ε > 0,
+if d = 1, Mn(x) = o(n
+1
+2 +ε) and, if d = 2, Mn(x) = o(nε).
+In the transient case, if there is a moment of order η for some η > 0, then Mn(x) = o(nε)
+for all ε > 0. For these estimates, in both cases, see [11].
+A question if to extend the previous results to a larger class of functions depending weakly
+on the far coordinates. For such an extension is that explicit bounds in the LLT are not
+always available.
+3.3. Step functions over rotations.
+Now we take X = Tr, r ≥ 1 endowed with µ, the uniform measure and we consider
+cocycles over rotations. When they are centered, such cocycles are strongly recurrent and
+therefore the associated quantities Vn and Mn are big. The difficult part is to bound them
+from above. We will give an example where an upper bound can be obtained.
+Let Tα be the rotation by an irrational α. For f : X → Zd, recall that the cylinder map
+(cf. Subsection 2.1) is ˜Tf,α = ˜Tα : X ×Zd → X ×Zd defined by ˜Tα(x, ℓ) = (x+α, ℓ+f(x)).
+Non centered step cocycles over a rotation.
+Let f be a non centered function with a finite number of values values in Zd. Suppose
+that f is Riemann integrable, which amounts to assume that, for the uniform measure of
+the torus, the measure of the set of discontinuity points of f is zero.
+
+EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS
+35
+Then by a remark in Subsection 2.2, Mn(x) is bounded uniformly in x and n. Therefore,
+for Vn(x), the bounds n ≤ Vn(x) ≤ Cn are satisfied.
+Centered step cocycles over a 1-dimensional rotation.
+The interesting situation is that of centered functions. We will consider the case r = 1
+and when the irrational number α has bounded partial quotients.
+Recall that an irrational α with continued fraction expansion [0; a1, a2, ..., an, ...] is said
+to have bounded partial quotients (bpq) if supn an < +∞. The set of bpq numbers has
+Lebesgue measure zero and Hausdorff dimension 1.
+In the sequel of this subsection, α will be an irrational bpq number (for instance a qua-
+dratic irrational) and f a centered function with values in Z and bounded variation.
+By Denjoy-Koksma inequality, there is a logarithmic bound for the cocycle (Tα, f): |fn(x)| ≤
+C ln n, for a constant C.
+The cocycle is strongly recurrent to 0 (and this is true for d ≥ 1 if f centered has values
+in Zd, when its components have bounded variation).
+This makes the corresponding
+maximum Mn(x) big. Nevertheless, we will see that condition (18) is satisfied, at least
+for a special example.
+Lower bound.
+Lower bound for Vn and variance, case d = 1.
+For a general sequence (zk), we can obtain a lower bound for Vn by an elementary method
+when there is an upper bound for the variance defined below.
+Lemma 3.5. Defining the mean mn and the variance σ2
+n by
+mn = 1
+n
+n
+�
+k=1
+zk, σ2
+n = 1
+n
+n
+�
+k=1
+(zk − mn)2,
+we have
+Vn ≥ 1
+9
+n2
+σn
+, if σn > 1.
+(48)
+Proof. Suppose that σn > 0. For λ > 1, let ∆λ := [−λσn + mn, λσn + mn] � Z. We have:
+σ2
+n ≥ 1
+n
+n−1
+�
+k=0
+(zk − mn)21zk∈∆c
+λ ≥ 1
+n
+n−1
+�
+k=0
+(1zk∈∆c
+λ) λ2σ2
+n.
+Therefore: �n−1
+k=0 1zk∈∆λ ≥ n(1 − λ−2). As Card(∆λ) ≤ 2λσn + 1. It follows by (24):
+Vn ≥ (1 − λ−2)2
+2λσn + 1 n2.
+For λ = 2 we get: Vn ≥
+9
+16
+4σn+1 n2 ≥
+9
+80
+n2
+σn, if σn > 1; hence (48).
+□
+
+36
+GUY COHEN AND JEAN-PIERRE CONZE
+If zk is given by ergodic sums, i.e., zk = fk(x) , let
+mn(x) := 1
+n
+n
+�
+k=1
+fk(x), σ2
+n(x) = 1
+n
+n
+�
+k=1
+(fk(x) − mn(x))2.
+By [5, Proposition 13], for α bpq and f with bounded vartion, it holds σ2
+n(x) ≤ C ln n.
+Using (48) and Vn(x) ≤ nMn(x), this gives a lower bound for Vn(x) and Mn(x):
+Vn(x) ≥ c
+n2
+√
+ln n
+, Mn(x) ≥ c
+n
+√
+ln n
+.
+(49)
+Below we will get an estimate from above in the following example.
+Example 3.6. f = 1[0, 1
+2 ) − 1[ 1
+2,1) and α bpq.
+Upper bound for the example (3.6).
+For f as above and α bpq, we have by [1], for some constant C1 > 0,
+∥Nn(·, 0)∥∞ = ∥ ˜Sn(1T1×{0})(·, 0)∥∞ ≤
+C1n
+√log n.
+(50)
+Remark that the bound (50) is obtained in [1] as the limit of ∥Nn(·, 0)∥p, the Lp-norm
+of Nn(·, 0), as p goes to ∞. Therefore the bound holds for the norm ∥.∥ess sup, but it can
+be easily replaced by the uniform norm as written above. Indeed, for any x, there is a
+neighborhood V (x) of x, such that for y ∈ V (x), |Nn(x, 0) − Nn(y, 0)| ≤ 1 (at most one
+jump in V (x)). As one can find y ∈ V (x) satisfying Nn(y, 0) ≤
+C1n
+√log n, the same inequality
+holds for x, with C1 replaced by 2C1.
+Using Remark 2.4, it follows:
+Mn(x) ≤ C1
+n
+√log n.
+(51)
+By (51) and since Vn(x) ≤ n Mn(x), we obtain
+Vn(x) ≤ C1
+n2
+√log n.
+(52)
+From (49), (51) and (52), it follows: Vn(x) ≍ n2/√log n and Mn(x) ≍ n/√log n, where
+an ≍ bn for two sequences (an) and (bn) means c an ≤ bn ≤ C an, ∀n ≥ 1, with two positive
+constants c, C.
+Therefore we get in this special example 3.6:
+M2
+n(x)
+Vn(x) ≤ ( C1n
+√log n)2/
+cn2
+√log n = C2
+1
+c
+1
+√log n → 0.
+(53)
+Condition (18) of Theorem 1.9 is satisfied in this example, as well as the condition of
+Theorem 1.6 a), hence a Glivenko-Cantelli theorem along (Snf(x)) for i.i.d. r.v.’s.
+
+EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS
+37
+But the sufficient conditions for the Glivenko-Cantelli theorems 1.5, 1.6 b), 1.8 are not
+satisfied by this cocycle and more generally, in view of the lower bound (49), by a cocycle
+defined by step functions over a bpq irrational rotation.
+4. About limit theorems along ergodic sums
+4.1. Glivenko-Cantelli theorem along ergodic sums.
+The Glivenko-Cantelli theorem recalled in the introduction is a (pointwise) law of large
+numbers uniform over a set of functions (here the indicators of intervals).
+When the
+r.v.’s Xk are i.i.d., the proof is an easy consequence of the strong law of large numbers
+applied to the sequence of i.i.d. bounded r.v.’s (1Xk≤s). Using Birkhoff’s ergodic theorem,
+the Glivenko-Cantelli theorem has been extended to the setting of a strictly stationary
+sequence (Xk) of random variables. More precisely, formulated in terms of dynamical
+systems, the following holds:
+Let (Y, A, ν) be a probability space and S an ergodic measure preserving transformation on
+Y . For any measurable function ϕ : Y → R, let us consider the strictly stationary sequence
+(Xk) defined by Xk = ϕ◦Sk, k ≥ 0. Then the sequence of empirical distribution functions
+satisfies: for ν a.e. y ∈ Y, sups | 1
+n
+�n−1
+k=0 1X(y)k≤s − F(s)| → 0, where F(s) = ν(ϕ ≤ s).
+Observe that the result is an application of Birkhoff’s theorem and Lemma 1.1 recalled
+in Section 1. Its extension to the non ergodic case has been formulated by Tucker [29],
+the distribution function F(s) being replaced by the conditional distribution function
+E(1ϕ≤s|J ), where J is the σ-algebra of S-invariant sets. In others words, we have:
+for ν a.e. y ∈ Y, lim
+n→∞ sup
+s | 1
+n
+n−1
+�
+k=0
+1ϕ(Sky)≤s − E(1ϕ≤s|J )(y)| = 0.
+The above formula relies on the ergodic decomposition which can be used in the proof.
+In the previous framework, for a process, a Glivenko-Cantelli like theorem sampled along
+a sequence generated by a dynamical system can be obtained as follows:
+As in Subsection 2, let T be an ergodic measure preserving transformation on a probability
+space (X, B, µ) and f a measurable function on X with values in Zd, d ≥ 1.
+Let us take a second system (Ω, P, θ), where θ = (θℓ)ℓ∈Zd is a Zd-action preserving P.
+The skew product associated to the cocycle (T, f) and θ is the map: Tθ,f : (x, ω) →
+(Tx, θf(x)ω) from X × Ω to itself. By iteration we get:
+T k
+θ,f(x, ω) = (T kx, θfk(x)ω).
+For example, as Zd-action, we can take a Zd-Bernoulli shift (Ω, P, (θℓ)ℓ∈Zd), with P a
+product measure and θ the shift on the coordinates. If X0 is the first coordinate map,
+then (Xℓ) = (X0 ◦ θℓ) is a family of i.i.d. r.v.’s indexed by Zd.
+
+38
+GUY COHEN AND JEAN-PIERRE CONZE
+In general, let Iθ,f denote the conditional expectation with respect to the σ-algebra of
+Tθ,f-invariant sets. The ergodic theorem for Tθ,f shows that, for ψ ∈ L1(µ × P),
+lim
+n
+1
+n
+n−1
+�
+k=0
+ψ(T kx, θfk(x)ω) = Iθ,f(ψ)(x, ω), for µ × P-a.e.(x, ω).
+(54)
+If ϕ is a measurable function on Ω, putting ψs(x, ω) = 1Is(ϕ(ω)), where Is is the half-line
+] − ∞, s], we have
+ψs(T k
+θ,f(x, ω)) = 1Is(ϕ(θfk(x)ω)).
+By the quoted Tucker’s result, the convergence in (54) for each ψs, s ∈ R, can be strength-
+ened into a uniform convergence with respect to s:
+for µ × P-a.e (x, ω), 1
+n sups | �n−1
+k=0 1Is(ϕ(θfk(x)ω)) − I(ψs)(x, ω)| → 0.
+Therefore, by the Fubini theorem, there is a “sampled” version of the Glivenko-Cantelli
+theorem for the empirical process of a stationary sequence:
+Proposition 4.1. For µ-a.e x, we have
+| sups
+1
+n
+�n−1
+k=0 1Is(ϕ(θfk(x)ω)) − I(ψs)(x, ω)| → 0, for P-a.e ω.
+When Tθ,f is ergodic, if ψ ∈ L1(µ × P), we have Iθ,f(ψ)(x, ω) =
+�
+ψ dµ dP, for µ ×
+P-a.e. (x, ω), and the centering I(ψs)(x, ω) is given by the distribution function F(s) =
+µ(ϕ ≤ s). In this case, for a.e. x, a Glivenko-Cantelli theorem with the usual centering
+holds for the empirical process sampled along the sequence (zn) given by zn = Snf(x)
+(with a set of ω’s of P-measure 1 depending on x).
+The lemma below shows, as it is known, that ergodicity of the cylinder map ˜Tf implies
+ergodicity of the skew map Tθ,f. Let us sketch a proof.
+Lemma 4.2. Suppose that the cocycle (T, f) is recurrent and the map ˜Tf ergodic. If the
+action of Zd by θ on (Ω, P) is ergodic, then Tθ,f is ergodic on (X × Ω, µ × P).
+Proof. : Let Φ be a Tθ,f invariant measurable function on X × Ω:
+Φ(Tx, θf(x)ω) = Φ(x, ω), for a.e. (x, ω).
+For a.e. x, there is a set Ω0
+x of full P-measure in Ω such that Φ(Tx, θf(x)ω) = Φ(x, ω), for
+all ω ∈ Ω0
+x. As Zd is countable, for a.e. x, there is a set Ωx of full measure such that
+Φ(Tx, θf(x)θℓω) = Φ(x, θℓω), for all ω ∈ Ωx.
+Let ω ∈ Ωx. The function ϕω(x, ℓ) := Φ(x, θℓω) on X × Zd is measurable, ˜Tf-invariant:
+ϕω( ˜Tf(x, ℓ))
+=
+ϕω(Tx, ℓ + f(x)) = Φ(Tx, θℓ+f(x)ω)
+=
+Φ(Tx, θf(x)θℓω) = Φ(x, θℓω) = ϕω(x, ℓ).
+It follows from the ergodicity of ˜Tf that there is a constant cω such that ϕω(x, ℓ) = cω for
+a.e. x. Therefore Φ coincides a.e. with a function ψ on Ω which is θ-invariant, hence a
+constant by the assumption of ergodicity of the action of Zd on Ω.
+□
+
+EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS
+39
+With Fubini’s argument, we get a Glivenko-Cantelli theorem for a.e. x, if we can show
+that the skew map Tθ,f is ergodic.
+There are many examples cylinder flows ˜Tf which are shown to be ergodic in the literature
+and so providing examples via Lemma 4.2. For instance, we can take for T an irrational
+rotation and f = 1[0, 1
+2) − 1[ 1
+2,1). The cocycle (T, f) is ergodic and the above version of
+Glivenko-Cantelli theorem applies for any stationary sequence (Xk) (with a conditional
+distribution if the stationary sequence is not ergodic). See also examples for which the
+skew map is ergodic in [25].
+4.2. Discussion: universal sequences.
+The weakness in the approach of the previous subsection for a sampled Glivenko-Cantelli
+theorem along ergodic sums (Skf(x), k ≥ 0) is that it yields a set of x’s of µ-measure
+1 depending on the dynamical system (Ω, P, θ) and on ϕ. One can try to reinforce the
+statement by introducing a notion of “universal property”.
+In this direction, the LLN for sums sampled along ergodic sums is closely related in the
+following way to the random ergodic theorems which have been studied in several papers.
+First, let us call “universally good” a sequence (zk) such that, for every dynamical system
+(Ω, P, θ), for every ϕ ∈ L1(P), the sequence 1
+n
+�n−1
+k=0 ϕ ◦ θzk converges P-a.e.
+We say that (T, f) a “(pointwise) good averaging cocycle” (or a universally representative
+sampling scheme) if, for µ-a.e. x, the sequence (Skf(x)) is universally good, i.e., for every
+dynamical system (Ω, P, θ), for every ϕ ∈ L1(P), 1
+n
+�n−1
+k=0 ϕ ◦ θSkf(x) converges P-a.e.
+The definition of a “mean good averaging cocycle” is similar, changing the above conver-
+gence into convergence in L2(P)-norm, for every ϕ in L2(P).
+A question which has been studied is to find mean or pointwise good averaging cocycles. In
+the first direction, examples and counterexamples of mean good averaging 1-dimensional
+cocycles are studied in [25],
+For pointwise convergence, there are 1-dimensional examples given by cocycles with a
+drift. In [23], the following result is shown: the cocycle defined by a random walk with a
+moment of order 2 is a pointwise good averaging cocycle if and only if it is not centered.
+Moreover it is shown that any ergodic integrable integer-valued stochastic process with
+nonzero mean is universally representative for bounded stationary processes. The proofs
+are based on the recurrence time theorem ([6]).
+Notice that a related, but different, notion can be introduced by restricting the dynamical
+system (Ω, P, θ) to belong to a given class C of dynamical systems.
+Let us call “pointwise good for a class C of dynamical systems”, a sequence (zk) such that,
+for every dynamical system (Ω, P, θ) in the class C, for every ϕ ∈ L1(P), limn
+1
+n
+�n−1
+k=0 ϕ ◦
+θzk =
+�
+ϕ dP, P-a.e. There is a similar property for the mean convergence.
+
+40
+GUY COHEN AND JEAN-PIERRE CONZE
+This can be also expressed for a class of random fields satisfying a condition on the decay
+of correlations.
+For example, by Remark 2.8, every cocycle with values in Zd which is not a coboundary is
+a mean good averaging cocycle for the stationary r.f.s on Zd such that �
+ℓ |⟨Uℓ, U0⟩| < +∞.
+If (zk) is pointwise universally good for a class C, clearly we get the Glivenko-Cantelli
+property for any dynamical system (Ω, P, θ) in C and every measurable function ϕ, i.e.:
+sups | 1
+n
+�n−1
+k=0 1Is(ϕ(θzkω)) − P(ϕ ≤ s)| → 0, for P-a.e ω.
+(55)
+As we see, there are two different approaches of the notion of universal sequences for a
+law of large numbers: either we ask for a LLN along such a sequence for every dynamical
+system (Ω, P, θ) and all functions in L1(P) or we fix a class of dynamical systems, or a
+class of functions in L1(P). In the latter case, the condition on the sequence (zk) may be
+expressed in a quantitative way. Let us give a known example and recall the proof.
+Proposition 4.3. Let (zk) be a strictly increasing sequence of positive integers. If the
+sequence satisfies: for a finite constant C, zk ≤ Ck, ∀k ≥ 1, then (zk) is a pointwise good
+averaging sequence for the class C of dynamical systems (Ω, P, θ) with Lebesgue spectrum.
+Proof. There is a dense set of functions ϕ ∈ L1(P) such that
+1
+n
+n−1
+�
+k=0
+ϕ(θzkω) converges P-a.e.
+(56)
+Indeed, by the SLLN for orthogonal random variables, (56) is satisfied by ϕ ∈ L2(P) such
+that ⟨ϕ, ϕ ◦ θk⟩ = 0, ∀k. The Lebesgue spectrum property implies that such functions
+span a dense linear space in L2(P), hence in L1(P).
+Moreover, the space of functions ϕ such that (56) holds is closed by the ergodic maximal
+lemma in view of the assumption on (zk). Therefore (56) is satisfied by every ϕ ∈ L1(P).
+□
+To finish, we recall the following example which shows that the behaviour may depend
+on the properties of the dynamical system (Ω, P, θ) (cf. [13]):
+Let (Ω, F, P) be the interval [0, 1] endowed with the Borel σ-algebra and the Lebesgue
+measure and take f = 1[0, 1
+2]. Denote by T the class of invertible measure preserving
+transformations on this space. It can be shown that there are increasing sequences (zk)
+of positive integers satisfying the conditions of the previous proposition such that, for a
+dense Gδ of elements in T with continuous spectrum, the ergodic means of f along (zk)
+do not converge P-a.e.
+References
+[1]
+Aaronson, J., Bromberg, M. and Nakada, H.: Discrepancy skew products and affine random
+walks, Israel J. Math. 221 (2017), no. 2, 973-1010.
+
+EMPIRICAL PROCESS SAMPLED ALONG A STATIONARY PROCESS
+41
+[2]
+Ambrose L.: Functional generalizations of Hoeffding’s covariance lemma and a formula for
+Kendall’s tau, Statistics and Probability Letters, 122 (2017), 218-226.
+[3]
+Billingsley P.: Convergence of probability measures, 1rst ed. John Wiley & Sons, Inc., 1968.
+[4]
+Birkel, T.: A note on the strong law of large numbers for positively dependent random variables,
+Statist. Probab. Lett. 7 (1988), no. 1, 17-20.
+[5]
+Borda, B.: On the distribution of Sudler products and Birkhoff sums for the irrational rotation,
+Arkiv 2021.
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+Bourgain J., Furstenberg H., Katznelson Y. and Ornstein D.: Appendix on return-time se-
+quences. Publ. Math. IHES, (69) 42-45, 1989.
+[7]
+Chung, K.L: A course in probability theory, 3rd ed. (2001), Acad. Press, Inc., San Diego, CA.
+[8]
+Chung, K. L., Fuchs, W. H. J. On the distribution of values of sums of random variables, Mem.
+Amer. Math. Soc. 6 (1951), 157-168.
+[9]
+Chung, K. L., Ornstein, D., On the recurrence of sums of random variables, Bull. Amer. Math.
+Soc. 68 (1962), 30-32.
+[10]
+Cohen, G., Conze, J.-P.: On the quenched functional CLT in random sceneries, to appear in
+Studia Matematica.
+[11]
+Cohen, G., Conze, J.-P.: CLT for random walks of commuting endomorphisms on compact
+abelian groups, J. Theoret. Probab. 30 (2017), no. 1, 143-195.
+[12]
+Conze, J. P., Remarques sur les transformations cylindriques et les ´equations fonctionnelles,
+Pub. s´em. math. info. de Rennes, 1976, fasc. 2, http://www.numdam.org/actas/PSMIR/
+[13]
+Conze, J.-P., Convergence des moyennes ergodiques pour des sous-suites, Bull. Soc. Math. Fr.,
+M´emoire 35, (1973), p. 7-15.
+[14]
+Deligiannidis G., Gouezel, S., Kosloff, Z.: Boundary of the range of a random walk and the
+Folner property, Electron. J. Probab. 26: 1-39 (2021). DOI: 10.1214/21-EJP667
+[15]
+Dembo, A.; Peres, Y.; Rosen, J.; Zeitouni, O.: Thick points for planar Brownian motion and
+the Erdos-Taylor conjecture on random walk. Acta Math. 186 (2001), no. 2, 239-270.
+[16]
+Erdos, P.; Taylor, S. J.: Some problems concerning the structure of random walk paths. Acta
+Math. Acad. Sci. Hungar. 11 (1960), 137–162.
+[17]
+Esary, J. D.; Proschan, F.; Walkup, D. W., Association of random variables, with applications,
+Ann. Math. Statist. 38 (1967), 1466-1474.
+[18]
+Gou¨ezel S.: Berry-Esseen theorem and local limit theorem for non uniformly expanding maps,
+Ann. Inst. H. H. Poincar´e PR 41 (2005), 997–1024
+[19]
+Guivarc’h, Y. et Hardy, J.: Th´eor`emes limites pour une classe de chaˆınes de Markov et appli-
+cations aux diff´eomorphismes d’Anosov, Ann. Inst. H. Poincar´e 24 (1) (1988), 73-98.
+[20]
+Hoeffding W.: Probability Inequalities for Sums of Bounded Random Variables, Journal of the
+AMS vol. 58 (1963), no. 301, 13-30.
+[21]
+Hewitt, E.; Ross, K. A.: Abstract harmonic analysis, Vol. II. Die Grundlehren der mathematis-
+chen Wissenschaften, Band 152 Springer-Verlag, New York-Berlin 1970.
+[22]
+Kesten, H.: An iterated logarithm law for local time. Duke Math. J. 32 (1965), 447-456.
+[23]
+Lacey, M.; Petersen, K.; Wierdl, M.; Rudolph, D.: Random ergodic theorems with universally
+representative sequences., Ann. I. H. Poincar´e Prob. Stat. 30 (1994), no. 3, 353-395.
+[24]
+Lehmann, E. L.: Some concepts of dependence, Ann. Math. Statist. 37 (1963), 1137-1153.
+[25]
+Lema´nczyk, M.; Lesigne, E.; Parreau, F.; Voln´y, D.; Wierdl, M.: Random ergodic theorems and
+real cocycles, Israel J. Math. 130 (2002), 285-321.
+[26]
+Newman, C.M.: Normal fluctuations and the FKG inequalities, Comm. Math. Phys. 74 (1980),
+no. 2, 119-128.
+
+42
+GUY COHEN AND JEAN-PIERRE CONZE
+[27]
+Newman, C.M. and Wright, A.L.: An invariance principle for certain dependent sequences, Ann.
+Probab. 9 (1981), no. 4, 671-675.
+[28]
+Spitzer, F.: Principles of random walk, The University Series in Higher Mathematics D. Van
+Nostrand Co., Inc., Princeton, N.J.-Toronto-London (1964). doi: 10.1007/978-1-4757-4229-9
+[29]
+Tucker, H.: A generalization of the Glivenko-Cantelli theorem, Annals of Math. Stat., vol. 20,
+no 3, p. 828-830 (1959).
+[30]
+Yu Hao, A Glivenko-Cantelli lemma and weak convergence for empirical processes of associated
+sequences, Probab. Theory Related Fields 95 (1993), no. 3, 357-370.
+[31]
+Zygmund, A.: Trigonometric series, Second edition, Cambridge University Press, London-New
+York 1968.
+Guy Cohen,
+School of Electrical Engineering,
+Ben-Gurion University, Israel
+Email address: guycohen@bgu.ac.il
+Jean-Pierre Conze,
+IRMAR, CNRS UMR 6625,
+University of Rennes, Campus de Beaulieu, 35042 Rennes Cedex, France
+Email address: conze@univ-rennes1.fr
+

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+arXiv:2301.01001v1  [math.DG]  3 Jan 2023
+On a Class of Generalized Berwald Manifolds
+A. Tayebi and F. Eslami
+January 4, 2023
+Abstract
+The class of generalized Berwald metrics contains the class of Berwald metrics. In this
+paper, we characterize two-dimensional generalized Berwald (α, β)-metrics with vanishing
+S-curvature. Let F = αφ(s), s = β/α, be a two-dimensional generalized Berwald (α, β)-
+metric on a manifold M. Suppose that F has vanishing S-curvature. We show that one of
+the following holds: (i) if F is a regular metric, then it reduces to a Riemannian metric of
+isotropic sectional curvature or a locally Minkowskian metric; (ii) if F is an almost regular
+metric that is not Riemannian nor locally Minkowskian, then we find the explicit form of
+φ = φ(s) which obtains a generalized Berwald metric that is neither a Berwald nor Lands-
+berg nor a Douglas metric. This provides a generalization of Szab´o rigidity theorem for
+the class of (α, β)-metrics. In the following, we prove that left invariant Finsler surfaces
+with vanishing S-curvature must be Riemannian surfaces of constant sectional curvature.
+Finally, we construct a family of odd-dimensional generalized Berwald Randers metrics.
+Keywords: Generalized Berwald metric, Berwald metric, (α, β)-metric, S-curvature.1
+1
+Introduction
+A Finsler metric F on a manifold M is called a generalized Berwald metric if there exists a
+covariant derivative ∇ on M such that the parallel translations induced by ∇ preserve the
+Finsler function F [18][19][20][26]. In this case, F is called a generalized Berwald metric on
+M and (M, F) is called a generalized Berwald manifold. If the covariant derivative ∇ is also
+torsion-free, then F reduces to a Berwald metric.
+Therefore, the class of Berwald metrics
+belongs to the class of generalized Berwald metrics. The importance of the class of generalized
+Berwald manifolds lies in the fact that these manifolds may have a rich isometry group [16][17].
+For some progress about the class of generalized Berwald manifolds, see [2], [4], [20], [25], [26],
+[27] and [28].
+To find generalized Berwald metrics, one can consider the class of (α, β)-metrics. An (α, β)-
+metric is a Finsler metric defined by F := αφ(s), s = β/α, where φ = φ(s) is a smooth function
+on a symmetric interval (−b0, b0) with certain regularity, α =
+�
+aij(x)yiyj is a Riemannian
+metric and β = bi(x)yi is a 1-form on the base manifold. The simplest (α, β)-metrics are the
+Randers metrics F = α+β which were discovered by G. Randers when he studied 4-dimensional
+general relativity. These metrics have been widely applied in many areas of natural sciences,
+including physics, biology, psychology, etc [5]. In [26], Vincze proved that a Randers metric
+F = α + β is a generalized Berwald metric if and only if dual vector field β♯ is of constant
+12010 Mathematics subject Classification: 53C60, 53C25.
+1
+
+On a Class of Generalized Berwald Manifolds
+2
+Riemannian length. In [20], Tayebi-Barzegari generalized Vincze’s result for (α, β)-metrics and
+showed that an (α, β)-metric satisfying the so-called sign property is a generalized Berwald
+metric if and only if β♯ is of constant Riemannian length. Then, Vincze showed that an (α, β)-
+metric satisfying the regularity property φ′(0) ̸= 0 is a generalized Berwald metric if and only
+if β♯ is of constant Riemannian length [25]. In [4], Bartelmeß-Matveev proved that a Finsler
+metric is a generalized Berwald metric if and only if it is monochromatic.
+Generally, two-dimensional Finsler metrics have some different and special Riemannian and
+non-Riemannian curvature properties from the higher dimensions. For example, Bartelmeß-
+Matveev showed that except for torus and the Klein bottle, the other closed 2-dimensional
+manifolds can not have non-Riemannian generalized Berwald metrics [4]. In [28], Vincze et al.
+showed that a connected generalized Berwald surface is a Landsberg surface if and only if it
+is a Berwald surface. The S-curvature is constructed by Shen for given comparison theorems
+on Finsler manifolds [5]. An interesting problem is to study generalized Berwald metrics with
+vanishing S-curvature. Here, we characterize the class of two-dimensional generalized Berwald
+(α, β)-metrics with vanishing S-curvature and prove the following.
+Theorem 1.1. Let F = αφ(s), s = β/α, be a two-dimensional generalized Berwald (α, β)-
+metric on a connected manifold M. Suppose that F has vanishing S-curvature. Then one of
+the following holds:
+(i) If F is a regular metric, then it reduces to a Riemannian metric of isotropic sectional
+curvature or a locally Minkowskian metric;
+(ii) If F is an almost regular metric that is not Riemannian nor locally Minkowskian, then φ
+is given by
+φ = c exp
+� � s
+0
+kt + q
+√
+b2 − t2
+1 + kt2 + qt
+√
+b2 − t2dt
+�
+,
+(1.1)
+where c > 0, q > 0, and k are real constants, and β satisfies
+rij = 0,
+si = 0.
+(1.2)
+In this case, F is neither a Berwald nor Landsberg nor a Douglas metric.
+It is remarkable that, for a generalized Berwald (α, β)-metric F = αφ(s), s = β/α, on an
+n-dimensional manifold M, we show that S = 0 if and only if F is Riemannian or β is a Killing
+form with constant length (see Lemma 3.2).
+The celebrated Szab´o rigidity theorem states that every 2-dimensional Berwald surface is
+either locally Minkowskian or Riemannian (see [13]).
+Every Berwald metric has vanishing
+S-curvature [22]. Then, Theorem 1.1 is an extension of Szab´o’s result for (α, β)-metrics.
+The condition of generalized Berwaldness can not be dropped from the assumption of The-
+orem 1.1. There are many two-dimensional (α, β)-metrics with vanishing S-curvature which
+are not Riemannian nor locally Minkowskian. For example, let us consider Shen’s Fish-Tank
+Randers metric F = α + β on R2 given by following
+F =
+�
+(xv − yu)2 + (u2 + v2)(1 − x2 − y2)
+1 − x2 − y2
++
+yu − xv
+1 − x2 − y2.
+F has vanishing S-curvature [14] while it is not a generalized Berwald metric (∥β∥α =
+�
+x2 + y2).
+This metric is not Riemannian. Also, F has vanishing flag curvature while it is not locally
+Minkowskian.
+In Theorem 1.1, the vanishing of S-curvature is necessary. For example, see the following.
+
+On a Class of Generalized Berwald Manifolds
+3
+Example 1. Let us consider G := {(x, y) ∈ R2|y > 0} and define a multiplication on G by
+(x1, y1) ∗ (x2, y2) := (x2y1 + x1, y1y2), for (xi, yi) ∈ G, i = 1, 2. (G, ∗) is a Lie group [8]. One
+can introduce a Randers metric F on G by
+F =
+�
+2dx2 + 2dxdx + 2dy2
+y
++ dx + dy
+y
+.
+(1.3)
+Then
+a11 = a22 = 2
+y2,
+a21 = a12 = 1
+y2,
+b1 = b2 = 1
+y,
+a11 = a22 = 2
+3y2,
+a12 = a21 = −1
+3y2,
+b1 = b2 = 1
+3y.
+It is easy to see that α is a positive-definite Riemannian metric. Also, we get
+b := ∥β∥α =
+�
+aijbibj =
+�
+bibi =
+�
+b1b1 + b2b2 =
+�
+2
+3
+It follows that F is a positive-definite Randers metric on G. In [26], Vincze showed that a
+Randers metric F = α + β is a generalized Berwald metric if and only if β is of constant length
+with respect to α. Then, the Randers metric defined by (1.3) is a generalized Berwald metric.
+We have
+dβ = 1
+y2dx ∧ dy ̸= 0.
+Thus β is not closed which implies that F can not be a Berwald metric. We have
+s11 = s22 = 0,
+s12 = 1
+y2 = −s21,
+s1 = − 1
+3y,
+s2 = 1
+3y.
+(1.4)
+where sij and si defined by (3.1) and (3.2). We claim that F has not vanishing S-curvature.
+On contrary, let S = 0. Then by Lemma 3.2.1 in [5], we have rij + bisj + bjsi = 0. Contracting
+it with bi yields
+rj + b2sj = 0.
+By considering ri + si = 0 and b2 = 2/3, we must have si = 0 which contradicts with (1.4).
+Then, the Randers metric (1.3) is a generalized Berwald metric with B ̸= 0, L ̸= 0, D ̸= 0,
+S ̸= 0 and E ̸= 0. We claim that F is not R-quadratic. On the contrary, let the Finsler metric
+F defined by (1.3) be R-quadratic. By Theorem 6.2.1 in [5], if the two-dimensional Randers
+metric (1.3) is R-quadratic then it has constant S-curvature S = 3cF, for some real constant
+c. In [23], Xu-Deng proved that every homogeneous Finsler metric of isotropic S-curvature has
+vanishing S-curvature. Thus, we must have S = 0 which is a contradiction. Therefore, the
+Randers metric (1.3) is not R-quadratic.
+Theorem 1.1 may not be held for an (α, β)-metric of constant S-curvature. For example, at
+a point x = (x1, x2) ∈ R2 and in the direction y = (y1, y2) ∈ TxR2, consider the Riemannian
+metric α = α(x, y) and one form β = β(x, y) as follows
+α(x, y) :=
+�
+(y1)2 + e2x1(y2)2,
+β(x, y) := y1.
+(1.5)
+Then sij = 0 and rij = b2aij − bibj which yield ri + si = 0. If φ = φ(s) satisfies
+Φ = −6k φ∆2
+b2 − s2,
+(1.6)
+
+On a Class of Generalized Berwald Manifolds
+4
+for some constant k, then F = αφ(β/α) has constant S-curvature S = 3kF (see [6]). Here,
+∆ = ∆(b, s) and Φ = Φ(b, s) defined by (3.3) and (3.4), respectively. The existence of regular
+solution of (1.6) for arbitrary k ∈ R, when α and β are given by (1.5), is proved in [10]. It is
+easy to see that F is a generalized Berwald metric while it is not a Berwald metric.
+We must mention that Theorem 1.1 does not hold for Finsler metrics of dim(M) ≥ 3.
+Denote generic tangent vectors on S3 as u∂/∂x + v∂/∂y + w∂/∂z. The Finsler functions for
+Bao-Shen’s Randers metrics F = α + β are given by the following
+F =
+�
+K(cu − zv + yw)2 + (zu + cv − xw)2 + (xv + cw − yu)2
+1 + x2 + y2 + z2
+±
+√
+K − 1 (cu − zv + yw)
+1 + x2 + y2 + z2
+,
+where K > 1 is a real constant. For these metrics, we have
+b := ∥β∥α =
+�
+1 − 1
+K
+One can see that, the one form β is not closed, and then F can not be a Douglas metric. This
+family of Randers metrics is generalized Berwald metrics with S = 0 which are not Berwaldian.
+In [4], it is proved that every 3-dimensional closed manifold admits a non-Riemannian
+generalized Berwald metric. However, the closeness condition is very restrictive. Indeed, for
+every manifold M with dim(M) ≥ 3, one can construct generalized Berwald (α, β)-metrics that
+are not Berwaldian.
+Example 2. The projective spherical metric on R3 is given by the following
+α :=
+�
+(1 + ||X||2)||Y ||2 − ⟨X, Y ⟩2
+1 + ⟨X, X⟩
+, X ∈ R3, Y ∈ TXR3,
+(1.7)
+where ⟨, ⟩ and ||.|| denote the Euclidean inner product and norm on R3, respectively.
+Put
+X = (x, y, z) and Y = (u, v, w). Suppose that β = κ(b1u + b2v + b3w) is a Killing 1-form of α,
+where 0 < κ < 1. By a simple calculation, we get
+b1 = A1
+2y + A1
+3z + C1
+1 + ⟨X, X⟩
+,
+b2 = A2
+1x + A2
+3z + C2
+1 + ⟨X, X⟩
+,
+b3 = A3
+1x + A3
+2y + C3
+1 + ⟨X, X⟩
+,
+(1.8)
+where A = (Ai
+j) is an antisymmetric real matrix, and C = (Ci) is a constant vector in R3. Let
+us put
+C = (0, 1, 0),
+A1
+2 = A2
+3 = 0,
+A1
+3 = 1.
+(1.9)
+In this case, β is a Killing 1-form with ∥β∥α = κ < 1 such that is not closed. According to
+Shen’s theorem in [12], a regular (α, β)-metric is a Berwald metric if and only if β is parallel
+with respect to α. Using the Riemannian metric (1.7) and 1-form β satisfying (1.8) and (1.9),
+one can construct generalized Berwald (α, β)-metrics which are not Berwaldian.
+Homogeneous Finsler manifolds are those Finsler manifolds (M, F) that the orbit of the
+natural action of I(M, F) on M at any point of M is the whole M. For the class of homogeneous
+generalized Berwald (α, β)-metrics, we prove the following.
+Corollary 1.1. Let F = αφ(s), s = β/α, be a two-dimensional homogeneous generalized
+Berwald (α, β)-metric on a manifold M. Then F has isotropic S-curvature if and only if it is
+a Riemannian metric of constant sectional curvature or a locally Minkowskian metric.
+
+On a Class of Generalized Berwald Manifolds
+5
+Let G be a connected Lie group with a bi-invariant Finsler metric ¯F, and H a closed
+subgroup of G. Denote the Lie algebras of G and H as g and h, respectively. Let ρ be the natural
+projection from G to M = G/H. Then there exists a uniquely defined G-invariant metric F
+on M such that for any g ∈ G, the tangent map ρ∗ :
+�
+TgG, ¯F(g, .)
+�
+→
+�
+Tπ(g)M, F(π(g), .)
+�
+is
+a submersion (see Lemma 3.1 in [24]). The G-invariant Finsler metric F is called the normal
+homogeneous metric induced by ¯F. The pair (M, F) is said the normal homogeneous space
+induced by ρ : G → M = G/H and ¯F. For normal homogeneous generalized Berwald metrics,
+we have the following.
+Corollary 1.2. Every two-dimensional normal homogeneous generalized Berwald (α, β)-metric
+is a Riemannian metric of non-negative constant sectional curvature or a locally Minkowskian
+metric.
+A Finsler metric F on a manifold M is called a generalized normal homogeneous metric
+if for every x, y ∈ M, there exists a δ(x)-translation of (M, F) sending x to y (see [30]). In
+this paper, we consider two-dimensional homogeneous generalized Berwald Randers metric of
+generalized normal-type and prove the following.
+Corollary 1.3. Every two-dimensional generalized normal homogeneous generalized Berwald
+Randers metric must be a Riemannian metric of non-negative constant sectional curvature or
+a locally Minkowskian metric.
+Let (G, ·) be a connected Lie group with identity element e, and λg denotes the left trans-
+lation by g ∈ G. A Finsler metric F on G is called a left invariant Finsler metric if it satisfies
+F ◦ (λg)∗ = F for any g ∈ G. In [2], Aradi proved that left invariant Finsler metrics are
+generalized Berwald metrics. For this class of Finsler metrics, we have the following.
+Theorem 1.2. Every left invariant Finsler surface has vanishing S-curvature if and only if it
+is a Riemannian metric of constant sectional curvature.
+By considering Theorems 1.1 and 1.2, it seems that two-dimensional generalized Berwald
+(α, β)-metrics with vanishing S-curvature may be only Riemannian. But we do not find a short
+proof for this conjecture.
+A Finsler metric F on a manifold M is called an isotropic Berwald metric, if its Berwald
+curvature is given by
+By(u, v, w) = cF −1�
+h(u, v)
+�
+w − gy(w, ℓ)ℓ
+�
++ h(v, w)
+�
+u − gy(u, ℓ)ℓ
+�
++h(w, u)
+�
+v − gy(v, ℓ)ℓ
+�
++ 2FCy(u, v, w)ℓ
+�
+.
+(1.10)
+where hy(u, v) = gy(u, v)−F −2(y)gy(y, u)gy(y, v) is the angular form in direction y, C denotes
+the Cartan torsion of F and c ∈ C∞(M). Every Berwald metric is an isotropic Berwald metric
+with c = 0. The Funk metrics are isotropic Berwald metrics with c = 1/2. As a straightforward
+conclusion of Theorem 1.2, one can get the following.
+Corollary 1.4. Every left invariant isotropic Berwald surface must be a Riemannian surface
+of constant sectional curvature.
+
+On a Class of Generalized Berwald Manifolds
+6
+2
+Preliminaries
+Let M be an n-dimensional C∞ manifold, TM = �
+x∈M TxM the tangent space and TM0 :=
+TM − {0} the slit tangent space of M.
+Let (M, F) be a Finsler manifold.
+The following
+quadratic form gy : TxM × TxM → R is called fundamental tensor
+gy(u, v) := 1
+2
+∂2
+∂s∂t
+�
+F 2(y + su + tv)
+�
+s=t=0,
+u, v ∈ TxM.
+Let x ∈ M and Fx := F|TxM. To measure the non-Euclidean feature of Fx, one can define
+Cy : TxM × TxM × TxM → R by
+Cy(u, v, w) := 1
+2
+d
+dt
+�
+gy+tw(u, v)
+�
+t=0 ,
+u, v, w ∈ TxM.
+The family C := {Cy}y∈TM0 is called the Cartan torsion.
+For a Finsler manifold (M, F), its induced spray on TM is denoted by G = G(x, y) which
+in a standard coordinate (xi, yi) for TM0 is given by G = yi∂/∂xi − 2Gi(x, y)∂/∂yi, where
+Gi := 1
+4gil� ∂2F 2
+∂xk∂yl yk − ∂F 2
+∂xl
+�
+.
+For a Finsler metric F on an n-dimensional manifold M, the Busemann-Hausdorff volume form
+dVF = σF(x)dx1 · · · dxn is defined by
+σF(x) :=
+Vol
+�
+Bn(1)
+�
+Vol
+�
+(yi) ∈ Rn �� F
+�
+yi ∂
+∂xi|x
+�
+< 1
+�.
+Let Gi denote the geodesic coefficients of F in the same local coordinate system. Then for
+y = yi∂/∂xi|x ∈ TxM, the S-curvature is defined by
+S(y) := ∂Gi
+∂yi (x, y) − yi ∂
+∂xi
+�
+ln σF(x)
+�
+.
+(2.1)
+For a vector y ∈ TxM0, the Berwald curvature By : TxM × TxM × TxM → TxM is defined
+by By(u, v, w) := Bi
+jkl(y)ujvkwl∂/∂xi|x, where
+Bi
+jkl :=
+∂3Gi
+∂yj∂yk∂yl .
+F is called a Berwald metric if B = 0. Every Berwald metric satisfies S = 0 (see [22]).
+For y ∈ TxM, define the Landsberg curvature Ly : TxM × TxM × TxM → R by
+Ly(u, v, w) := −1
+2gy
+�
+By(u, v, w), y
+�
+.
+A Finsler metric F is called a Landsberg metric if L = 0.
+Taking a trace of Berwald curvature B give us the mean of Berwald curvature E which is
+defined by Ey : TxM × TxM → R, where
+Ey(u, v) := 1
+2
+n
+�
+i=1
+gij(y)gy
+�
+By(u, v, ∂i), ∂j
+�
+.
+(2.2)
+where {∂i} is a basis for TxM at x ∈ M. In local coordinates, Ey(u, v) := Eij(y)uivj, where
+Eij := 1
+2Bm
+mij.
+
+On a Class of Generalized Berwald Manifolds
+7
+Taking a horizontal derivation of the mean of Berwald curvature E along Finslerian geodesics
+give us the H-curvature H = H(x, y) which is defined by Hy = Hijdxi ⊗ dxj, where
+Hij := Eij|mym.
+Here, “|” denotes the horizontal covariant differentiation with respect to the Berwald connection
+of F.
+For a non-zero vector y ∈ TxM0, one can define Dy : TxM × TxM × TxM → TxM by
+Dy(u, v, w) := Di
+jkl(y)uivjwk∂/∂xi|x, where
+Di
+jkl :=
+∂3
+∂yj∂yk∂yl
+�
+Gi −
+2
+n + 1
+∂Gm
+∂ym yi
+�
+.
+(2.3)
+D is called the Douglas curvature. F is called a Douglas metric if D = 0.
+For a non-zero vector y ∈ TxM0, the Riemann curvature is a family of linear transformation
+Ry : TxM → TxM which is defined by Ry(u) := Ri
+k(y)uk∂/∂xi, where
+Ri
+k(y) = 2∂Gi
+∂xk −
+∂2Gi
+∂xj∂yk yj + 2Gj ∂2Gi
+∂yj∂yk − ∂Gi
+∂yj
+∂Gj
+∂yk .
+(2.4)
+The family R := {Ry}y∈TM0 is called the Riemann curvature.
+For a flag P := span{y, u} ⊂ TxM with the flagpole y, the flag curvature K = K(P, y) is
+defined by
+K(x, y, P) :=
+gy
+�
+u, Ry(u)
+�
+gy(y, y)gy(u, u) − gy(y, u)2.
+(2.5)
+The flag curvature K(x, y, P) is a function of tangent planes P = span{y, v} ⊂ TxM. A Finsler
+metric F is of scalar flag curvature if K(x, y, P) = K(x, y) is independent of P. Also, F is
+called of isotropic and constant flag curvature if K = K(x) and K = constant, respectively.
+Throughout this paper, we use the Berwald connection on Finsler manifolds. The pullback
+bundle π∗TM admits a unique linear connection, called the Berwald connection. Let (M, F)
+be an n-dimensional Finsler manifold. Let {ej} be a local frame for π∗TM, {ωi, ωn+i} be the
+corresponding local coframe for T ∗(TM0) and {ωi
+j} be the set of local Berwald connection forms
+with respect to {ej}. Then the connection forms are characterized by the structure equations
+as follows
+• Torsion freeness:
+dωi = ωj ∧ ωi
+j.
+(2.6)
+• Almost metric compatibility:
+dgij − gkjωk
+i − gikωk
+j = −2Lijkωk + 2Cijkωn+k,
+(2.7)
+where ωi := dxi and ωn+k := dyk + yjωk
+j.
+The horizontal and vertical covariant derivations with respect to the Berwald connection re-
+spectively are denoted by “|” and “, ”. For more details, one can see [13].
+
+On a Class of Generalized Berwald Manifolds
+8
+3
+Proof of Theorems 1.1 and 1.2
+For an (α, β)-metric, let us define bi;j by bi;jθj := dbi − bjθj
+i , where θi := dxi and θj
+i := γj
+ikdxk
+denote the Levi-Civita connection form of α. Let
+rij := 1
+2(bi;j + bj;i),
+sij := 1
+2(bi;j − bj;i),
+ri0 := rijyj,
+r00 := rijyiyj,
+rj := birij,
+(3.1)
+si0 := sijyj,
+sj := bisij,
+si
+j = aimsmj,
+si
+0 = si
+jyj,
+r0 := rjyj,
+s0 := sjyj.
+(3.2)
+Put
+Q :=
+φ′
+φ − sφ,
+∆ := 1 + sQ + (b2 − s2)Q′,
+Θ := Q − sQ′
+2∆
+,
+(3.3)
+Φ := −(Q − sQ′)(n∆ + 1 + sQ) − (b2 − s2)(1 + sQ)Q′′,
+(3.4)
+Ψ :=
+φ′′
+2
+�
+(φ − sφ′) + (b2 − s2)φ′′�,
+where b := ∥β∥α. Let Gi = Gi(x, y) and ¯Gi
+α = ¯Gi
+α(x, y) denote the coefficients of F and α,
+respectively, in the same coordinate system. By definition, we have
+Gi = Gi
+α + αQsi
+0 + α−1(r00 − 2Qαs0)(Θyi + αΨbi).
+(3.5)
+Clearly, if β is parallel with respect to α, that is rij = sij = 0, then Gi = Gi
+α = γi
+jk(x)yjyk are
+quadratic in y. In this case, F reduces to a Berwald metric.
+For an (α, β)-metric F = αφ(s), s = β/α, on an n-dimensional manifold M, the S-curvature
+is given by
+S =
+�
+2Ψ − f ′(b)
+bf(b)
+�
+(r0 + s0) −
+Φ
+2α∆2(r00 − 2αQs0),
+(3.6)
+where
+f(b) :=
+� π
+0 sinn−2 t T(b cos t)dt
+� π
+0 sinn−2 tdt
+,
+T(s) := φ(φ − sφ′)n−2�
+(φ − sφ′) + (b2 − s2)φ′′�
+.
+For more details, see [6].
+To prove Theorem 1.1, we need the following key lemma.
+Lemma 3.1. ([4][20]) Let F = αφ(s), s = β/α, be an (α, β)-metric on a manifold M. Then
+F is a generalized Berwald metric if and only if β has constant length with respect to α.
+Proof. In [20], Tayebi-Barzegari showed that every (α, β)-metric F = αφ(s), s = β/α, with
+sign property is a generalized Berwald metric if and only if the dual vector field β♯ is of
+constant Riemannian length. In [9], Ivanov proved that a two-dimensional Finsler metric is a
+generalized Berwald metric if and only if it is monochromatic, i.e., Finsler metrics such that
+each two tangent spaces are isomorphic as normed spaces. In [4], Bartelmeß-Matveev extended
+his result for n-dimensional Finsler metric. It follows that an (α, β)-metric is a generalized
+Berwald metric if and only if dual vector field β♯ is of constant Riemannian length.
+In [6], Cheng-Shen characterized (α, β)-metrics with isotropic S-curvature on a manifold M
+of dimension n ≥ 3. Soon, they found that their result holds for the class of (α, β)-metrics with
+constant length one-forms, only. Here, we give a characterization of the class of generalized
+Berwald metrics with vanishing S-curvature.
+
+On a Class of Generalized Berwald Manifolds
+9
+Lemma 3.2. Let F = αφ(s), s = β/α, be a generalized Berwald (α, β)-metric on an n-
+dimensional manifold M. Then S = 0 if and only if F is Riemannian or β is a Killing form
+with constant length.
+Proof. Let b := ∥β∥α =
+�
+amjbjbm = √bmbm. Then, the following holds
+∂b
+∂xi = 1
+bbmbm|i = 1
+b(ri + si).
+(3.7)
+By Lemma 3.1, we have b = constant. Then, by (3.7) we obtain ri + si = 0. In this case, (3.6)
+reduces to following
+S = −
+Φ
+2α∆2(r00 − 2αQs0),
+(3.8)
+By (3.8), S = 0 if and only if Φ = 0 or β satisfies
+r00 = 2αQs0.
+(3.9)
+In the case of Φ = 0, F reduces to a Riemannian metric (see Proposition 2.2 in [11]). Now,
+let (3.9) holds. We are going to simplify (3.9). For this aim, one can change the y-coordinates
+(yi), i = 1, · · · , n, at a point to the polar coordinates (s, uA), where A = 2, · · · , n (see [6]).
+For an arbitrary and fix point x ∈ M, let us take an orthonormal basis ei at x such that the
+Riemannian metric is written as α =
+��n
+i=1(yi)2 and its related one-form is given by β = by1,
+where b := ||β||α. Let us fix an arbitrary number s such that |s| < b. Define
+¯α =
+�
+�
+�
+�
+n
+�
+A=2
+(yA)2.
+Then, by β = sα we get
+y1 =
+s
+√
+b2 − s2 ¯α,
+yA = uA.
+(3.10)
+Also, we have
+α =
+b
+√
+b2 − s2 ¯α,
+β =
+bs
+√
+b2 − s2 ¯α.
+(3.11)
+Let us put
+¯r10 :=
+n
+�
+A=2
+r1AyA,
+¯s10 :=
+n
+�
+A=2
+s1AyA,
+¯r00 :=
+n
+�
+A,B=2
+rAByAyB,
+¯r0 :=
+n
+�
+A=2
+rAyA,
+¯s0 :=
+n
+�
+A=2
+sAyA.
+Then we get the following useful relations
+r1 = br11,
+rA = br1A,
+s1 = 0,
+sA = bs1A,
+(3.12)
+r00 =
+s2
+b2 − s2 ¯α2r11 +
+2s
+√
+b2 − s2 ¯α¯r10 + ¯r00,
+(3.13)
+r10 =
+s
+√
+b2 − s2 ¯αr11 + ¯r10,
+s0 = ¯s0 = b¯s10.
+(3.14)
+Using (3.11)-(3.14), the equation (3.9) can be written as follows
+¯r00 +
+s2
+b2 − s2 ¯α2 r11 =
+2
+√
+b2 − s2
+�
+b2Q¯s10 − s¯r10
+�
+¯α.
+(3.15)
+
+On a Class of Generalized Berwald Manifolds
+10
+By (3.15), we get two following relations
+¯r00 +
+s2
+b2 − s2 ¯α2r11 = 0,
+(3.16)
+b2Q¯s10 − s¯r10 = 0.
+(3.17)
+On the other hand, (3.11) implies that
+s2
+b2 − s2 ¯α2 − 1
+b2β2 = 0.
+(3.18)
+By (3.16) and (3.18), we get
+b2¯r00 + β2r11 = 0.
+(3.19)
+The following hold
+∂¯r00
+∂y1 = 0,
+∂β
+∂y1 = b.
+Then by differentiating (3.19) with respect to y1 we have β/br11 = 0. Thus r11 = 0 and by
+putting it in (3.19), we get ¯r00 = 0. Putting these relations in (3.13) and (3.14) imply that
+r00 =
+2s¯α
+√
+b2 − s2 ¯r10 = 2β
+b ¯r10,
+(3.20)
+r10 = ¯r10.
+(3.21)
+Now, by considering (3.20) and (3.21), we divide the problem into two cases: (a) ¯r10 = 0 and
+(b) ¯r10 ̸= 0.
+Case (a):
+¯r10 = 0.
+In this case, by (3.20) we get rij = 0.
+Putting it in (3.9) implies
+that si = 0. In this case, β reduces to a Killing one-form of constant length with respect to α.
+Case (b): ¯r10 ̸= 0. We have ∂¯r10/∂y1 = 0 and ∂¯s10/∂y1 = 0. Thus, differentiating (3.17) with
+respect to y1 yields
+(s)y1¯r10 = b2(Q)y1¯s10.
+(3.22)
+Contracting (3.17) with (s)y1 give us
+s(s)y1¯r10 = b2Q(s)y1¯s10.
+(3.23)
+By (3.22) and (3.23), we get
+�
+(s)y1Q − s(Q)y1
+�
+¯s10 = 0.
+(3.24)
+According to (3.24), we get ¯s10 = 0 or Q(s)y1 = s(Q)y1. Let ¯s10 = 0 holds. Then (3.17) reduces
+to s = 0, which is impossible. Then, we have
+Q(s)y1 = s(Q)y1,
+which is equal to
+(Q)y1
+Q
+= (s)y1
+s
+.
+(3.25)
+
+On a Class of Generalized Berwald Manifolds
+11
+Using sy1 ̸= 0 and (Q)y1 = sy1(Q)s, then (3.25) give us
+(Q)s
+Q
+= 1
+s
+which yields
+ln(Q) − ln(s) = c,
+where c is a real constant. Thus Q = ks, where k is a non-zero real constant. In this case, we
+get φ =
+√
+1 + ks2 which shows that F is a Riemannian metric. This completes the proof.
+Here, we solve an ODE which will appear in the proof of Theorem 1.1.
+Lemma 3.3. Let F = αφ(s), s = β/α, be an (α, β)-metric on a manifold M. Suppose that φ
+satisfies following
+αΘ2 + 2Λ1Θ1 + Λ2Q = 0,
+(3.26)
+where
+Λ1 := biαyi,
+Λ2 := bibjαyiyj,
+Θ1 := biQyi,
+Θ2 := bibjQyiyj.
+Then F is a singular Finsler metric given by
+φ = c exp
+� � s
+0
+k1t + k2
+√
+b2 − t2
+1 + t
+�
+k1t + k2
+√
+b2 − t2�dt
+�
+,
+(3.27)
+where k1 and k2 are real constants, and c > 0 is a non-zero constant.
+Proof. Let us put
+Ajk := α2ajk − yjyk.
+Then, the followings hold
+αyi = 1
+αyi,
+αyjyk = 1
+α3Ajk,
+αyjykyl = − 1
+α5
+�
+Ajkyl + Ajlyk + Alkyj
+�
+.
+Also, one can obtain the following
+Λ1 = s,
+(3.28)
+Λ2 = 1
+α(b2 − s2),
+(3.29)
+Θ1 = 1
+α(b2 − s2)Q′,
+(3.30)
+Θ2 = 1
+α2(b2 − s2)
+�
+(b2 − s2)Q′′ − 3sQ′�
+.
+(3.31)
+Suppose that (3.26) holds. Putting (3.28)-(3.31) into (3.26) yield
+Q′′ −
+s
+b2 − s2Q′ +
+1
+b2 − s2Q = 0,
+
+On a Class of Generalized Berwald Manifolds
+12
+which implies that
+Q = k1s + k2
+√
+b2 − s2,
+(3.32)
+where k1 and k2 are real constants. Considering (3.3), the equation (3.32) is equal to following
+�
+1 + k1s2 + k2s
+√
+b2 − s2
+�
+φ′ =
+�
+k1s + k2
+√
+b2 − s2
+�
+φ.
+(3.33)
+By (3.33), we get (3.27). It is an almost regular (α, β)-metric, namely, it is singular in two
+directions y = (±b, 0, 0) ∈ TxM at any point x (for more details, see [12]).
+The function φ in (3.27) is specifically has been seen for the first time in Asanov’s paper [3]
+as follows
+φ = c exp
+� � s
+0
+k
+√
+b2 − t2
+1 + tk
+√
+b2 − t2dt
+�
+.
+(3.34)
+Then, its more general form (3.27) found by Shen in [12], where he looked for unicorn metrics,
+namely the Landsberg metrics which are not Berwaldian.
+Shen realized that the function
+φ = φ(b, s) in (3.27) can be expressed in terms of elementary functions. See (7.2) in [12].
+Here, we show that the Douglas curvature of Finsler surfaces satisfies a special relation.
+More precisely, we prove the following.
+Lemma 3.4. The Douglas curvature of any Finsler surface (M, F) satisfies
+Di
+jkl|sys = Djklyi
+(3.35)
+for some tensor D = Dijkdxi ⊗ dxj ⊗ dxk which are homogeneous of degree -1 in y.
+Proof. By definition, the Douglas curvature of a two-dimensional Finsler metric is given by
+Di
+jkl = Bi
+jkl − 2
+3
+�
+Ejkδi
+l + Eklδi
+j + Eljδi
+k + Ejk,lyi�
+.
+(3.36)
+Taking a horizontal derivation of (3.36) along Finslerian geodesics and using yi
+|s = 0 give us
+the following
+Di
+jkl|mym = Bi
+jkl|mym − 2
+3
+�
+Hjkδi
+l + Hklδi
+j + Hljδi
+k + Ejk,l|mymyi�
+.
+(3.37)
+Every Finsler surface is of scalar flag curvature K = K(x, y). Then, by (11.24) in [13] we have
+Bi
+jml|kyk = 2KCjlmyi − 1
+3
+�
+ylδi
+m + ymδi
+l − 2glmyi�
+Kj − 1
+3
+�
+yjδi
+m + ymδi
+j − 2gjmyi�
+Kl
+−1
+3
+�
+yjδi
+l + ylδi
+j − 2gjlyi�
+Km − 1
+3F 2�
+Kjmhi
+l + Kjlhi
+m + Klmhi
+j
+�
+, (3.38)
+where yi = FFyi, Kj := Kyj and Kjk := Kyjyk. Taking a trace of (3.38) yields
+Hjl = −1
+2
+�
+ylKj + yjKl + F 2Kjl
+�
+.
+(3.39)
+By putting (3.38) and (3.39) in (3.37) we get
+Di
+jkl|mym = 1
+3
+�
+6KCjkl + 2
+�
+gklKj + gkjKl + gjlKk
+�
++
+�
+yjKkl + ykKjl + ylKkj
+�
+−2Ejk,l|mym�
+yi.(3.40)
+(3.40) give us (3.35).
+
+On a Class of Generalized Berwald Manifolds
+13
+Now, we show that the covariant derivative of Berwald curvature of 2-dimensional gener-
+alized Berwald (α, β)-metric with vanishing S-curvature satisfies an interesting relation which
+will play an important role in the proof of Theorem 1.1.
+Lemma 3.5. The Berwald curvature of any non-Riemannian 2-dimensional generalized Berwald
+(α, β)-metric with vanishing S-curvature satisfies following
+hm
+p Bp
+jkl|sys =
+sm
+0|0(αjklQ + αjkQl + αlkQj + αljQk + αQjkl + αlQjk + αjQlk + αkQjl)
++sm
+0(αjklQ|0 + αjkQl|0 + αlkQj|0 + αljQk|0 + αQjkl|0 + αlQjk|0 + αjQlk|0
++αkQjl|0) + Am
+lXjk + Am
+jXlk + Am
+kXjl + Bm
+l Yjk + Bm
+jYlk + Bm
+kYjl = 0, (3.41)
+where
+Xjk := Q|0αjk + Qk|0αj + Qj|0αk + Qjk|0α,
+Yjk := Qαjk + Qkαj + Qjαk + αQjk,
+Am
+l := sm
+l − F −2s0
+lym,
+Bm
+l := Am
+l|sys = sm
+l|0 − F −2s0
+l|0ym.
+Proof. By Lemma 3.2, we have rij = 0 and sj = 0. In this case, (3.5) reduces to following
+Gi = Gi
+α + αQsi
+0.
+(3.42)
+Taking three vertical derivations of (3.42) with respect to yj, yl and yk give us
+Bi
+jkl =
+si
+0
+�
+αQjkl + αlQjk + αjQlk + αkQjl + αjklQ + αjkQl + αlkQj + αljQk
+�
++ si
+j
+�
+Qαlk + Qkαl + Qlαk + αQlk
+�
++ si
+k
+�
+Qαjl + Qjαl + Qlαj + αQjl
+�
++si
+l
+�
+Qαjk + Qkαj + Qjαk + αQjk
+�
+.
+(3.43)
+Contracting (3.43) with hm
+i implies that
+hm
+i Bi
+jkl =
+�
+αjklQ + αjkQl + αlkQj + αljQk + αQjkl + αlQjk + αjQlk + αkQjl
+�
+sm
+0
++
+�
+Qαjk + Qkαj + Qjαk + αQjk
+�
+(sm
+l − F −2s0
+lym)
++
+�
+Qαlk + Qkαl + Qlαk + αQlk
+�
+(sm
+j − F −2s0
+jym)
++
+�
+Qαjl + Qjαl + Qlαj + αQjl
+�
+(sm
+k − F −2s0
+kym).
+(3.44)
+On the other hand, by taking a horizontal derivation of Douglas curvature along Finslerian
+geodesics and contracting the result with hm
+i , we get the following
+hm
+i Di
+jkl|sys = hm
+i Bi
+jkl|sys − 2
+3
+�
+Hjkhm
+l + Hklhm
+j + Hljhm
+k
+�
+.
+(3.45)
+Contracting (3.35) with hm
+i yields
+hm
+i Di
+jkl|sys = 0.
+(3.46)
+Since S = 0, then by definition we get H = 0. Thus, (3.45) and (3.46) imply that
+hm
+i Bi
+jkl|sys = 0.
+(3.47)
+We have hm
+i|s = 0. Then, by considering (3.47), we have
+�
+hm
+i Bi
+jkl
+�
+|sys = hm
+i Bi
+jkl|sys = 0.
+(3.48)
+Therefore, taking a horizontal derivation of (3.44) along Finslerian geodesic and considering
+(3.48) give us (3.41).
+
+On a Class of Generalized Berwald Manifolds
+14
+Proof of Theorem 1.1: Taking a horizontal derivation of ymsm
+0 = 0 with respect to the
+Berwald connection of F implies that
+ym|0sm
+0 + ymsm
+0|0 = 0.
+(3.49)
+Since ym|0 = 0, then (3.49) reduces to following
+ymsm
+0|0 = 0.
+(3.50)
+By contracting (3.41) with ym and considering (3.50), we get
+s0
+lXjk + s0
+jXlk + s0
+kXjl + s0
+l|0Yjk + s0
+j|0Ylk + s0
+k|0Yjl = 0.
+(3.51)
+Since sj = 0, then one can get
+0 = (sj)|0 = (rm0 + sm0)sm
+j + bmsm
+j|0
+which considering rij = 0, it reduces to following
+bmsm
+j|0 = −sm0sm
+j.
+(3.52)
+Multiplying (3.52) with yj yields
+bisi
+0|0 + si0si
+0 = 0.
+(3.53)
+Also, contracting (3.52) with bj implies that
+bisi
+j|0bj = 0.
+(3.54)
+Multiplying (3.51) with bjbkbl and considering (3.53) and (3.54) give us
+(αΘ2 + 2Λ1Θ1 + Λ2Q)sm0sm
+0 = 0.
+(3.55)
+By (3.55), we have two cases: if sm
+0sm0 = 0, since α is a positive-definite metric, then we
+find that β is closed. Therefore, by (3.43) we conclude that F reduces to a Berwald metric.
+By Szabo’s rigidity result for Finsler surfaces, F reduces to a locally Minkowskian metric
+or a Riemannian metric. On the other hand, every Finsler surface has scalar flag curvature
+K = K(x, y). According to Akbar-Zadeh theorem in [1], a Finsler manifold (M, F) of scalar
+flag curvature K = K(x, y) has isotropic flag curvature K = K(x) if and only if it has vanishing
+H-curvature H = 0. Thus, the obtained Riemannian metric has isotropic sectional curvature.
+Now, suppose that F is not a Riemannian metric nor a locally Minkowskian metric. Then,
+by (3.55) we have αΘ2 + 2Λ1Θ1 + Λ2Q = 0. By Lemma 3.3 we obtain (1.1). In this case, since
+S = 0, then F can not be a Douglas metric. On the other hand, the Berwald curvature of
+2-dimensional Finsler manifold is given by
+Bi
+jkl = − 2
+F 2Ljklyi + 2
+3
+�
+Ejkhi
+l + Eklhi
+j + Ejlhi
+k
+�
+.
+(3.56)
+See the relation (15) in [21]. If F is a Landsberg metric then by considering S = 0, (3.56) implies
+that F is a Berwald metric. This is a contradiction. Then, (1.1) is a generalized Berwald metric
+which is not Berwald, Landsberg nor Douglas metric.
+Proof of Corollary 1.1: In [23], Xu-Deng proved that every homogeneous Finsler metric
+of isotropic S-curvature has vanishing S-curvature. Then, by assumption we get S = 0. The
+
+On a Class of Generalized Berwald Manifolds
+15
+Akbar-Zadeh theorem in [1] stated that a Finsler manifold (M, F) of scalar flag curvature
+K = K(x, y) has isotropic flag curvature K = K(x) if and only if it has vanishing H-curvature
+H = 0. On the other hand, every Finsler surface has scalar flag curvature K = K(x, y). Thus,
+by Akbar-Zadeh theorem we get K = K(x). Every scalar function on M which is invariant
+under isometries of (M, F) is a constant function. The homogeneity of (M, F) and invariancy
+of the flag curvature under isometries of F imply that K = constant. Then, by Theorem 1.1
+we get the proof.
+Proof of Corollary 1.2: Let F = αφ(s), s = β/α, be a two-dimensional normal homogeneous
+generalized Berwald (α, β)-metric. In [24], Xu-Deng proved that every normal homogeneous
+manifold has vanishing S-curvature and non-negative flag curvature. By Theorem 1.1 and the
+same method used in the proof of Corollary 1.1, it follows that F is a Riemannian metric of
+non-negative constant sectional curvature or a locally Minkowskian metric.
+Proof of Corollary 1.3: Let (G/H, F) be a generalized normal homogeneous Randers man-
+ifold. In [30], Zhang-Deng proved that F has vanishing S-curvature (Corollary 3.11). Also,
+they showed that any generalized normal homogeneous Randers metric has non-negative flag
+curvature (see Proposition 3.13 in [30]). Then, by Theorem 1.1 we get the proof.
+According to Theorem 1.1, every two-dimensional generalized Berwald (α, β)-metric with
+vanishing S-curvature is Riemannian or locally Minkowskian. Every left invariant Finsler metric
+is a generalized Berwald metric [2]. Here, we prove Theorem 1.2 which states that left invariant
+Finsler metrics with vanishing S-curvature reduce to Riemannian metrics, only. The approach
+of the proof of Theorem 1.2 is completely different from Theorem 1.1.
+Proof of Theorem 1.2: To prove a homogeneous surface with S = 0 is Riemannian, we only
+need to consider the nontrivial case, namely, a 2-dimensional non-Abelian Lie group G with a
+left invariant Finsler metric F. At each y ∈ g with F(y) = 1, there is a gy orthonormal basis
+e1 = y and e2 tangent to F = 1. At almost all non-zero y, the spray vector field η is nonzero,
+i.e., η(y) is a nonzero multiple of e2. By the homogenous S-curvature formula, we have
+S(y) = −I(η) = −Cy(η, e2, e2) = 0.
+Then, Cy(e2, e2, e2) = 0 everywhere. The speciality of 2-dimensional spaces implies C = 0
+everywhere, so F is Riemannian. By the same method used to prove Corollary 1.1, one can
+conclude that the Riemannian metric is of constant sectional curvature. This completes the
+proof.
+Proof of Corollary 1.4: By assumption, F has isotropic Berwald curvature
+Bi
+jkl = cF −1�
+hjkhi
+l + hjlhi
+k + hklhi
+j + 2Cjklyi�
+.
+(3.57)
+where c = c(x) is a scalar function on M. In [22], it is proved that every Finsler surface of
+isotropic Berwald curvature (3.57) metric has isotropic S-curvature S = 3cF. By Xu-Deng’s
+result in [24], F has vanishing S-curvature S = 0. Then, by Theorem 1.2, F reduces to a
+Riemannian metric.
+
+On a Class of Generalized Berwald Manifolds
+16
+4
+Some Examples of Generalized Berwald Manifolds
+In this section, we are going to give some important examples of the class of generalized Berwald
+manifolds. First, by using trans-Sasakian structure, we construct a family of odd-dimensional
+generalized Berwald Randers metrics.
+Example 3.
+�
+Odd-dimensional generalized Berwald Randers metrics
+�
+Let M be a differen-
+tiable manifold of dimension 2n + 1.
+Suppose that η = ηi(x)dxi, ξ = ξi∂/∂xi and ϕ =
+ϕi
+j∂/∂xi ⊗ dxj are a 1-form, a vector field, and a (1, 1)-tensor on M, respectively. The triple
+(η, ξ, ϕ) is called an almost contact structure on M if it satisfies
+ϕ(ξ) = 0,
+η(ξ) = 1,
+ϕ2 = −I + η ⊗ ξ.
+A differentiable manifold of odd dimension 2n+ 1 with an almost contact structure is called an
+almost contact manifold. Let a manifold M with the (η, ξ, ϕ) structure admits a Riemannian
+metric g such that
+g(ϕX, ϕY ) = g(X, Y ) − η(X)η(Y ).
+Then M is called an almost contact metric structure and g is called a compatible metric. In this
+case, (η, ξ, ϕ, g) is called almost contact metric structure. An almost contact metric structure
+(η, ξ, ϕ, g) on M is called a trans-Sasakian structure if it satisfies
+(∇Xϕ)Y = c1
+�
+g(X, Y )ξ − η(Y )X
+�
++ c2
+�
+g(ϕX, Y )ξ − η(Y )ϕX
+�
+for some scalar functions c1 = c1(x) and c2 = c2(x) on M, where ∇ denotes the Levi-Civita
+connection of g.
+Now, let (η, ξ, ϕ, g) be a trans-Sasakian structure on M. Define α, β : TM → [0, ∞) by
+∀(x, y) ∈ TM,
+α(x, y) :=
+�
+gx(y, y),
+β(x, y) := ǫ ηx(y),
+(4.1)
+where 0 < ǫ < 1 be a constant. Then, for the Randers metric F := α + β, we have
+||β||α = ǫ.
+It follows that the class of Randers metrics induced by trans-Sasakian manifolds (M, η, ξ, ϕ, g)
+are (2n + 1)-dimensional generalized Berwald metrics on M.
+Here, we give a two-dimensional Randers metric F = α +β with vanishing S-curvature. We
+show that if F is a generalized Berwald metric then it reduces to a Riemannian metric.
+Example 4. Let y = u∂/∂x + v∂/∂y ∈ T(x,y)R2. Consider the Randers metric F = α + β,
+where α = α(y) and β = β(y) are given by
+α :=
+��
+1 + (1 − ǫ2)(x2 + y2)
+�
+(u2 + v2) +
+�
+1 + ǫ2 + x2 + y2
+�
+(xv − yu)2
+�
+1 + (1 − ǫ2)(x2 + y2)
+��
+1 + x2 + y2
+,
+β := −
+ǫ(xv − yu)
+1 + (1 − ǫ2)(x2 + y2),
+
+On a Class of Generalized Berwald Manifolds
+17
+and ǫ is a real constant (see [15]). F is defined on the whole sphere for |ǫ| < 1. It is remarkable
+that, one can rewrite F in a polar coordinate system, x = r cos(θ), y = r sin(θ). Express
+α =
+�
+a11µ2 + a12µν + a21νµ + a22ν2,
+β = b1µ + b2ν,
+where
+a11 =
+1
+(1 + r2)
+�
+1 + (1 − ǫ2)r2�,
+a12 = a21 = 0,
+a22 =
+r2(1 + r2)
+�
+1 + (1 − ǫ2)r2�2,
+b1 = 0,
+b2 = −
+ǫr2
+1 + (1 − ǫ2)r2.
+Let us put A := det(aij). Then, we get
+a11 =
+r2(1 + r2)
+A
+�
+1 + (1 − ǫ2)r2�2,
+a22 =
+1
+A(1 + r2)
+�
+1 + (1 − ǫ2)r2�,
+a12 = a21 = 0,
+b1 = 0,
+b2 = −
+ǫr2
+A(1 + r2)
+�
+1 + (1 − ǫ2)r2�2.
+Therefore, we obtain
+∥β∥2
+α = aijbibj = bibi =
+ǫ2r4
+A(1 + r2)
+�
+1 + (1 − ǫ2)r2�3 ̸= constant.
+(4.2)
+(4.2) means that F is not a generalized Berwald metric. On the other hand, a direct computa-
+tion yields
+r11 = r22 = 0,
+r12 = r21 =
+ǫ3r3
+(1 + r2)
+�
+1 + (1 − ǫ2)r2�2,
+s11 = s22 = 0,
+s12 =
+ǫr
+�
+1 + (1 − ǫ2)r2�2 = −s21
+s1 =
+ǫ2r
+(1 + r2)
+�
+1 + (1 − ǫ2)r2��,
+s2 = 0,
+r1 = −
+ǫ4r5
+A(1 + r2)2�
+1 + (1 − ǫ2)r2�4,
+r2 = 0.
+It is easy to find that rij + bisj + bjsi = 0. Then by Lemma 3.2.1 in [5], we get S = 0. Also,
+one can see that the following holds
+ri + si = ǫ2r
+�
+A(1 + r2)(1 + (1 − ǫ2)r2)3 − ǫ2r4�
+A(1 + r2)2�
+1 + (1 − ǫ2)r2�4
+.
+(4.3)
+According to (4.3), F is a generalized Berwald metric (equivalently, ri + si = 0) if and only if
+ǫ = 0 or the following holds
+�
+1 + (1 − ǫ2)r2�4 = 0.
+(4.4)
+(4.4) contradicts with |ǫ| < 1. Therefore, F is a generalized Berwald metric if and only if ǫ = 0
+or equivalently β = 0. In this case, F reduces to the standard Riemannian metric F = α.
+
+On a Class of Generalized Berwald Manifolds
+18
+Example 5. (Xu) It is proved that a Finsler metric F = F(x, y) is of Randers type F = α+β if
+and only if it is a solution of the navigation problem on a Riemannian manifold (M, h) (see [5]).
+Zermelo navigation is an efficient method to study of Randers metrics with certain Riemannian
+and non-Riemannian curvature properties. More precisely, any Randers metric F = α + β on
+a manifold M is a solution of the following Zermelo navigation problem
+h
+�
+x, y
+F − Wx
+�
+= 1,
+where h =
+�
+hij(x)yiyj is a Riemannian metric and W = Wi(x)∂/∂xi is a vector field such
+that
+h(x, −Wx) =
+�
+hij(x)Wi(x)Wj(x) < 1.
+In fact, α and β are given by
+α =
+√λh2 + W0
+λ
+,
+β = −W0
+λ ,
+respectively and moreover,
+λ := 1 − ∥W∥2
+h,
+W0 := hijWiyj.
+For more details, see [5]. Then, F can be written as follows
+F =
+�
+λh2 + W2
+0
+λ
+− W0
+λ .
+(4.5)
+In this case, the pair (h, W) is called the navigation data of F.
+Now, let G/H be any homogeneous manifold and g = h + m is its reductive decomposition.
+Suppose m = m0 + m1 be an Ad(H)-invariant decomposition, in which m0 is 1-dimensional
+and the Ad(H)-action on m0 is trivial. Let h be a G-invariant Riemannian metric on G/H,
+such that m0 and m1 are h-orthogonal to each other. Let W be a G-invairant vector field on
+G/H, such that W(o) ∈ m0\{0}. Then, the navigation process with the data (h, W) provides
+a G-invariant generalized Berwald Randers metric with S = 0 (see [7]).
+Example 6. (Xu) As we mentioned in Introduction, the Bao-Shen’s Randers metrics on S3 are
+concrete generalized Berwald metrics, namely they are not Berwaldian. Any non-Riemannian
+homogeneous Randers sphere S3 = SU(3)/SU(2) (including Bao-Shen’s Randers metrics) sat-
+isfies S = 0 with constant pointwise ∥β∥α-norms. Then, every non-Riemannian homogeneous
+Randers sphere is a generalized Berwald metric. An S3 × S1, in which S3 = SU(3)/SU(2) and
+the navigation field is tangent to the S3-factor, is a 4-dimensional generalized Berwald Randers
+metric (see [29]).
+Acknowledgments: The authors are so grateful to Ming Xu for his valuable comments on
+this manuscript. Likewise, we thank him for providing us with examples 5 and 6 which improve
+the quality of our manuscript. Also, we are thankful to Behzad Najafi, Mansoor Barzegari and
+Libing Huang for their reading of this manuscript and their comments.
+
+On a Class of Generalized Berwald Manifolds
+19
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+Akbar Tayebi and Faezeh Eslami
+Department of Mathematics, Faculty of Science
+University of Qom
+Qom. Iran
+Email: akbar.tayebi@gmail.com
+Email: faezeh.eslami70@gmail.com
+

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+British Journal of Social Policy (2022), 2, 1–18
+doi:–
+British
+Journal of
+Social
+Policy
+ARTICLE
+Quantify how space mission influence geopolitical
+dynamics? A security and social policy approach.
+Andrea Russo*,1,2 and Davide Coco3
+1University of Catania, Italy
+2University College Dublin, Ireland
+3University of Rome, Sapienza University, Italy
+*Corresponding author. Email: Andrea.russo@phd.unict.it
+(Received –; revised –; accepted –; first published online –)
+Abstract
+We present a computational method to quantify the geopolitical impact of a space mission, based on the
+national budget and data logs of previous mission, and evidencing how even if some missions succeed,
+they can bring negative effect to the sponsored country. The objective of this research is to study how the
+success (or failure) of a space mission can bring an economical and political benefit (or loss) to a country.
+By retrieving various data, including sentiment from #hashtags related to the considered space missions,
+national budgets for space exploration, and the reliability of space launch systems, from social networks,
+public institutions, and online repositories, we propose an equation to evaluate the geopolitical importance
+of a space mission for a particular country or space agency. The geopolitical equation can be used by public
+institutions or private companies to estimate the potential impact of a space mission on the public opinion
+and international relationships, which can be either positive or negative, as even successful missions may
+negatively affect the international relationships and negotiation with some countries and their partners.
+Also we combine the ideology of classic social policy with a security and space mission point of view,
+to enlighten cultural, institutional, and political limits in public spending decisions.
+Keywords: Geopolitical dynamics, Space Mission, Social policy, Political dynamics and Computational methods
+1.
+Introduction
+———————————-
+Since the end of World War II, the proposal of ambitious space programs by governments and
+national space agencies has always been a means not only to push forward space exploration and
+research but also to alter the prestige and geopolitical influence in the international context. For
+instance, the social policy action by the 35th United States president John Fitzgerald Kennedy to
+invest $25 billion (1961 US dollar value) in the Apollo mission [Bozzo 2018] was not only a social
+investment to the increase public work with high qualification skills, but also a geopolitical plan
+action against the URSS space expansionism.
+Geopolitical value appears since the first space missions, for instance, after Russian first space satellite
+"Sputnik 1" successfully orbited the Earth. The Space Race that characterized the 20th century, was
+actually a geopolitical and propaganda race to determine which country would have finally had
+© Cambridge University Press 2022. This is an Open Access article, distributed under the terms of the Creative Commons Attribu-
+tion licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium,
+provided the original work is properly cited.
+arXiv:2301.03538v1  [physics.soc-ph]  9 Jan 2023
+
+2
+Andrea Russo et al.
+access (and conquered) the “new and endless world above us”. This geopolitical space race has been
+sustained by a huge effort from a social policy prospective. The $25 billion USD for the Apollo
+mission on 12th September 1962 (same day of the “Address at Rice University on the Nation’s Space
+Effort” speech by United States President John F. Kennedy to further inform the public about his plan
+to land a man on the Moon before 1970), are the equivalent of $231 billion USD on 7th February
+2022 [Appendix 1].
+Nowadays, superpowers like the United States, China, or the Russian federation, have increased
+the frequency of space missions to show their presence and value in a geopolitical and international
+perspective. The surge of space missions’ proposals in the last decades was due also to the cost of
+access to space, which significantly decreased thanks to the development of reusable launch systems,
+performing hardware, and IT and IoT improvement.
+Space missions have attracted huge money investments by public and private actors, with a social
+and business impact, due to the their potential economic return and their socioeconomic impact, as
+the design of a space mission encourages public high quality work and many public services originate
+from space activity (GPS, global mapping, high speed connection, global communication, and many
+others) [ASI 2020].
+Given the aforementioned result, thus the spin-offs of space services to the population, can we
+define space missions (both research and security missions) as social policy, given the socio-economic
+effects and the infrastructure-related services?
+The security policy of the US government in early 2000s launched by the “Pentagon” (i.e., the
+headquarters building of the United States Department of Defense) highlights how the difference
+between social policy and security policy became less evident. Indeed, the financial investment was
+aimed at developing a security program to defend social and public infrastructures and resulted in
+the X37-B program, a small shuttle that could defend other satellites, which provide social services,
+from Russian and Chinese physical and cyber-attacks, thus interlacing the security and social aspects
+[Spagnulo 2020]. Today, the relationship among social policy, security, space missions, and geopoli-
+tics is more intricate and complex than we could have imagined years ago.
+Designing a space mission is an extremely difficult task, with a high probability of failure due
+to the complexity of aerospace systems and the harsh conditions under which they are supposed to
+operate. The launch system plays an essential role in a space mission, as rockets must be “perfect”
+systems that respond seamlessly to all the perturbations that they experience during the atmospheric
+ascent and the exoatmospheric flight [Benedikter et al. 2021, 2022]. Every phase of the ascent
+trajectory must be carefully studied and planned before flight, as the margin of error is extremely
+small, even for the apparently simple scenarios. A first distinction by difficulty in space launches can
+be made by considering suborbital flights and orbital flights. In the former, the greatest difficulty
+is reaching a high altitude and then re-enter the atmosphere before completing a full revolution
+around the Earth. Suborbital flights generally cross what is called the Karman line, an imaginary line
+at an altitude of 100 km (330’000 ft) above sea level that conventionally marks the boundary between
+the Earth’s atmosphere and outer space. Although it is not a necessary requirement, the apogee (i.e.,
+the maximum altitude) is a benchmark for more or less complex suborbital flights, as it indirectly
+determines the speed that the vehicle will reach during re-entry and therefore the thermal and
+mechanical stresses that it will undergo. For orbital flights, instead, the inherent difficulty depends
+on the target orbit that the payload must be released into. The range of possible scenarios increases
+enormously when it comes to orbital flights from one planet (or celestial body) to another. For
+instance, a flyby, that is, the short passage of a high-speed probe near a celestial body, is seen as
+an extremely critical moment compared to the simple time spent cruising. On the other hand, the
+orbit insertion of a probe around a celestial body is a more critical moment than the flyby because it
+
+British Journal of Social Policy
+3
+requires several active maneuvers that involve multiple simultaneously operating systems. In a similar
+way, landing on a planet or asteroid is even a more critical accomplishment, as it involves much more
+complex operations.
+This paper is inspired by the growing amount of usable data from social and space science.
+During recent years, social science has acquired methods and skills to collect data and use it like hard
+science to highlight and identify social patterns or social dynamics. Information technology has also
+increased the quality of social research, thanks to huge advancements in algorithms and data analysis.
+This social-informatics improvement allows social science to connect with others disciplines, such as
+space science, engineering, and complex systems. Social science can finally prove or evidence social
+complex interactions in political science and others social disciplines. For example, in this work, we
+try to evidence a complex interaction between space missions and international cooperation, based
+on risk-cost-benefit analysis and political affinity.
+Due to the great inherent complexity of aerospace projects, international cooperation allows for
+mitigating the risks and costs (both financial and time-related) of space missions. There are several
+examples that show that the cooperation among national space agencies or research institutes has
+brought benefits to all the parties involved, not only relative to the economic return of scientific
+discoveries and patented technologies, but also to a positive outcome in terms of reputation and
+geopolitical prestige associated with these missions.
+Besides the technical aspects, the organization and management of a space mission are also
+quite challenging because every political interaction or action during the mission has a series of
+emerging behaviors in international affairs, and nonlinear interactions affect also reactions on political,
+economical, and security layers, which go beyond the space system [Dittmer 2014]. Complexity
+science tries to explain the dynamics of a system (e.g., physical, biological, social, or economical) that
+bring a different result than the expected one. A simple way to define a complex system is when the
+whole system is bigger than the sum of its parts. Since space missions are related not only to social
+and security policies but also to international relationships and geopolitical dynamics, then they can
+be considered as complex systems.
+Social policy, social network activity, and space exploration are just a slice of the whole part,
+but sufficient to understand how space missions influence the geopolitical strategy. In complexity
+sciences, the online social network activity patterns consist of successive, spike-like perturbations,
+generated by consecutive shocks bearing conceptual similarities to the tremors preceding or following
+an earthquake [Lymperopoulos 2017]. Indeed, the earthquake’s dynamics follow the Self-Organized
+Critically (SOC) model [Bak and Chen 1991], present not only in nature but also in social systems
+[Dmitriev, Dmitriev, and Balybin 2019].
+These dynamics could have been studied only with an hard science methodology and high quality
+data, which, in the pasts years, could have been very hard to collect.
+In this paper, we used computational method to collect high quality data with a rigorous
+methodology to understand geopolitical and public policy effect from space mission. More precisely,
+we acquired data from social networks like Twitter, to evaluate the sentiment of space missions, which
+evidences the social reaction about the related space event. The sentiment analysis has been used by
+many others scientist over different research areas, and, if combined with a high quality computational
+method, it could highlight important patterns related to social events. This methodology has been
+called Computational Social Science (CSS) [Cioffi-Revilla 2014].
+1.1
+Related Work
+Computational Social Science, since its introduction, provided a valuable approach to asses and
+explain social dynamical events. In this short section, we present some noteworthy works, relative to
+
+4
+Andrea Russo et al.
+computational social complexity and sentiment analysis, helpful to understand our project.
+Joseph Downing and Richard Dron [Downing and Dron 2020] have contributed to the under-
+standing of constructivist security by analysing social media outputs to understand who is influential
+in the security debate. Jie Yin et al. [Yin et al. 2015] have focused on analyzing Twitter messages
+generated during humanitarian crises and disasters. They presented relevant methods for burst
+detection, tweet filtering and classification, online clustering, and geotagging to classify whether or
+not a tweet is talking about a disastrous event. A disaster type classifier groups tweets according to
+representative disaster types: earth-quake, flooding, fire, storm, other disasters (e.g., traffic accident
+and civil disorders), and non-disaster.
+Sancheng Peng et al. [Peng et al. 2017] have worked on a framework that quantifies social
+influence in mobile social networks (Phones). The social influence of users was measured by analyzing
+the SMS/MMS-based communication behaviors among individuals. They have also revealed and
+characterized the social relations among mobile users through the analysis of the entropy of friend
+nodes and the entropy of interaction frequency. The extensive analytical results demonstrate that
+the influence spread of their proposed method outperforms a random method and a degree-based
+method.
+Kolli et al. [Kolli, Balakrishnan, and Ramakrishnan 2017] have quantified the predictability of
+cascade volumes in online social media communications, and tried to understand to what degree
+are future cascade trajectories predictable and to what degree are they random. The predictability
+analysis in their work reveals that for methods that combine information on frequency with temporal
+correlation of the trajectory, provides the theoretical limit on predictability. Hence, their methods
+such as AR and ARMA models that rely on the time order of data have the potential to achieve
+prediction accuracy as high as 83% in the MemeTracker dataset and 87% in the Twitter Hashtag
+dataset. Correspondingly, these methods have the potential to achieve prediction accuracy as high as
+94% in the MemeTracker dataset.
+Cihon and Yasseri [Cihon and Yasseri 2016], have written a short review that considers a small
+number of selected papers on computational social science related to sociology and social science; they
+analyze their theoretical approaches, review their methodological innovations, and offer suggestions
+given the relevance of their results to political scientists and sociologists. They evidence how the
+sentiment analysis content using semantic and sentiment analytic algorithms on individual users
+opinions from the 15M movement in Spain ( analyzing up to 200 tweets on the topic per user) is a
+promising technique for future studies of political activity, and, indeed, any activity, on Twitter.
+Vidgen and Yasseri [Vidgen and Yasseri 2020] built a multi-class classifier that distinguishes
+between non-Islamophobic, weak Islamophobic, and strong Islamophobic content. Accuracy is
+77.6% and balanced accuracy is 83%. They applied the classifier to a dataset of 109 488 tweets
+produced by far right Twitter accounts during 2017. While most tweets resulted not Islamophobic,
+weak Islamophobia was considerably more prevalent (36 963 tweets) than strong (14 895 tweets).
+1.2
+Geopolitical Efects of Space Missions
+Sentiment analysis has been used in different research areas and with different goals. In this paper, we
+apply it to assess the geopolitical effect of space missions by collecting data from latest and legacy space
+mission that have or have not been successful. Improving the success rate of space missions implies,
+from a geopolitical standpoint, an improvement of the international status. However, on the other
+hand, a failure can damage the relationship among international partners. The success of the Apollo
+11 mission by NASA made the United States the winner of the space race and raised its geopolitical
+value even though many milestones were reached earlier by the URSS (first orbiting satellite and first
+human in space, to name a couple). Recently, numerous space missions were successfully launched and
+fulfilled their planned goals or even performed beyond expectations, receiving positive reception from
+the public opinion and altering (or consolidating) the international status of the involved countries.
+
+British Journal of Social Policy
+5
+Noteworthy examples of recent successful missions are the Ingenuity helicopter, the first helicopter
+to fly outside the planet Earth, which was sent to Mars together with the Perseverance rover as part
+of NASA’s Mars2020 mission, the Rosetta mission, which carried Philae, the first spacecraft ever to
+accomplish a soft landing on the surface of a comet (67P/Churyumov-Gerasimenko), launched in
+2014 by ESA (European Space Agency), the latest James Webb Space Telescope, placed in a very
+challenging orbit (Sun–Earth Lagrangian point L2) in 2021 by NASA and ESA, and the Chang’e 4
+mission, featuring the first soft landing rover on the far side of the Moon, lunched in 2018 by CNSA
+(China National Space Administration).
+Even when a mission succeeds, there may be criticism from the society or even consequences
+and repercussions from others international actors, affecting the geopolitical status and international
+relationships of the sponsoring country. A computational social science methodology [Lazer et
+al. 2020] [Lazer et al. 2009] [Cihon and Yasseri 2016] could assess how much space missions could
+increase or decrease (in case of failures or partially successful missions) the geopolitical status of each
+country. In particular, the reaction of people on social networks generally gives a "feel strong" status
+to each country during international negotiations. Also, a statistical qualitative evaluation of the
+success-to-failure ratio of space missions and rocket launches indicates the reliability of space launch
+systems and mission design and management of every country. The geopolitical value depends on
+other people’s perception of strength. Therefore, if many people make negative comments (or have a
+negative sentiment) toward something that has been accomplished by some entity, implicitly it will
+come to change other people’s perception of it, and thus also its geopolitical value.
+Over the past decades, the goal of numerous space missions was not only to meet the social
+policy agenda of each country or related to scientific investigations, but also to reinforce the strength
+of international relationships between countries. For instance, the ExoMars mission has helped
+reinforcing the relationship between ESA and Roscosmos, the Russian space agency, before the
+Russian-Ukrainian war. Likewise, the DART mission is a cooperation between ASI (Italian Space
+Agency) and NASA, which strengthened the bond between European countries and the United States
+in space operations. The number of cooperative space missions holds the promise to significantly
+increase in the near future, not only to strengthen and seal international relationships, but also to
+reduce the cost and time needed to design and accomplish a space mission.
+The United States was the first country to understand this new political frontier [Bozzo 2018],
+and its actions have paved the way (from the industrial side) for the design of new engines and
+the world’s first reusable rocket, giving a huge advantage from the other space competitors. The
+objective of this paper is to quantify the geopolitical score of each state, and evaluate how much it
+may fluctuate depending on space missions.
+1.3
+Paper Outline
+This paper is organized as follows. Section 2 introduces the research hypotheses on space missions
+and geopolitical dynamics. Section 3 describes the methodological approach. In Section 4, we present
+the considered data sets and discuss the results. Section 5 explores the opportunity for a Space Mission
+Proposal by ESA, which can improve its geopolitical score. A conclusion section ends the manuscript.
+2.
+Research hypotheses
+Our research hypothesis, is to establish whether it is possible to create an equation that gives a
+geopolitical scores inherent to space missions, since, we think that as a ’social policy’, are included all
+those services deriving from space missions that also increase the security and geopolitical prestige
+of the state. To equate the outcome of space missions, as a public policy, is not so wrong given the
+increases in public services and highly skilled labour that are initiated at the beginning and end of
+each space mission. Defence missions can also be part of an increase in work and services to citizens.
+Indeed, the US, Russian Federation, China, and EU countries use defense budgets to organize space
+
+6
+Andrea Russo et al.
+missions for security and communication operation. Space missions can provide useful information
+to the state, companies and citizens, such as weather alerts to prevent catastrophes, prediction and
+prevention of potentially dangerous events, increased communication areas (internet and telephones)
+and also precise information on possible antagonists of the state to which they belong. In fact, over
+the past decades, many administrations have financed security space mission to obtain data about
+their competitors (using for example spy satellites) or just to amplify the satellite communication
+network. Anyway, the political tension about the dynamical and unpredictable security situation
+between US and China, did not touch only the military and economic sector, but have reached also
+the space and intelligence system [Giannangeli 2022]. As a matter of fact, Admiral Rob Bauer, the
+Dutchman who since June 2021 chairs the NATO Military Committee, the highest military body
+of the Atlantic Alliance, did not turn around when it comes to describing the framework of the
+challenges pressing on the Euro-Atlantic area. He said that "Russia and China are increasing their
+respective military capabilities, conventional and nuclear, space and cyber: it’s a fact, not an opinion."
+[Pioppi 2021]
+However, space missions can also be used to improve conditions between states. Indeed, inter-
+national events, such as space missions, are often used as a time to reconcile or strengthen relations
+between states. In fact, projects between the US and ESA are not only tools for providing jobs and
+services to citizens, thus moving the economy, but are also areas of cultural inclusion and knowledge,
+which is jealously guarded, especially the last one since it can be copied by foreign competitors.
+There can therefore be real international as well as economic strategies to get a space project off the
+ground.
+In order to calculate, how space missions have a geopolitical impact, hence on security and citizen
+services, we took a large part of the space missions inherent to the various countries.
+2.1
+Considered Space Missions
+We searched which kind of recent (i.e., after social networks existence) space missions had had
+repercussions in the geopolitical and security domains, and we collected data for the missions
+reported in Table 1. The list includes only missions sponsored by countries that develop launch
+vehicles, as launch success and failure data are used as a way to estimate the country’s experience in
+space missions (see Section Methodology or Table 4).
+Table 1. Space mission data collected
+Mission
+Country
+Result
+Tianwen-1
+China
+Succeeded
+Tianhe
+China
+Succeeded
+ASAT
+Russia
+Succeeded
+Mars 2020
+US
+Succeeded
+James Webb
+US/EU
+Succeeded
+Rosetta
+EU
+Semi-Succeeded
+The Tianwen-1 is a Chinese interplanetary mission to send a robotic spacecraft on Mars. It was
+launched July 23, 2020 [CNSA.gov 2020], and it landed on Mars on May 14, 2021 [CNSA 2021].
+The mission consisted of an orbiter, a lander, and a rover called "Zhurong". The mission succeeded,
+and NASA’s associate administrator Thomas Zurbuchen and the director general of Roscosmos
+Dmitry Rogozin congratulated the China National Space Administration (CNSA).
+China has also lunched the Tianhe core module, which is the first module of the Tiangong space
+station, on April 29, 2021 atop a Long March 5B rocket [S. CNSA 2021]. The core stage of the
+
+British Journal of Social Policy
+7
+LM-5B crashed back to Earth on Saturday, May 8, 2021, after 10 controversial days that captured
+the world’s attention and started a wider conversation about orbital debris and the responsibility for
+the return of spent stages.
+On November 15, 2021, Russia conducted a direct-ascent anti-satellite test (ASAT), destroying
+one of its own space objects, a defunct satellite, in low-earth orbit [BBC 2021]. The test captured
+international attention and was quickly and widely condemned as threatening and irresponsible
+action, not least for the cloud of uncontrollable debris it created, which will endanger both space assets
+and human spaceflight for years to come. Other countries in the past have already organised ASAT
+missions, like the Mission Shakti (India), the ASM-135 ASAT (US) and the 2007 Chinese anti-satellite
+missile test (China). Others to object in the wake of the test included Australia, the European Union,
+Japan, NATO, and South Korea. China and India – the two countries other than Russia and the
+US that have previously conducted destructive ASAT tests —- are yet to comment publicly. Also,
+following the Russian ASAT mission, the International Space Station started emergency procedures
+due to the debris, closing its security hatches while the crew sheltered.
+Mars 2020 is a Mars rover mission [NASA.gov 2021] that includes the rover Perseverance and
+a small helicopter called Ingenuity. It was launched from Earth on July 30, 2020 and landed in
+Martian crater Jezero on the 18th of February of the following year. The Mars 2020 mission is
+forming part of NASA’s Mars Exploration Program, which will continue with a sample return from
+Mars. Ingenuity is a robotic helicopter that demonstrated the technology for rotor-craft flight in the
+extremely thin atmosphere of Mars, becoming the first controlled helicopter on another planet. The
+budget for the Perseverance rover was US$2.8 billion in 2020 and was cheaper than its predecessor,
+the Curiosity rover, which costed $3.2 billion. Wikipedia 2022
+The James Webb Space Telescope [J. NASA 2020] is a space telescope designed primarily to conduct
+infrared astronomy. NASA led the development of the telescope in collaboration with ESA and
+CSA (Canadian Space Agency). The mission duration is expected to be about 20 years and, the time
+planning for all the missions was 20 years (10 for planning and 10 for realization). It was Launched
+on December 25, 2021 by the contractor Arianespace from the Centre Spatial Guyanais, with an
+Ariane 5 rocket. The James Webb space telescope had a total budget of USD ∼ 9.70 billion (2002
+to 2021) and has several scientific goals, including the search for light coming from the first stars
+and galaxies that formed in the universe after the Big Bang, the study of the galaxy, star, and planet
+formation and evolution, and studying planetary systems and the origins of life.
+The Rosetta mission [ESA 2014] was a space probe built by the ESA (European Space Agency)
+launched on March 2, 2004. The mission’s goal was to perform a detailed and comprehensive study
+of comet 67P/Churyumov–Gerasimenko (67P). On August 2014, the spacecraft reached the comet
+and performed a series of manoeuvers to eventually orbit the comet at distances of 30 to 10 kilometres.
+The probe also housed a lander called Philae, which unfortunately was unable to last long on the
+comet’s surface after a less-than-perfect landing. This was, indeed, the first mission landing on a
+comet. Yet, despite the problems, the probe’s instruments obtained the first images from a comet’s
+surface, and several instruments made the first direct analysis of a comet, sending back data that
+would be analysed to determine the composition of the surface. The mission cost was about ∼ 1.3
+billion € (US ∼ $ 1.8 billion).
+It is worth noting that even when the mission succeeds, it may happen that the mission goal has
+external or side effects that provoke a social or political reaction. Such reactions is spread, when there
+is something that does not belong to the common day-order, or that may provoke instability in
+
+8
+Andrea Russo et al.
+some country-system. We collect data from Twitter to see the social reaction to the space missions in
+Table 1. In all those cases, the missions succeeded, but, in some cases, the mission can cause a social
+reaction due to side effects.
+3.
+Methodology
+Online social networks have evolved into valuable sources of information and pervasive commu-
+nication platforms where people, businesses, and organizations generate and share content, build
+relationships, and join public conversations. In this online ecosystem, social networks are where
+information propagation is affected by external sources of influence, such as mass media, socioe-
+conomic circumstances, advertising, or events, giving rise to collective intelligence or collective
+behaviour patterns.
+We have collected social reactions from Twitter, for every mission in Table 1 and compared them
+with the difficulty and quality of the national space organization and the country that launched the
+space mission to quantify the "Authority and the status of power" status in geopolitical interaction
+dynamics. To simplify this operation, we create an equation called GSS, to quantify how space
+mission can influence the geopolitical feelings during international affairs dynamics.
+The geopolitical space score (GSS) index is the geopolitical score value from each country,
+depending on the result of the spaceflight mission, related to statistical and social events [Equation (1)].
+GSS =
+S
+G ∗ B ∗ (F + Q)
+(1)
+S is the Sentiment value from the Twitter event; B is the amount of money invested (budget); G is a
+difficulty rate associated with the country that launched the space mission, "not because they are
+easy, but because they are hard" that imply Geopolitical effects; F is related to the success or Failure
+of the mission; and Q stands for the statistical Quality and difficulty of the country in spaceflight
+launch organization.
+The S and Q parameters are evaluated by data scientists through statistical methods. The G score
+is the only parameter that needs a subjective value because it depends on personal evaluation and
+hypotheses to evaluate the difficulty rate of reaching that goal.
+3.1
+S Factor
+To collect data for the selected topic, we use the public API by Twitter and Tweepy. Both services
+use permission from Twitter to obtain and gather data. We collected a total of ∼ 7000 tweets, but
+any downloaded topic needs revisions and a cleaning process to increase the quality of the research.
+This has significantly reduced the volume of the tweets. We used the same methodology for each
+topic to obtain standard and quality data. In addition, to obtain the correct amount of tweets for
+each day we use getdaytrends.com, a specific site where it is possible to monitor every topic in real
+time as well as aged topics.
+To calculate the sentiment for the selected topic, we used the VADER sentiment analysis tools
+provided by MIT. The VADER sentiment quantifies each selected post or sentence’s negative, neutral,
+or positive sentiment, giving in the end of the analysis, a compound score, i.e. the average between
+all sentence.
+However, to evaluate the sentiment for every mission, it is necessary that each mission be very
+important to the general public or, at least, sufficiently viral among the space community.
+
+British Journal of Social Policy
+9
+3.2
+G Factor
+As mentioned above, the G factor is the only parameter without computational or statistical data.
+This parameter evaluates the country that launches the space mission, corresponding to a difficulty
+rate implying geopolitical effects. For example, if a small state succeeds in a mission with the same
+budget and other factors in comparison to a big state such as the US etc., the small state will get a
+bigger bonus.
+Unfortunately, there is no universal value factor that gives a score to quantify the difficulty of space
+missions.
+Geopolitics is a social evaluation, therefore deriving from the perceptual fluctuations of people.
+Since people make up a complex system, they cannot give a univocal value to the G factor, because it
+always depends on the value of the individual and on the oscillations (provoked by news, newspapers,
+friends, etc.) that modify the perception and evaluation of individuals and society on certain topics.
+Due to this problem, we decided to self-evaluate the difficulty of a space mission, even if it is
+hard to make an assessment that takes into account everything. There are many pros and cons, and,
+in each case, we know that people cannot evaluate sufficiently well the difficulty of a space mission.
+Yet, we believe that people can understand whether it is a surprise if a very small country alone (like
+Ireland or Pakistan) can accomplish, for example, to build a Martian base, and the US could not do it.
+We estimate a self-score from 0.1 to 1, where 0.1 is the maximum and 1 is the minimum possible
+score. For example, a sub-orbital lunch mission could have a 1 score, an orbital mission could be a 0.8
+score. The Tianwen-1 could be evaluated as 0.3, like the Martian Ingenuity helicopter on Mars. A
+full Moon base could be evaluated as 0.2 (or 0.1 if it is on Mars). All other space missions beyond the
+state-of-the-art technology level cannot be evaluated, because, we do not have sufficient information
+to be able to carry out the mission successfully, like a human base on the surface of Mercury, Titan
+or Europa, for example.
+3.3
+B Factor
+In the equation, we thought it was important to evaluate, then quantify, the level of resources
+invested versus the expected outcome (successful or failed mission). We thought this based on the
+logic "why should a mission cost a lot of money, while it is possible doing the same things while
+spending fewer resources?". To evaluate and quantify this factor, we collected data from the most
+important space agencies of each country. Since the experience gained in the design and management
+of past missions can be exploited in newer missions, we chose not to use a specific budget for each
+mission in the equation, but, rather, the annual funds dedicated to space missions. In this logic, it
+is hard to quantify how much money the oldest mission have helped (with knowledge, moneys,
+competence, technology) the newest space mission. Also, huge space missions like the JWST, or the
+Rosetta mission’s budget, grows up during years. The budget for the mission is spread over years,
+and we cannot only take a single space mission budget, because there are also many other funded
+missions different from the JWST in the same year. Therefore, we thought that the Budget factor,
+imply also public opinion logic. Public opinion is usually skeptic about spending money for space
+mission, because they did not (unfortunately) see the huge policy investment on research, security
+and works employments, "Rockets don’t run on cash". And nowadays, is still di��cult to quantify
+the investment return (knowledge, moneys, competence, technology) form the policy investment
+from space mission. In addition, the budget factor imply also a economic strength from the country
+that invest on space mission, For example, in the equation, if a small country achieve a successful
+space mission with a low budget, the GSS score will be higher than a the same country (or a bigger
+country) will achieve the same mission with a higher budget.
+However, to give an insight into space investment, in 2020, the policy plan for the major gov-
+ernment in space mission amount of a $73.98 billion, and it is the ∼ 0,927 % as a share of gross
+
+10
+Andrea Russo et al.
+domestic product (GDP), with a medium of ∼0,115875 % for each country. The Organisation for
+Economic Co-operation and Development (OECD) as show the total value of space budgets from
+the G20 country [Table 2], and we add in comparison, the years military budget for each country.
+Table 2. G20 government space and military budgets (2020)
+Country
+2020 Space Budget in Billions
+∼ National Space Budget %
+∼ National Military Budget %
+US
+22.62
+0.480%
+3.74 %
+Russia
+3.58
+0.210 %
+4.26 %
+France
+4.04
+0.122%
+2.07 %
+Japan
+3.32
+0.076 %
+1 %
+Saudi Arabia
+2.1
+0.076 %
+8.45 %
+China
+8.85
+0.075%
+1.75 %
+Italy
+2.0
+0.069 %
+1.56 %
+Germany
+2.40
+0.049 %
+1.4 %
+% in billion U.S. dollars [Appendix 2]
+3.4
+F Factor
+The factor F, was needed to the equation to weigh/ponder the space mission, and it shows if the
+mission succeeded or failed. The F factor is equal to 1 if the mission succeeded, or 0 if it failed.
+3.5
+Q Factor
+The factor Q, evidence the statistical risk factor and the reliability for each Country about spaceflight
+missions. Usually, every space mission have a risk factor, like the Apollo mission had the 95% of failure
+[NASA 1965], but this information (if they still made it) is not available to the public. Therefore, since
+we cannot obtain data for every single mission, we hypothesize a different evaluation method, so we
+rely as a risk factor on collecting data on the success/failure of each country’s space launch. Those
+data give a specific statistical risk and reliability factor to each country, since year 2010. [Appendix 3]
+It can happen that some space mission fail. In this scenario, we had imagined a failure factor that
+influence as a feature on GSS equation. The failures factor arises when you make mistakes over and
+over again, and in this case you always lose trust from others. The "Success/Authority improvement"
+comes from maintaining your own (high) standard for as long as possible. We chose to not put the
+Failures Factor in the equation, because factor Q evidence all the space mission (satellite mission,
+supplying mission, scientific mission etc.), and not only the scientific space mission, like those we
+have chosen as subject of this paper (Table 1).
+A well know example for a Failures Factor, could be the Tianhe mission from China, that had
+unfortunately an uncontrolled stage reentry, and the vector crashed back to Earth without having
+the possibility to calculate the final crash site, due the amount of variables on the descent stage. Sadly,
+this inconvenience will arise often by the China, any time when they decide to add a core module
+on his space station, because the Long March 5B (Y2) cannot claim to get the core module in orbit
+(hooked to the Tianhe core stage), without losing control of the rocket on re-entry. [Jones 2021]
+The Long March 5B re-entry had provoke concern about the security for some city, because
+the rocket had an orbital inclination of 41.5 degrees, means the rocket body passes a little farther
+north than New York, Madrid and Beijing and as far south as southern Chile and Wellington, New
+Zealand, and could make it is reentry at any point within this area. With obvious concerns for those
+country.
+
+British Journal of Social Policy
+11
+4.
+Data & Result
+According to our model/equation, the most important and determinant score are the sentiment
+score, which refers as the reaction and evaluation about the space mission’s result. The resulting
+online activity give us a social input valuable to quantify and qualify the international social reaction
+about space mission. Through the analysis of online activity topics, we identified the sentiment that
+describing the dynamic between space activity mission and geopolitical dynamics. In table 3 we
+show the sentiment results (S) from the most recently and most know space mission of the last decade.
+Table 3. S factor - Sentiment analysis
+Mission
+Hashtag
+Sentiment
+Result
+Tianwen-1
+Tianwen-1
+0,46447
+Succeeded
+Tianhe*
+ChineseRocket*
+-0,05151
+Succeeded (Sides efects)
+ASAT
+ASAT
+-0,16607
+Succeeded (Sides efects)
+Mars 2020*
+Perseverance*
+0,428263
+Succeeded
+Mars 2020
+Mars2020*
+0,487525
+Succeeded
+James Webb
+JWST*
+0,480994
+Succeeded
+Rosetta
+Rosetta
+0,429542
+Semi-Succeeded
+* Actually, some hashtags are derived not from the mission itself, but from the ef-
+fect achieved (losing control of the Chinese rocket on re-entry) or the main sub-
+ject of the mission (perseverance).
+The results clearly evidence that an increase of side effect during mission, increases the negative
+score sentiments from the space mission community.
+Regarding the economical resources invested (B) we collect data from different source, and
+arranged it on USD Billions dollars. Table 4 was the difficult one to make, because it is hard to obtain
+data from not-direct-democratic state, like China, and also because the inclusion of both civilian and
+military space budget for security missions.
+Table 4. B factor - Space budgets
+Country
+2010
+2011
+2012
+2013
+2014
+2015
+2016
+2017
+2018
+2019
+2020
+2021
+Japan
+1.67
+1.59
+1.68
+1.6
+1.76
+1.56
+1.33
+1.32
+3.06
+1.34
+3.32
+4.14
+Russia
+2.4
+3.8
+-
+5.6
+4.39
+2.42
+3.18
+-
+4.17
+3.58
+3.58
+1.92
+EU
+4,19
+4,52
+4,56
+4,85
+4,85
+4,65
+5,95
+6,52
+6,35
+6,49
+5,52
+5,16
+China
+-
+-
+-
+-
+2.66
+-
+4.91
+-
+5.83
+8.00
+8.85
+10.28
+US
+18.72
+18.44
+17.77
+16.86
+17.64
+18.01
+19.3
+19.50
+20.73
+21.5
+22.629
+23.27
+In billion U.S. dollars [Appendix 2] [Appendix 3]
+Therefore, the data from the statistical failures presence in our equation, that mark the quality of
+space launch system for each country (Q) are shows in Table 5. The data are collected since the year
+2010 to 2021 (the 2022 is not finished yet), show the total number of Core Stage Manufacture send
+in space, between overall launch log outside brackets and failures inside brackets. In the end, we
+shows also the total launch and failures between 2010 and 2021 and the failures percentage. The
+failures percentage is the Q value in our equation, evidencing the failures probability to each space
+launch system-country for each launch.
+It is possible to see that the Chinese government have many more launch than the US and Europe,
+this is due to the low presence of space satellite (for communication and security mission) orbiting
+the Earth by the Chinese government. The Chinese government had invested a huge amount of
+
+12
+Andrea Russo et al.
+Table 5. Q factor - Statistical failures
+Country
+2010
+2011
+2012
+2013
+2014
+2015
+2016
+2017
+2018
+2019
+2020
+2021
+TOT
+Failures %
+China
+55(3)
+39(4)
+34(2)
+39(1)
+18(2)
+22(2)
+19(0)
+16(0)
+15(1)
+19(0)
+19(1)
+15(0)
+310 (16)
+5,2
+Russia
+25(2)
+17(0)
+25(0)
+20(1)
+20(1)
+19(1)
+27(3)
+35(3)
+32(1)
+26(2)
+32(4)
+31(1)
+309(19)
+6,1
+US
+43(2)
+35(3)
+19(0)
+29(0)
+28(0)
+21(0)
+20(2)
+20(0)
+17(0)
+13(1)
+15(1)
+15(0)
+275(9)
+3,3
+Europe
+6(0)
+5(1)
+6(1)
+8(1)
+9(0)
+9(0)
+8(0)
+7(0)
+5(0)
+8(0)
+7(0)
+6(0)
+84(3)
+3,6
+India
+2(1)
+2(0)
+6(0)
+7(0)
+5(1)
+7(0)
+5(0)
+4(0)
+3(0)
+2(0)
+3(0)
+3(2)
+49(4)
+8,2
+Japan
+3(0)
+4(0)
+2(0)
+6(0)
+7(1)
+4(0)
+4(0)
+4(0)
+3(0)
+2(0)
+3(0)
+2(0)
+42(1)
+2,4
+New Zeal.
+6(1)
+7(1)
+6(0)
+3(0)
+1(1)
+-
+-
+-
+-
+-
+-
+-
+23(3)
+13,6
+Iran
+1(1)
+2(1)
+2(2)
+-
+-
+-
+1(0)
+-
+-
+3(2)
+1(0)
+-
+10(6)
+60
+Total launch by year (total launch failures by year) [Appendix 4]
+money to get enough satellites to make public infrastructures work. Also, Russian launch operation
+have been increased since the 2011, due to the retirement of the space shuttle (last mission 21th July
+2011), and as result, the US astronauts had to traveling by Russian Soyuz spacecraft to get to the
+international space station. We have compared the data from the equation (S, T, R and Q parameters),
+and in Figure 1 and Table 6 we show the score after the mission succeeded or failed.
+Figure 1. GSS’s info-graphic of mains space mission
+As is easiest to see, there is a negative score (we have highlight it with red on Figure 1), due
+to the risk of the ASAT and the space debris rocket had on population on heart and on the ISS.
+Tianwen-1 and Rosetta mission have a high GSS score also because it was a first huge milestone
+for the respectively space agency (ESA and CNSA). These data evidence the difference between a
+successful mission, achieving its goal; wile a successful mission, still achieving is goal but with side
+effect.
+4.1
+GSS trend in time
+We have tried to quantify a GSS value for space Research Mission only between 2010 and 2021, for
+each country, due the fact that ASAT is a military space mission. So we have gather data from all
+
+Geopolitical Space Score
+Rosetta
+JWST
+Mars 2020
+Perseverance
+ASAT
+ChineseRock
+Tianwen-1
+-2
+0
+2
+4
+6
+8
+10British Journal of Social Policy
+13
+Table 6. Geopolitical Space Score
+Mission
+Geopolitical Space Score
+Tianwen-1
+9,547438889
+ChineseRock
+-0,208492857
+ASAT
+-0,659007937
+Perseverance
+2,198416733
+Mars 2020
+2,502628333
+JWST
+2,469102533
+Rosetta
+7,588575333
+factor for the equation, but unfortunately we did not get all the Sentiment Factor (S), because usually
+unknown space mission does not have much reaction on social network; As said before to evaluate
+the sentiment for each missions, those missions should be very important for the Humankind, or at
+least "virality" for the space community, and many mission unfortunately become known to major
+society only from newspaper and occasionally from TV news. To solve this problem, we estimate a
+medium 0,425 S factor from successful mission, and a medium -0,125 for failed missions. For the
+other well know mission (Table 1) we have used the original sentiment. Regarding the economical
+resources invested (B) toward the data accumulated, we did not get the full budget planning over
+years for China and Russia, so we estimate a progressive linear regression between the missing data
+on table 4.
+Appendix 5 - 6
+Figure 2. GSS - Research missions
+As Figure 2 shows, it is possible to see the fluctuations characterised by the missions. In addition,
+it is also possible to see the type of strategy and activity for each country/agencies.
+For example, ESA, which is characterised by the limitation of not having a launch station on the
+east coast, has specialised in international collaboration, and it is possible to see how ESA manages
+missions with fluctuations of about one year (given the many collaborations, especially with NASA).
+
+6
+5
+4
+.
+3
+GSS
+2
+1
+0
+-1
+2009
+2010
+2011
+2012
+2013
+2014
+2015
+2016
+2017
+2018
+2019
+2020
+2021
+China
+Russia
+·USA
+ESA14
+Andrea Russo et al.
+Moreover, it should be noted that the Rosetta mission, like the Tianwen-1 mission, could be designed
+as a "baptism of independence", derived from "not because they are easy, but because they are hard"
+philosophy; Thy was indeed, the first to do that, showing his competence and skill upon the others
+superpowers space country like US and Russia. China, on the other hand, is concentrating on a few
+space research missions, since it is a new superpower and has yet to stabilise its infrastructure and
+network telecommunication on space. Instead, the US have/had many active missions for a long
+time, in fact the score is relatively lower than the others precisely because of their resilience. People
+expect them to fail less than the others, so the astonishing/sensational felling can only be found in
+very difficult missions, or when they failed badly.
+Appendix 5 - 6
+Figure 3. GSS - Research missions cumulative
+5.
+Analysis
+The analyses support the expectation that: 1) that it is possible to numerically assess geopolitical
+factors; 2) that even in the case of a successful mission, there is the possibility of a negative geopolitical
+feedback; 3) that public spending on space missions can also positively improve the geopolitical level
+of the investing state; 4) that the geopolitical level can vary over time, depending on the success of
+the missions, proportional to the investment.
+An increase in the social policy inherent in space missions, in addition to providing public
+employment with low-high qualification (and therefore removing people from the poverty line),
+would improve the risk factor (Q Factor) decreasing it to each future space mission, thus improving
+the geopolitical sentiment and also the geopolitical weight of the state.
+In fact, as already mentioned, the most resilient state is the US. They are not first on the geopolitical
+level despite having a high number of missions both quantitatively and qualitatively, in indeed, this
+score is given by the fact that they rarely miss, thus showing a high quality work, given the quality
+of the workers.
+6.
+Conclusions
+Our study shows that can be possible using computational social method to quantify geopolitical
+dynamics for every country. In our case we have shown how space mission influence geopolitical
+
+20
+15
+10
+GSS
+5
+0
+-5
+2009
+2010
+2011
+2012
+2013
+2014
+2015
+2016
+2017
+2018
+2019
+2020
+2021
+China
+Russia
+·USA
+ESABritish Journal of Social Policy
+15
+dynamics, supported by a security and social policy approach. In a more specific way, we have
+demonstrated (1) A socio-physical method for assessing the geopolitical level of any country, based
+on its goal and its system-organization; (2) That space missions, even if successful, can bring negative
+sentiment, which goes to reflect on the geopolitical value to the state itself; (3) That social policy is
+also an investment in the defense and security system, and that increased investment of government
+spending on space missions, also has a spillover effect on geopolitical value. From a sociological and
+socio-physical point of view, through the contribution of a Computational social science model, the
+GSS equation can be used to evaluate the geopolitical level not only in the spatial domain, but also in
+other "competitions", such as sports competitions (Olympics, world sports championships) or music
+competitions or international wars, where there are enough social reactions (S Factor) and enough
+statistical data (Q - B Factor) to evaluate both the event and the organization/country participating
+in the event. Therefore, our work would emphasise the most general Political policy, and in the
+most specific way, the Social policy, as not only a economical financial aids, but also a more complex
+systems, related to complex dynamics. As we demonstrated that Social policy is also funding the
+defense, security, and geopolitical systems. It has become not only an aid to the population, but much
+more. Moreover, an increase in the spending policy for space missions (which we define as part of
+social policy), also has a geopolitical impact on the state that promotes them, and by reflection, to its
+society.
+7.
+Declaration of interest statement
+The authors declare no conflict of interest.
+8.
+Authors details
+8.1
+Andrea Russo
+Is a PhD candidate in Complex Systems at the University of Catania. He is currently working at the
+Department of Physics and Astronomy. He collaborated with CNR Ibam, he also has worked purely
+on projects involving technology and society. His main research field and interests are focused on
+the study and the development of Computational social method to explain social complexity, in
+particular field like Politics - Economics - Business and Defense-Security sector applications. ORCID:
+0000-0003-3816-0539
+Corresponding author. Email: Andrea.russo@phd.unict.it
+8.2
+Davide Coco
+M.Sc. in Space and Astronautical Engineering at University of Rome "Sapienza", with several years
+of experience in mission design and preliminary studies of space missions.
+His main goal is to further specialize and be involved in conceptual studies, system requirements
+definition, mission proposal writing, space system modeling and simulation, launcher or mission
+design and operations.
+He is currently leading the design and development of Cubesat’s subsystems in a small Italian startup,
+Human4Research.
+ORCID: 0000-0001-8010-9468
+Email: davide.coco@outlook.it
+Acknowledgement
+We thank Taha Yasseri for for his crucial help in the equation.
+
+16
+Andrea Russo et al.
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+Appendix 1.
+Equivalence of the 1962 USD to 2022 USD
+More specifically: 231 408 697 113.64 USD $ .
+Inflation over the period: 825.63 %
+Index used: USCPI31011913 (Bureau of Labor Statistics).
+Initial Index: 309.12, Final Index: 2 861.33
+Link=https://fxtop.com/it/conversione-valute-passato.php
+Appendix 2.
+Space annual budget (∼ approximately) over years and percentage of the na-
+tional budget
+It is very hard to collect data for the most important Space Agency, this is due to the different money
+value (Yen - EURO - USD), and also the difficult to obtain data from not-democratic state, like
+China, due to information security and also due to the inclusion of both civilian and military space
+spending global security.
+Source:
+1. https://www.statista.com/statistics/745717/global-governmental-spending-on-space-programs-
+leading-countries/
+2. https://www.euroconsult-ec.com/press-release/government-space-budgets-driven-by-space-exploration-
+and-militarization-hit-record-92-billion-investment-in-2021-despite-covid-with-1-trillion-forecast-
+over-the-decade/
+3. https://spacenews.com/op-ed-global-government-space-budgets-continues-multiyear-rebound/
+4. https://stacker.com/stories/2524/countries-spend-most-space-exploration
+5. https://www.oecd.org/sti/inno/space-forum/space-economy-for-people-planet-and-prosperity.pdf
+6. https://global,jaxa,jp/about/transition/index,html
+7. esa.int
+8. https://en.wikipedia.org/wiki/Budget_of_NASA
+9. https://global.jaxa.jp/
+[Salas 2021]
+Military budget:
+1. https://databank.worldbank.org/reports.aspx?source=2&type=metadata&series=MS.MIL.XPND.GD.ZS#
+2. https://www.defensenews.com/global/2021/04/26/the-world-spent-almost-2-trillion-on-defense-
+in-2020/
+Appendix 3.
+Billion U.S. dollars
+Exchange rate 1.1269 at 27/02/2022 02:50 28 Feb 2020 - 25 Feb 2022
+Appendix 4.
+Statistical failures
+Annual space reports:
+From 2010 to 2021 Launch Log
+Source = https://www.spacelaunchreport.com/index.html
+
+18
+Andrea Russo et al.
+Appendix 5.
+GSS - Research Missions
+Table 7. GSS related to space research missions
+Country
+2009
+2010
+2011
+2012
+2013
+2014
+2015
+2016
+2017
+2018
+2019
+2020
+2021
+China
+-
+-
+-0,541666667
+4,969230769
+5,383333333
+-
+-
+-
+-
+-
+1,615
+1,490577956
+-
+Russia
+-
+-
+-0,334429825
+-
+-
+-
+-
+2,389937107
+-
+-
+1,75719055
+-
+-
+USA
+-
+0,808621288
+0,464438894
+1,026104163
+-0,040508492
+0,492783694
+0,512710274
+0,976252159
+-
+0,776134433
+0,876356589
+0,011896792
+0,942333613
+ESA
+-
+4,647391567
+-
+-
+2,648339061
+-
+2,762246117
+1,128012708
+-
+2,654855643
+-
+1,903820817
+2,11289354
+Japan
+-
+4,542027057
+-
+-2,976190476
+-
+4,277597403
+-
+-
+-
+5,740740741
+-7,462686567
+-
+-
+TOT
+-
+3,332679971
+-0,137219199
+1,006381485
+2,663721301
+2,385190548
+1,637478196
+1,498067325
+-
+3,057243606
+-0,803534857
+1,135431855
+1,527613577
+GSS - Research missions
+Appendix 6.
+Country Succeeded and Failed Space Research Mission
+Table 8. Country Succeeded(Failed) Space Research Mission
+Year
+China
+Russia
+USA
+ESA
+Japan
+India
+2010
+-
+-
+Deep Impact
+Rosetta
+Akatsuki
+-
+-
+-
+Stardust
+-
+IKAROS (Shin’en)
+-
+2011
+(Yinghuo-1)
+(Fobos-Grunt)
+Dawn
+-
+-
+-
+2012
+Chang’e 2
+-
+MSL Curiosity
+-
+(PROCYON)
+-
+2013
+Chang’e 3
+-
+(Deep Impact)
+Gaia
+-
+-
+2014
+-
+-
+MAVEN
+-
+Shin’en 2
+Mangalyaan
+2015
+-
+-
+DSCOVR
+LISA Pathfinder
+-
+-
+-
+-
+New Horizons
+-
+-
+-
+-
+-
+Dawn
+-
+-
+-
+2016
+-
+ExoMars 2016 (Schiaparelli EDM lander)
+Juno
+ExoMars 2016 (Schiaparelli EDM lander)
+-
+-
+2017
+-
+-
+-
+-
+-
+-
+2018
+-
+-
+Parker Solar Probe
+MASCOT
+Hayabusa2
+-
+-
+-
+MarCO A "WALL-E"
+BepiColombo
+BepiColombo
+-
+-
+-
+MarCO B "EVE"
+-
+-
+-
+-
+-
+OSIRIS-REx
+-
+-
+-
+-
+-
+InSight
+-
+-
+-
+2019
+Chang’e 4
+Spektr-RG
+New Horizons
+-
+(Minerva II-2)
+-
+-
+-
+Spektr-RG
+-
+-
+-
+2020
+Chang’e 5
+-
+Mars 2020
+Solar Orbiter
+-
+-
+Tianwen-1
+-
+-
+-
+-
+-
+Beidou
+-
+-
+-
+-
+-
+2021
+-
+-
+James Webb
+James Webb
+-
+-
+GSS - Research missions
+

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+JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, AUGUST 201X
+1
+Multi-task Weakly Supervised Learning for
+Origin–Destination Travel Time Estimation
+Hongjun Wang, Zhiwen Zhang, Zipei Fan, Jiyuan Chen,
+Lingyu Zhang, Ryosuke Shibasaki and Xuan Song
+Abstract—Travel time estimation from GPS trips is of great importance to order duration, ridesharing, taxi dispatching, etc. However,
+the dense trajectory is not always available due to the limitation of data privacy and acquisition, while the origin-destination (OD) type
+of data, such as NYC taxi data, NYC bike data, and Capital Bikeshare data, is more accessible. To address this issue, this paper starts
+to estimate the OD trips travel time combined with the road network. Subsequently, a Multi-task Weakly Supervised Learning
+Framework for Travel Time Estimation (MWSL-TTE) has been proposed to infer transition probability between roads segments, and the
+travel time on road segments and intersection simultaneously. Technically, given an OD pair, the transition probability intends to recover
+the most possible route. And then, the output of travel time is equal to the summation of all segments’ and intersections’ travel time in
+this route. A novel route recovery function has been proposed to iteratively maximize the current routes’ co-occurrence probability, and
+minimize the discrepancy between routes’ probability distribution and the inverse distribution of routes’ estimation loss. Moreover, the
+expected log-likelihood function based on a weakly-supervised framework has been deployed in optimizing the travel time from road
+segments and intersections concurrently. We conduct experiments on a wide range of real-world taxi datasets in Xi’an and Chengdu
+and demonstrate our method’s effectiveness on route recovery and travel time estimation.
+Index Terms—Travel Time Estimation, Urban Computing, Weakly Supervised Learning
+!
+1
+INTRODUCTION
+With the emergence of newly-developed applications, es-
+timating travel time has become one of the hottest top-
+ics, which is of great importance to route planning, taxi
+dispatching, and ride-sharing in recent years. In the early
+phase, the data of real traffic state is mainly collected from
+loop sensors, which can only provide the individual travel
+time in a certain road segment and usually face the sparse
+issue. Recently, an alternative solution is to use floating
+car data. The floating cars equipped with GPS receivers,
+including taxis, buses, private cars, and online ride-hailing,
+record time stamps, longitude, latitude, speed, and other in-
+formation at regular intervals, which can reflect the vehicle’s
+operation status.
+As a result, a good deal of travel time estimation tech-
+niques based on floating car data have been proposed in
+different scenes, such as dense trajectory [1], [2], [3], low-
+•
+Hongjun
+Wang,
+Jiyuan
+Chen
+and
+Xuan
+Song
+are
+with
+(1)
+SUSTech-UTokyo
+Joint
+Research
+Center
+on
+Super
+Smart
+City,
+Department
+of
+Computer
+Science
+and
+Engineering
+(2)
+Research
+Institute of Trustworthy Autonomous Systems, Southern University
+of
+Science
+and
+Technology
+(SUSTech),
+Shenzhen,
+China.
+E-mail:
+wanghj2020,11811810@mail.sustech.edu.cn and songx@sustech.edu.cn.
+•
+Zipei Fan, Ryosuke Shibasaki and Zhiwen Zhang are The University
+of Tokyo, 5-1-5 Kashiwanoha, Kashiwa-shi, Chiba, 277-8561, Japan;
+emails: zhangzhiwen@csis.u-tokyo.ac.jp, fanzipei@iis.u-tokyo.ac.jp, and
+shiba@skl.iis.u-tokyo.ac.jp
+•
+Lingyu Zhang is Research Institute of Trustworthy Autonomous Systems,
+Southern University of Science and Technology (SUSTech), Shenzhen,
+China. emails: zhanglingyu@didiglobal.com
+•
+Corresponding to Zipei Fan, Xuan Song;
+•
+Hongjun Wang, Zhiwen Zhang equal contribution;
+sampling-rate trajectory [4], [5], [6]. However, due to the
+privacy concern and data acquisition problems, extensive
+works focus on inferring the travel time from OD data,
+which only gives the origin-destination location, such as
+finding nearby neighbors [7], such as distance-based [8] or
+representation-based [9] neighbors. In general, the OD type
+of data is more available than the dense type, and multiple
+sources of OD data have been released, for example, the
+NYC taxi data1, NYC bike data2 and Capital Bikeshare
+Data3. However, as far as we know, previous literature omits
+the factor of the road network, which often leads to a high
+estimating error. Since the total travel time of the trajectory
+is equal to the sum of the travel time of all road segments
+and intersections (e.g., waiting traffic signal). Each traffic
+condition in road segment change will affect the total travel
+time. With the road network introduced, here we face three
+intractable problems:
+1) How to recover the route when only OD pairs are given.
+2) How to effectively estimate the travel time when the route
+has been obtained.
+3) How to learn features from complex road network.
+At first glance, given a pair of origin-destination, the short-
+est path algorithm (e.g., Dijkstra’s algorithm) is a natural
+choice for problem 1) because people usually choose paths
+that are similar to the shortest path with less number of
+turns. However, the shortest path in the geometry aspect
+may not always match the definition of the ’shortest path’
+in the driver’s route choice. For example, some resident or
+1. https://www1.nyc.gov/site/tlc/about/tlc-trip-record-data.page
+2. https://data.cityofnewyork.us/Transportation/Bike-Data/374u-
+5ie7
+3. https://www.capitalbikeshare.com/system-data
+arXiv:2301.05336v1  [cs.AI]  13 Jan 2023
+
+JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, AUGUST 201X
+2
+tertiary types of road are shorter than primary and trunk
+types of road, but they are more vulnerable to congestion,
+since the complex traffic state (many pedestrians), or nar-
+rowness of road width. How to encode the road features
+into the road search procedure? One way can be done by
+learning the transition probability between road segments
+and inferring the route via the search for the maximum
+route probability, where the superiority of this approach is
+that the character of the road will be considered in every
+search process. Inspired by [10], this article employs the
+graph neural network (GCN) to learn the features, such
+as road type, road length, road sign, and road lanes, of
+each road segment. Consequently, the problem of ignoring
+natural road networks in the shortest-path algorithm can be
+alleviated. Fig. 1 shows an example of searching candidate
+path through transition probability. Based on the Markov
+assumption, the routing probability can be obtained by
+multiplying the probabilities and equal to the sum of the
+log probabilities. The candidate paths r1, r2, r3 are acquired
+by Depth First Searching (DFS) algorithm with pruning
+operation.
+As we mentioned, this paper infers the overall travel
+time of a given route by summing up the travel time of
+all the road segments and intersections on that route. This
+raises the question of how to estimate the travel time of road
+segments and intersections reasonably. One concern is how
+to model the differences in individual driving behaviors,
+since given a specific OD pair, the travel time in the same
+time interval is varied. To address this problem, we here
+model the travel time with uncertainty, which means that
+each road/intersection travel time follows certain distribu-
+tion (e.g., Lognormal). Those roads/intersections tend to
+be provided a large variance σ, for example, with large
+crowd flow. In conclusion, the uncertain travel time of
+route generated by route searching has been obtained. To
+effectively optimize the distribution, this paper formulates
+it as the inexact supervision learning problem [11]. One of
+the most well-known examples of inexact supervision is the
+drug activity prediction problem [12], which predicts if a
+molecule induces a given effect. Inexact supervision deals
+with training data arranged in sets called bags, and the labels
+are only annotated on bags. For modeling the uncertainty,
+given a pair of origin and destination, we consider the
+travel time label annotated on the unobserved routes (bags)
+and use the normal distribution [13], [14], [15] to model
+the uncertainty for each road segment and the interaction
+travel time in bags. We drive the objective function (Eq.
+(2)) based on the assumption of aggregated observation and
+Markov chain, and solve it with a general inexact learning
+probabilistic framework [16] and an iterative route recovery
+algorithm.
+After solution 1) and 2) has been discussed, we finally
+introduce how to learn the meaningful features from a
+complicated road network, since multiple factors will affect
+the traffic condition, such as road types, road lanes, speed
+limitations, and traffic signals. To learn the intricate relation
+of road networks, many existing works modeling the prob-
+lem of estimating travel time from either road segments [1],
+[3] or intersections [17], but do not assemble those features
+simultaneously. However, we argue that this will cause error
+accumulation with the road segments increasing. To fill this
+������������7
+Source
+Target
+������������1
+������������2
+������������3
+0.4
+0.6
+0.2
+0.8
+1
+0.9
+0.5
+0.5
+1
+0.1
+0.9
+Candidate paths:              
+������������1: ������������1 → ������������3 → ������������5 → ������������7 , ������������1 = log 2.5
+������������2: ������������1 → ������������3 → ������������8 → ������������5 → ������������7, ������������2 = log 2.2
+������������3: ������������1 → ������������2 → ��������������4 → ������������6 → ������������7, ������������3 = log 2.3
+������������4
+������������5
+������������6
+������������8
+������������9
+0.1
+Road-based graph
+Intersection-based graph
+������������1
+������������2
+������������8
+Road segment
+������������3
+������������5
+Intersection ������������2
+������������1
+������������2
+Intersection ������������1
+Road Network
+Fig. 1. The motivation in this paper is illustrated above. Given any OD
+pair, we want to recover the path by searching the maximum route
+probability learned by the model, and construct a dual graph from road-
+based and intersection-based aspects to estimate the travel time of road
+segment and intersection respectively.
+gap, in this work, we construct a dual graph comprised of
+the road-based graph and the intersection-based graph, and
+estimate the travel time by summing up all road segments
+and intersections on one route. In the meanwhile, we also
+take the connected relation4 into consideration from one
+road segment to one road segment, such as primary →
+secondary, or trunk → residential, and one intersection to one
+intersection, such as tertiary and residential. We introduce the
+Relational Graph Convolutional Networks (R-GCNs) [18] to
+learn the complex connected relation of the road network.
+Moreover, to solve the problem of losing local patterns
+when expanding the receptive neighborhood in GCN, we
+combine the Relational Graph Convolutional Networks, and
+gated recurrent unit (GRU) [19] together as the stacked
+architecture to capture both global and local features [20].
+Fig. 1 represents the procedure to construct the dual graph
+(gray box) of the road network and to recover the route
+from the candidate set that is searched from the transition
+probability.
+The main contributions of this paper can be summarized
+as follows.
+• We propose a multi-task framework to estimate the
+travel time of road segments and intersection, and
+transition probability simultaneously. To the best of
+our knowledge, it is the first attempt to recover the
+route and jointly model the factor of intersection and
+road segments in the general OD travel time estimation
+problem.
+• For the first time, we consider the estimation of the OD
+travel time as the weakly supervised learning problem,
+since the observation of the OD travel time is annotated
+with a bag of unobserved routes. This paper aims to
+infer each road segment and the intersection travel time
+distribution from the aggregation observation.
+• We validate the effectiveness of travel time estimation
+and route recovery using large-scale datasets from the
+real world in Chengdu and Xi’an, respectively, which
+significantly outperform current methods.
+Here, we list the organization in this paper: Sec. 2 gives
+the related works, including weakly supervised learning,
+travel time estimation, as well as route estimation. Sec. 3
+introduces the preliminary knowledge, such as the road net-
+4. https://wiki.openstreetmap.org/wiki/Key:highway
+
+JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, AUGUST 201X
+3
+work, the origin-destination. Sec. 4 provides the definition of
+our formulation, assumptions, and objective function. Sec.
+5 gives the methodology of our MWSL-TTE. Sec. 6 and
+Sec. 7 conduct the qualitative and quantitative experiments
+respectively to demonstrate the superiority of MWSL-TTE.
+Sec. 8 gives a summary of this paper and future work.
+2
+RELATED WORK
+In this section, we will discuss several relevant topics about
+weakly supervised learning, travel time estimation, as well
+as route estimation.
+2.1
+Weakly Supervised Learning
+Since the general supervised learning method requires each
+data in the training set to be labeled, this expensive la-
+beling consumes a lot of manpower and time. Therefore,
+learning under the condition of weakly supervised infor-
+mation has become a hot research topic in the field of
+machine learning in recent years [21], [22], [23]. The weakly
+supervised learning methods focus on addressing the low-
+quality labels scenarios 1) incomplete supervision [24] : only
+part of data can be labeled. 2) inexact supervision [12]:
+only have coarse-grained labels . 3) inaccurate supervision
+[25] : only part of the data owns true labels. The task of
+estimating travel time from OD can be considered as the
+inexact supervision belonging to the category of Weakly
+Supervised Learning, where the only observations are the
+total travel time and OD locations, but the accurate routes
+are unknown. Different from traditional approaches [7],
+[26] searching similar historical trajectories from data but
+ignoring the city road network structure, in this paper, we
+aim to infer the potential route from OD by using transition
+probability between road segments, where a set of potential
+routes can be seen as a bag, and the OD travel time can be
+obtained by summing up the estimated times of the road
+segments in bag. To the best of our knowledge, we are the
+first to introduce the problem of travel time estimation into
+the inexact supervision framework.
+2.2
+Travel Time Estimation
+Various TTE implementations were classified into three
+groups, traditional approaches, deep learning-based ap-
+proaches and graph neural network-based approaches. Tra-
+ditional approaches for TTE include the road-segment-
+based and path-based methods. The road segment-based
+methods [27], [28] coarsely forecast the route travel time
+by summing up all estimated times of roads by using the
+data collected from sensors like magnetometer detectors or
+highway cameras, which omitted the necessary factor of
+intersection and relationship among road segments. And the
+path-based methods address the above challenges mainly by
+nearest neighbors search [7], [26] and trajectory regression
+[28], [29], [30]. Nearest neighbors search (NNS) finds nearby
+historical trajectories according to the assumption that the
+routes with similar origins and destinations own close travel
+time. Trajectory regression methods predict the whole route
+travel time based on the given historical trajectories.
+Recently, deep learning-based approaches have become
+especially important in the task of TTE. These approaches
+can be divided into two groups, classical deep learning-
+based methods and graph deep learning-based methods.
+Some classical methods such as deep neural networks
+(DNNs) [1] and convolutional neural networks (CNNs)
+[2], [31] have been successfully applied in TTE. For exam-
+ple, Deep-TTE [1] proposed a CNN-based framework to
+integrate various types of attribute information (such as
+weather, time ID and driver ID) for TTE. However, most
+of these methods model the road network as a grid-based
+map, but they ignore the graphical structure of real-world
+road network.
+To fully utilize spatial information, GNN is an emerging
+tool to analyze the topological relations of graph-structured
+traffic data. Especially, Spatial-Temporal Graph Neural Net-
+work (STGNN) [32] is a framework that integrates GNN
+and temporal processing modules, which can handle spa-
+tial relations and temporal trends simultaneously. Due to
+the spatio-temporal characteristics of the real-world road
+network, STGNN are widely adopted in TTE. For example,
+diffusion convolutional recurrent neural network (DCRNN)
+[33] modeled the graph-structured traffic data as a diffu-
+sion process on a directed graph and transformed spatio-
+temporal features into a seq2seq framework. ASTGNN [34]
+proposes a trend-aware multi-head attention mechanism to
+capture multiple potential correlations in traffic forecasting.
+However, these works only consider the spatio-temporal at-
+tributes of road segments but ignore the interactive correla-
+tions between intersections and road segments. Meanwhile,
+both the real route of OD pairs and road condition also have
+an important influence on TTE.
+2.3
+Route Estimation
+Another bunch of research in travel time estimation is to
+solve the issue of sparse trajectory due to the privacy, busi-
+ness competition [4], [5], [6], and limitation of GPS devices,
+or in the scene of ETC [35], [36] and surveillance cameras
+[37]. A common strategy for solving the sparse trajectory is
+to infer the potential route based on the information of the
+road network. Reference [4] applies the inverse reinforce-
+ment learning to learn the latent cost (reward) of a road
+through historical data, and proposed Exact Route Search
+approach to find the maximum probabilistic route based on
+dynamic programming. However, route search-based algo-
+rithms are only adapted in low-sampling rate trajectories,
+but not OD problem, due to the heavy computational cost.
+Because of the large distance between a pair of toll stations
+or surveillance cameras, a frequently used path inferring
+algorithm is based on the Depth First Searching algorithm
+to find all possible simple paths that the one road segment can
+only appear at most once. However, those methods omitting
+the real traffic condition tend to generate the unreal route
+in the path inferring procedure. To this end, in this paper,
+we combine the transition probability and route search
+approach together to find the optimal route based on the
+real travel time and road network structure.
+3
+PRELIMINARIES
+We start with giving the definition about the road network,
+Origin-Destination, simple path as well as route.
+
+JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, AUGUST 201X
+4
+X
+A
+OD
+Z
+T
+W
+K nodes and links
+Fig. 2. The graphical model of the data generating process.
+Definition 3.1. Road Network. A road network is a directed
+graph G = (V, E, π), where V denotes the set of nodes, E ⊆ V×V
+is the set of directed edges, and π is a node’s feature set. This paper
+uses Gv and Ge to denote the node-wise and link-wise graphs
+respectively. For the π in Gv, the features of V can be such as,
+junction types, and traffic signals. For the π in Ge, the features of
+v can be such as, road types, road lanes, and speed limits.
+Definition 3.2. Origin-Destination (OD). In this paper, a OD
+pair represents a tuple with {ei, ej, tth}, where ei and ej denote
+the start and end road segment, respectively, and tth is the start
+time interval of a day (e.g., 8:00am-9:00am). Note that we assume
+the traffic conditions for all road segments and intersections are
+invariable within the same time slot.
+Definition 3.3. Simple Path. A simple path tr can be presented
+as a series of time-ordered links. We have tr : e1 → e2 → · · · →
+e|tr|, where each link satisfied ei ̸= ej.
+Definition 3.4. Route. A route r in this paper is a sequence
+alternating with links and intersections. We have r : e1 → v2 →
+e3 → · · · eK−1 → vK, where vi is the intersection of edges pair
+(ei−1, ei), and K is the length of r.
+4
+PROBLEM FORMULATION
+To overcome the previous issue with ignoring the road
+network in the OD-TTE problem, we here intend to give
+a formulation under the weakly supervised learning. This
+solution is motivated by the advance in weakly supervised
+learning and GCN path inference.
+Given a pair of origin and destination, estimating the
+travel time is to infer the total time cost. Traditionally,
+the formulation of TTE can be divided into two parts: 1)
+inferring the future traffic through historical state [3], [38],
+2) online infer traffic through real-time trajectories [39]. The
+previous one is mainly related to the robust traffic state,
+which is calculated on either dense trajectories [3] or loop
+detectors [32]. Another one intends to resolve the sparse
+issue by estimating (imputation) citywide-level traffic state
+from fewer real-time trajectories. Since this paper is under
+the OD scenario and hard to give a valid historical traffic
+state, we here follow the online TTE formulation. Formally,
+this paper follows the assumption that the traffic state at
+one road segment in a specific time interval ∆t = 60mins,
+e.g., 10:00 am-11:00 am, under the same distribution, such
+as, Gaussian distribution N(µ, σ) with µ = 60s and σ = 1.
+Subsequently, suppose the current time is t, we train the
+real-time OD pairs in time slot [t, t + ∆t1), the online traffic
+state will be completed and evaluated with the OD pairs in
+[t + ∆t1, t + ∆t).
+TABLE 1
+Partial Symbols Description.
+Notation
+Description
+A
+Links inner transition probability matrix
+Q
+Aggregate function
+ΩOD
+Set of candidate paths from O to D
+f
+Multi-task function
+X
+Features of links and nodes
+Z
+Unobservable travel time of links and nodes
+Gv and Ge
+Node-wise and link-wise graph respectively
+T and �T
+Ground truth and estimated of travel time
+T
+Time embedding vector
+W
+Learnable parameters
+About the training procedure of MWSL-TTE, Fig. 2 il-
+lustrates the data generating process with graphical repre-
+sentation. OD are the features of origin-destination loca-
+tions. X1:K5 stands for the features vectors of unobserved
+K nodes and links in route r : e1 → v2 → e3 →
+· · · eK−1 → vK, and Z1:K is the unobservable travel time
+of nodes and links. We assume that r is conditioned on OD
+and transition matrix A, where Ai,j indicates the transition
+probability from ei to ej. Therefore, we have p(r|A; OD) =
+p(X1:K|A; OD), where A is generated by multi-task func-
+tion f with A = f(WA; Ge), Ge is the link graph, and
+W is the learnable parameters. Meanwhile, Z also under
+a parametric distribution p(Z1:K|θz = f(X1:K; WZ; )) =
+p(Z1:K|(µ, σ) = f(X1:K; WZ; )) on the factors of X1:K and
+WZ, where µ and σ are the mean and variance of Gaussian
+distribution. For the relation between Z1:K and aggregate
+observation of travel time T, we have the following defini-
+tion:
+Definition 4.1. Given a route ri : e1 → v1 → e2 →
+· · · v|ri|−1 → e|ri| under a pair of OD, and it’s unobserved
+travel time Z1:K = {˜te1, ˜tv2, ˜te3, · · · ˜tvK−1, ˜teK}. The aggregate
+functions Q can be defined as
+Q(Z) = ˜te1 + ˜tv1 + ˜te2 · · · + ˜tvK−1 + ˜tK,
+(1)
+where an aggregate function Q : Z
+→ T is a mapping
+function from unobserved variable Z to observation T. Since
+we assume Z follows a Gaussian distribution, we can writ-
+ten T
+= Q(Z) = �
+i ˜ti = �
+i µi according to the na-
+ture of additivity: X + Y
+∼ N
+�µ1 + µ2, σ2
+1 + σ2
+2
+�
+, where
+X ∼ N
+�µ1, σ2
+1
+� Y ∼ N
+�µ2, σ2
+2
+�
+.
+Subsequently, we summarize our assumptions below.
+Assumption 1(Aggregate observation assumption) p(T |
+X1:K, Z1:K) = p(T | Z1:K)
+We here assume that the observation T is conditionally
+independent X1:K when given Z1:K (Def. 4.1). Intuitively,
+given Z, in fact, the travel time T can be obtained by
+summing up all components in Z.
+Assumption
+2
+(Markov
+chain
+assumption) p(Z1:K
+|
+X1:K) = p(Z1 | X1) �K
+i=2 p(Zi+1 | Xi; Zi)
+We assume that the road travel times Zi+1 are mutually
+independent except for Zi, which is consistently under the
+5. The subscript for example X1:K denotes an abbreviation for the
+set {X1, X2, · · · , XK}
+
+JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, AUGUST 201X
+5
+Nodes Embedding
+Links Embedding
+Road Attributes 
+Embedding Layer
+Link Embedding
+Node Embedding
+Link-wise Stacked 
+R-GCNs 
+Node-wise Stacked 
+R-GCNs 
+Multi-task Learning Model
+Multi-task Learning Module 
+Spatial GCN module 
+Transition 
+Probability
+Travel Time 
+Distribution of Nodes
+Travel Time 
+Distribution of Links
+Update
+Path Search
+(b) Detailed Structure of 
+Stacked R-GCNs 
+R-GCN
+GRU
+GRU
+Concatenation
+R-GCN
+GRU
+…
+(a) Overview of Our Model
+Concatenation
+R-GCN
+…
+GRU
+Source 
+Target  
+OD Pair
+Initialize Transition 
+Probability
+Candidate paths
+Path Travel Time 
+Estimation
+Road Features 
+Embedding
+Link-wise Stacked 
+Relational GCN 
+Node-wise Stacked 
+Relational GCN 
+MLP
+SoftMax
+Positional Vector 
+(N paths*dim)
+Transition Probability Generative Layer
+Current
+Path
+Travel Time 
+Distribution of Links
+Travel Time 
+Distribution of Nodes
+Transition 
+Probability
+(c) Detailed Structure of Multi-task 
+Learning Module 
+ReLU
+MLP
+Candidate Route 
+Posterior Probability
+⊗
+⊗
+Node-wise Graph
+Link-wise Graph
+Time Embedding
+Selection
+F
+F
+F
+…
+Route Search Layer
+Fig. 3. Framework design of MWSL-TTE. The overview of our proposed model is depicted in (a), which includes the route search layer, road
+attributes embedding layer, spatial GCN module, and multi-task learning module. (b) is the architecture of the stacked R-GCNs layer, which attempt
+to capture both global and local relation by modeling the GRU and R-GCNs together. (c) is our multi-task learning layer, which will estimate the
+travel time of links, intersections, and transition probability simultaneously. The transition probability will be updated when the multi-task learning
+layer is output, and then, the path search algorithm will work to find the candidate paths.
+assumption of Markov chain and extensive applications in
+trajectory data mining [40], [41]. Furthermore, since T can
+be determined by function Q, the conditional probability for
+p(T | Z1:K) = δQ(Z1:K)(T), where δ(·) represents the Dirac
+delta function.
+To sum up, the objective function in this paper can be
+defined as
+max
+X1:K p (T; X1:K | A; OD)
+=p (T | X1:K) p(X1:K|A; OD)
+=
+�
+ZK p (T; Z1:K | X1:K) dZ1:K p(X1:K|A; OD)
+=
+�
+ZK δQ(z1:K)(T)p(Z1:K | X1:K)dZ1:K p(X1:K|A; OD)
+=
+E
+Zi∼p(Zi|Xi−1;Zi−1)
+�δQ(Z1:K)(T)
+� p(X1:K|A; OD)
+(2)
+Therefore, according to Eq. (2), our training procedure
+could be split into two stages: 1) maximizes the posterior
+probability p(X1:K|A; OD). 2) maximizes the conditional
+probability p(Z1:K | X1:K) to estimate each road segments
+and intersection travel time, which can be optimized by
+expected log-likelihood [16]:
+ℓ(W) = E [log p (T | X1:K; W)]
+(3)
+For ease of reference, some important notations are summa-
+rized in Table 1.
+We here give a brief summarization of our problem for-
+mulation. In this paper, we aim to solve the two challenges
+in OD travel time estimation, which are uncertain routes and
+uncertain travel time. We intend to infer the potential route
+r between source and destination by transition probability
+Fig. 4. In the speed distributions of neighbor road segments during
+the 7 days of National Day, we observe that the speed distribution is
+highly consistent with the connected type. The color indicate the road
+correspondingly.
+A. Subsequently, we optimize the travel time distribution
+Z1:K = {˜te1, ˜tv2, ˜te3, · · · ˜tvK−1, ˜teK} at r via weakly su-
+pervised learning (Eq. (3)). We assume ˜t under Gaussian
+distribution N(µ, σ). Therefore, transition probability A and
+parameter at Gaussian distribution can be generated by
+(A, µ, σ) = f(W; Gv; Ge), where µ, σ are the mean and
+variance in Gaussian distribution, respectively.
+5
+METHODOLOGY
+In this section, the MWSL-TTE will be detailedly introduced.
+Specifically, the overview of MWSL-TTE has been depicted
+in Fig. 3 (a) including with four components included road
+attributes embedding layer, spatial GCN module, route
+search layer, and multi-task learning module. Fig. 3 (b)
+shows the inner structure of the stacked R-GCNs layer, and
+Fig. 3 (c) represents the multi-task learning module.
+
+80
+70
+Xiuyuan East Road
+60
+(km/
+Second Section of Jianshe
+speed
+Guoguang Road
+North Road
+40
+30
+Jianshe Road
+20
+10
+00:00
+04:00
+08:00
+12:00
+16:00
+20:00
+TimeJOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, AUGUST 201X
+6
+5.1
+Road Attributes Embedding layer
+Let each latent variable ˜ti ∈ Z belong to Gaussian dis-
+tribution N(µi, σi), which is a common assumption and
+widely be used in modeling the travel time distribution
+[13], [14], [15]. Given a pair of OD, we have the conditional
+probability p(Z1:K|θz = f(X1:K; WZ)) = p(Z1:K|(µ, σ) =
+f(X1:K; WZ)). Therefore, one of the tasks for neural net-
+work f is to estimate the distribution parameters µ, σ for
+each road segment and intersection. Since the road features
+X1:K could affect the travel time estimation Z, we consider
+the follow statistical spatial factors as the important matters
+for road segments:
+• Road types: e.g., primary, primary link, secondary, sec-
+ondary link, tertiary, residential, service road, etc.;
+• Number of lanes: how many traffic lines in the road;
+• Otherwise features: e.g., road length, whether it is a one
+way or not, limiting velocity, unique ID;
+and intersections, such as
+• Node tags: e.g., speed camera, traffic signal, crossing sign,
+turn circle, stop sign;
+• Node street count: e.g., T-juction X-junction, and 5-way
+junction:
+• Otherwise features: e.g., unique ID, GPS coordinate.
+To obtain the feature representations of both links and
+nodes, we use the embedding method [42] to transform each
+categorical attribute into a low-dimensional feature vec-
+tor by multiplying the spatial feature embedding matrices
+E ∈ Rn(s)×d(s). Here, n(s) represents the number of possible
+values of the categorical features, and d(s) represents the
+embedding dimension. This allows us to share efficient
+information among different road segments or intersections,
+so that rarely traveled segments could be learned from
+those frequently traveled with similar semantic meaning.
+Besides the categorical road attributes, we concatenate the
+obtained embedded feature vectors together with other road
+attributes (e.g., road length and GPS coordinate). Based on
+the above feature representations of the dual graph (link-
+wise and node-wise), we can obtain the corresponding input
+for the subsequent spatial GCN module.
+5.2
+Spatial GCN Module
+After the embedding characteristics of the road attributes
+have been obtained, we next introduce the spatial GCN
+module serving as modeling the complex spatial relations
+from the dual graph. The motivation for introducing R-
+GCNs in the travel-time estimation problem has been rep-
+resented in Fig. 4. We can observe that the travel speed is
+highly similar with the connected types. For specifically,
+even though the Second Section of JianShe North road is
+the neighbor of Xiuyuan East and Guoguang road, its speed
+distributions are more related to JianShe road, where their
+road types are the same. Therefore, we are concerned that
+the features of road types, the connected types, for example,
+resident → secondary (link level), and secondary (node level),
+are also important. To this end, here we introduce the Re-
+lational Graph Convolutional Networks [18] in our model,
+which can be defined as
+h(l+1)
+i
+= σ
+�
+� �
+r∈R
+�
+j∈N r
+i
+1
+ci,r
+W (l)
+r h(l)
+j
++ W (l)
+0 h(l)
+i
+�
+� ,
+(4)
+where h(l)
+i
+∈ Rd(l) is the hidden state of road ri in the lth
+layer of model with dimensionality d(l). N r
+i denotes the set
+of neighboring road segment/intersection indices under the
+relation r. W (l)
+r , W (l)
+0
+are the learnable parameters, ci,r is the
+normalization constant, and σ(·) is the activation function.
+In this paper, we set ci,r = |N r
+i |.
+Next, we will introduce the stacking operation based on
+R-GCNs. The stacking operation has recently been demon-
+strated to prevent local information loss [20], [43]. Thus, we
+model the temporal trends in the stacked GCN architecture
+combined with the spatial feature representations of both
+nodes and links. In this paper, we use a gated recurrent unit
+(GRU) [19] as a temporal processing module to incremen-
+tally concatenate multi-scale features, which can be written
+as
+c(0) = GRU
+�
+h(0), c(−1)�
+,
+h(1) = σ
+�
+� �
+r∈R
+�
+j∈N r
+i
+1
+ci,r
+W (0)
+r
+h(l)
+j
++ W (0)
+0
+h(0)
+�
+� ,
+h(2) = σ
+�
+� �
+r∈R
+�
+j∈N r
+i
+1
+ci,r
+W (1)
+r
+�
+h(0)
+j
+, h(1)
+j
+�
++ W (1)
+0
+�
+h(0), h(1)�
+�
+� ,
+h(l+1) = σ
+�
+� �
+r∈R
+�
+j∈N r
+i
+1
+ci,r
+W (l)
+r
+�
+c(l−1)
+j
+, h(l)
+j
+�
++ W (l)
+0
+�
+c(l−1), h(l)�
+�
+� ,
+(5)
+l = 2, 3, . . . , n − 1,
+where c(l) is the hidden state of the output of GRU, and
+c(−1) is initialized with 0. h(l) is the latent state of the link
+and node at lth hop. The detailed architecture of the stacked
+R-GCNs has been shown in Fig. 3 (b), and the formula of
+GRU can be expressed as
+o = σ (Wo1h(t) + Oo1c(t − 1) + bo1) ,
+q = σ (Wq1h(t) + Oq1c(t − 1) + bq1) ,
+c′(t) = tanh (Wh1h(t) + Oh1(q ⊙ c(t − 1)) + bh1) ,
+c(t) = o ⊙ h(t − 1) + (1 − o) ⊙ c′(t)
+(6)
+where Wo1, Oo1, Wq1, Oq1, Wh1, Oh1 are the learnable pa-
+rameters, and bo1, bq1, bh1 are biases.
+5.3
+Multi-task Learning Module
+In this section, we will introduce the productions of MWSL-
+TTE and the route recovery algorithm together.
+5.3.1
+Generating nodes and links travel time
+As we mentioned in Sec. 4, the task of TTE can be formu-
+lated as given the real-time OD pairs and the corresponding
+observation T in [t, t + ∆t1), we aim to complete the travel
+time for all links and nodes, and evaluate them using the OD
+pairs in [t + ∆t1, t + ∆t). To address the data sparse issue,
+this paper formulates it as the problem of tensor completion
+[44] by tensor decomposition technique, a popular method
+
+JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, AUGUST 201X
+7
+for traffic missing value imputation method. Since urban
+travel time has typical temporal and spatial distribution
+characteristics, it can frequently be divided into two levels:
+one is the modeling of road segments or intersections in
+space, and the other is temporal embedding in time (such
+as Weather ID and Holiday ID). Based on the above spatio-
+temporal embedding, we finally employ the 1st order CP
+decomposition to reconstruct the travel time distribution as
+µe = (h(l+1)
+e
+Wµe + bµe) ⊗ Ti, σe = (h(l+1)
+e
+Wσe + bσe) ⊗ Ti,
+µv = (h(l+1)
+v
+Wµv + bµv) ⊗ Ti, σv = (h(l+1)
+v
+Wσv + bσv) ⊗ Ti,
+where µe, σe ∈ R|E|×1, and µv, σv ∈ R|V|×1 are the mean
+and variance in Gaussian distribution for link and node
+respectively. Wµe, Wσe, Wµv, Wσv ∈ Rd(l)×d(l+1), bµe, bσe ∈
+R|E|, and bµv, bσv ∈ R|V| are the parameters in the fully
+connected layer (FC). T ∈ Rd(l+1)×I is the embedding tensor
+of time, and Ti ∈ Rd(l+1)×1 denotes the embedding vectors
+in real-time interval. We discretize the day of time into I
+time slots (e.g., ∆t=15 minutes). According to expected log-
+likelihood in Eq. (3), since the normal distribution is closed
+with addition, the loss function can be derived as
+Lµ,σ = −(Q(Z1:K) − Q(µ1:K))2
+2 �
+i(σ2)i
+− 1
+2 log
+�
+2π
+�
+i
+(σ2)i
+�
+= −(T − Q(µ1:K))2
+2σ2
+− 1
+2 log
+�2πσ2� ,
+(7)
+where Q is the aggregation function defined in Def. 4.1, and
+K denotes the number of samples in bag. In this paper, bag
+is equal to route (Def. 3.4) between origin-destination.
+5.3.2
+Transition Probability Generative Layer
+We thereafter introduce the detailed structure of the transi-
+tion probability generative layer to generate the link tran-
+sition probability by using edges features h(l+1) (Eq. (5)).
+Technically, for the last two layers, we use the multi-layer
+perceptron (MLP) to produce the weights of links
+ai→j = MLP
+�
+h(l+1)
+i
+|| h(l+1)
+j
+�
+,
+where || is the operator of features-wise concatenation, and
+then apply the softmax layer over outgoing links
+p(ej | ei) =
+exp(ai→j)
+�
+vn∈Vi exp(ai→n)
+(8)
+After that, the transition probability in Eq. (8) will be em-
+ployed in the Route Search Layer. For simplicity, we use
+the transition matrix A ∈ R|V|×|V| to represent all links’
+possibilities; for example, we have A[i, j] = P(ej | ei).
+5.3.3
+Route Search Layer
+As aforementioned, the shortest route algorithms omit the
+condition of the road in practice. To solve this problem, we
+intend to construct the transition probability between road
+segments and combine the transition probability to infer the
+route. However, considering the complex highway graph,
+there are a tremendous number of routes for any OD pair.
+It would be reasonable to prune the routes through some
+thresholds. Therefore, in this paper, we prune the route from
+two aspects: 1) The lengths of the simple route r from origin
+o to destination d should satisfy r.length < rshort + δlens,
+where rshort is the shortest simple route and δlens is the
+distance threshold. 2) the co-occurrence probability of a
+simple route P(r|OD; A) should also meet the criteria
+P(r|OD; A) > δprobs, where δprobs is the probability thresh-
+old. Next, we will introduce the definition of probability
+P(r|OD; A). According to Assumption 2 and Eq. (8), the
+co-occurrence route probability
+p(r) = P(e1, e2, · · · eK) = p(e0)
+K
+�
+i=1
+p(ei | ei−1),
+(9)
+where e0 is the origin location, and we have p(e0) = 1.
+After the strategy of pruning has been introduced, the
+candidate routes can be obtained by the Depth First Search
+(DFS) algorithm. Specifically, we generate the candidate
+route set ΩOD via posterior probability p(r|OD; A) and
+choose the route with maximum probability as the optimal
+solution. Formally, given an OD pair and observation T, the
+optimal route r∗ can be written as
+r∗ = arg max
+ri∈ΩOD p(ri | OD; A; T; Z)
+= arg min
+r
+|T −
+�
+ei∈rj
+µ(i)
+e
+−
+�
+vi∈rj
+µ(i)
+v |, ∀rj ∈ ΩOD. (10)
+Eq. 10 selects the most suitable route r∗ regarding current
+travel time µ.
+5.3.4
+Model Training
+Next, we will introduce the optimizing procedure of MWSL-
+TTE. As we discussed in Sec. 5.3.3, the top m maximum
+routes have been obtained, and we chose the most satisfied
+one by Eq. (10). However, such a choice may fall into a local
+solution, and other candidate routes might never be picked.
+To address this problem, we here introduce the ϵ-greedy
+algorithm, which means that the route satisfied Eq. (10) will
+be chosen in 1 − ϵ probability. Otherwise, randomly select
+the routes with top m maximum probability in ϵ probability.
+Moreover, we wish that the transition probability could help
+us infer the most possible route based on the ground truth
+(observation travel time). So, we here adopt the Kullback-
+Leibler (KL) divergence to measure the coherence, which
+can be written as
+DKL(P∥Q) = −
+�
+i
+P(i) ln Q(i)
+P(i),
+(11)
+where P is the probability distribution of each route in the
+candidate set, and Q is the inverse estimation loss distri-
+bution between Q(µ) and ground truth. In other words,
+the optimization direction is towards both higher route
+probability and more accurate travel time estimation. For
+the route picked up through Eq. (10), the Negative Log
+Likelihood (NLL) loss has been employed to minimize the
+negative log likelihood function, which can be defined as
+Ltp = −
+|tr|
+�
+i=2
+log(p(ei | ei−1, θ)),
+(12)
+where θ is the model’s trainable parameters to represent
+the posterior probability. By fusing all objective functions
+
+JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, AUGUST 201X
+8
+together, our model is trained to minimize the weighted
+combination of three loss terms
+L = αLµ,σ + βLtp + (1 − α − β)DKL(P∥Q)
+(13)
+where α and β are the const parameter to balance three loss
+terms Lµ,σ, Ltp and DKL(P∥Q). The training pseudocode
+of MWSL-TTE has been depicted in Algorithm 1.
+Algorithm 1: Training Procedure of MWSL-TTE
+Input: OD datasets D, node-wise graph Gv, link-wise
+graph Ge, and number of candidates routes N
+Output: OD TTE estimation function f
+1 while not convergence do
+2
+(µ, σ, A) = f(W; Gv; Ge)
+3
+for OD ∈ D do
+1. Generating Top N candidates route set ΩOD
+2. Select the route ri from ΩOD through ϵ-greedy
+3. Calculate the loss by Eq. (13) and update the
+parameters through back-propagation.
+4
+end
+5 end
+6 return f
+6
+EXPERIMENTS
+In this section, various experiments will be conducted based
+on a wide range of public real-world taxi dataset in Xi’an,
+and Chengdu to evaluate the superiority of MWSL-TTE in
+TTE and route recovery aspects.
+6.1
+Datasets
+Road Networks. We use two road networks: Chengdu
+Road Network and Xi’an Road Network. Both of them are
+extracted from OpenStreetMap [45], and include nine road
+types (trunk, trunk link, freeway link, primary, primary
+link, secondary, secondary link, tertiary, tertiary link). Here,
+Chengdu road network contains 8221 edges and 5182 nodes,
+which ranges from 30.63° to 30.69° in latitude and 104°
+to 104.07° in longitude. And Xi’an road network contains
+4780 edges and 3782 nodes, ranging from 34.20° to 34.29° in
+latitude and 108.90° to 108.99° in longitude.
+Taxi OD Orders. We use two public taxi trajectory datasets
+come from the Didi Express platform to generate the OD
+orders. Each generated order corresponds to a trip record
+that consists of the time-stamps and locations of an OD.
+Here, we implement Xi’an dataset that is from 10/10/2016
+- 10/22/2016, and Chengdu Dataset is from 08/18/2014 -
+08/24/2014 (a whole week from Monday to Sunday). The
+GPS points of both two datasets have been tied to the road
+and the interval of sample trajectory points is 2-4s, ensuring
+that the vehicle trajectory can correspond to the actual road
+information. Especially, we generate the ground truth route
+of the original vehicle trajectories for the route recovery task
+via a map-matching tool FMM [46].
+6.2
+Baseline Methods and Metrics
+We first compare our models with six baseline methods for
+the task of OD travel time estimation:
+• TEMP: Temporally weighted neighbors [7] is a nearest-
+neighbor-based approach that estimates the OD travel
+time by averaging the travel time of all historical trajec-
+tories falling in the same time slot with a similar origin
+and destination.
+• GBDT: Gradient boosting decision tree [47] is used for the
+regression of OD travel time estimation.
+• STNN: Spatio-temporal deep neural network [8] is a deep
+neural network-based approach that first predicts the
+travel distance given an OD pair, and then combines this
+prediction with the departure time to estimate the travel
+time.
+• MURAT: Multi-task representation learning [9] is a deep
+neural network-based approach that jointly predicts the
+travel distance and the travel time for taxi orders by
+learning representations of road segments and the origin-
+destination information.
+• DCRNN: It exploits GCN to capture spatial dependency,
+and then uses recurrent neural networks to model tempo-
+ral dependency [33]. We implemented this model based
+on OD estimation prediction of road network. The hidden
+vector size of GCN and GRU are set as 20 and 128,
+respectively.
+• ConSTGAT: This model adopts a graph attention mech-
+anism to explore the joint relations of spatio-temporal
+information [48]. The parameter setting is basically same
+with the original model. In the integration module, we
+also use two-layer MLP.
+• ASTGNN: This model consider multiple factors in traffic
+forecasting, such as, periodicity, spatial heterogeneity by
+leveraging a trend-aware multi-head attention mechanism
+[34]. The number of layers for both encoder and decoder
+is set to 3. And the kernel size of convolution is set to 5.
+Moreover, for the task of OD route recovery, we compare
+with two representative baselines of route recovery from
+sparse trajectories. Both of them are based on inverse rein-
+forcement learning to capture the spatial transition proba-
+bilities, and the difference between these two models is that
+the temporal components:
+• STRS: Spatio-temporal-based route recovery system [4]
+seeks to recover the route from sparse trajectories.
+The temporal components of STRS comprise a matrix
+factorization-based method.
+• DeepGTT-STRS: Li et al. [3] proposes a deep genera-
+tive travel time estimation model named DeepGTT that
+replaces the temporal component of STRS.
+Evaluation Metrics. For the OD travel time estimation
+of our MWSL, we evaluate the performance with RMSE
+(Root Mean Square Error), MAE (Mean Absolute Error), and
+MAPE (Mean Absolute Percentage Error). Then we adopt
+the accuracy of route recovery as the main performance
+metric for the route recovery task. It is defined as the ratio
+of the length of a correctly inferred route to the length of the
+ground truth route RG or the inferred route RI whichever
+is longer, i.e., accuracy =
+(RG∩RI).len
+max{RG.len,RI.len}.
+6.3
+Experimental Settings
+The experiments are implemented with PyTorch 1.6.0 and
+Python 3.6, and trained with a RTX2080 GPU. The platform
+ran on Ubuntu 16.04 OS. We trained the models using Adam
+
+JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, AUGUST 201X
+9
+TABLE 2
+Performance of MWSL and its variants for OD travel time estimation, compared with other baseline methods.
+Models
+Xi’an
+Chengdu
+RMSE (sec)
+MAE (sec)
+MAPE
+RMSE (sec)
+MAE (sec)
+MAPE
+TEMP
+398.95
+277.56
+34.24%
+446.98
+327.99
+32.00%
+GBDT
+365.72
+250.63
+31.27%
+435.88
+303.83
+30.35%
+STNN
+353.06
+241.19
+30.43%
+425.53
+293.10
+28.25%
+MURAT
+538.23
+512.65
+127.87%
+519.20
+503.36
+118.62%
+DCRNN
+282.54
+191.41
+24.94%
+392.45
+263.91
+25.95%
+ConSTGAT
+283.89
+195.31
+25.32%
+403.31
+280.90
+28.16%
+ASTGNN
+259.46
+179.08
+23.86%
+362.48
+244.03
+23.52%
+N-Node
+253.62
+173.71
+23.04%
+367.04
+236.18
+23.69%
+N-GRU
+259.52
+181.37
+24.25%
+371.34
+240.45
+24.03%
+N-R-GCN
+263.46
+184.73
+24.60%
+364.58
+234.37
+23.46%
+N-PathUpdate
+257.62
+178.25
+23.59%
+358.09
+229.56
+23.15%
+MWSL-TTE
+238.86
+162.37
+21.33%
+341.02
+215.03
+22.27%
+TABLE 3
+Inference time cost comparison for GCN-based travel time estimation
+models. The time unit is second.
+Models
+Xi’an
+Chengdu
+Time (sec)
+Time (sec)
+DCRNN
+0.15
+0.16
+ConSTGAT
+0.41
+0.45
+ASTGNN
+0.34
+0.37
+MWSL-TTE
+0.23
+0.25
+optimizer with an initial learning rate of 0.001 on both
+Chengdu and Xi’an datasets, and early stopping is used on
+the validation dataset. Especially, we run each experiment
+for three times.
+The main hyper-parameter settings of our proposed
+method are described as follows:
+• In the generation of a candidate route set Ωa,b, m
+candidate routes are selected between the OD pairs.
+Here, m = 6 is used for Xi’an, and m = 8 for
+Chengdu, respectively. Both two hyper-parameters can
+ensure that over 90 percents of ground truth route can
+be acquired from ΩOD.
+• The number of stacked R-GCNs is set to 3.
+• In the road attributes embedding layer, the embedding
+sizes of link feature representation (road ID, road types,
+number of lanes and one way or not) are set to 128, 8,
+4 and 2, respectively. And the embedding size of node
+feature representation (node ID, node type and node
+street count) are set to 96, 2 and 2, respectively.
+• In the temporal embedding component, we embed
+Weather ID and Holiday ID in R8 and R4, respectively.
+6.4
+Experimental Results
+We compare our MWSL-TTE with other baseline methods
+under two datasets.
+Performance on Travel Time Estimation. Table 2 shows
+the overall performance of estimating the travel times of
+Taxi OD orders. From the performance comparison, we find
+that our MWSL-TTE achieves the best performance than
+TABLE 4
+Performance of MWSL and STRS-based baseline methods for route
+recovery. The column T.time refers to the training time of the model and
+its unit is hour.
+Models
+Xi’an
+Chengdu
+Acc
+T. time
+Acc
+T. time
+STRS
+82.71%
+3.18
+71.64%
+4.74
+DeepGTT-STRS
+79.39%
+3.37
+68.72%
+5.02
+MWSL-TTE
+86.25%
+0.52
+77.03%
+0.69
+Congested
+Unblocked
+(a) 9:00 AM, Monday
+(b) 22:00 PM, Monday
+crossing
+traffic signal
+bus stop
+Fig. 5. The time-consuming ratio for some nodes in the Xi’an road
+network. Here, we select three types of nodes including ”crossing”,
+”traffic signal” and ”bus stop”.
+other methods in terms of all three metrics. The better
+prediction results can be explained in two aspects. First,
+our model implements an effective graphical structure to
+capture the prior information of road network. Although
+all DCRNN, ConSTGAT and ASTGNN model the link-
+wise adjacency of road segments, the node-wise features
+especially traffic signal junctions are ignored. Thus, these
+three models without considering the node-wise adjacency
+cannot achieve higher accuracy. Second, given an OD pair,
+our weak supervision-based method can study the travel
+time distributions of the links and nodes. Compared with
+Taxi OD estimation baselines such as STNN and MURAT,
+in-process information of OD pairs improves our estimation
+results.
+Ablation Study. In Table 2, except for the comparison
+experiment with the six baseline methods, we also con-
+
+0
+4 
+9
+8
+0
+2
+4
+6
+8
+0.0
+0.2 
+0.4
+0.6
+0.8
+1.0Kangjia
+油家村
+莲湖区
+新城区
+长乐西路
+环城西路
+东关南街
+张家村
+乐居场
+太乙路
+文艺路
+安路
+李家村
+雁塔区
+小寨路eiGuan
+Kangjia
+潘家村
+新城区
+莲湖区
+长乐西路
+环城西路
+东关南街
+张家村
+乐居场
+太乙路
+文艺路
+安路
+李家村
+小寨路
+雁塔区iGuan
+Kangjia
+潘家村
+新城区
+莲湖区
+长乐西路
+环城西路
+东关南街
+中
+张家村
+太乙路
+乐居场
+文艺路
+李家村
+路
+#
+中
+#
+小寨路
+中:国JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, AUGUST 201X
+10
+TABLE 5
+Performance of MWSL under different combinations of α and β based on Xi’an and Chengdu dataset for both OD travel time estimation and route
+recovery.
+Datasets
+Xi’an/Chengdu
+Parameters
+TTE
+Route recovery
+RMSE (sec)
+MAE (sec)
+MAPE (%)
+Acc (%)
+(α=1, β=0)
+240.62/354.36
+161.12/233.27
+21.62/23.17
+\
+(α=0.8, β=0.2)
+243.63/347.48
+164.15/221.19
+20.86/22.64
+85.94/73.36
+(α=0.8, β=0.1)
+238.86/341.02
+162.37/215.03
+21.33/22.27
+86.25/77.03
+(α=0.6, β=0.2)
+248.41/352.29
+167.56/232.34
+22.06/22.83
+86.92/78.14
+(α=0.6, β=0.3)
+247.46/358.75
+166.55/239.49
+22.12/23.39
+84.17/73.22
+(α=0.6, β=0.1)
+249.00/365.27
+166.59/237.92
+21.84/23.78
+80.83/67.50
+(b) East part of second ring south road (type: “trunk”)
+(a) Xiaozhai east road (type: “primary”)
+Fig. 6. Learned speed distributions of two sample links by MWSL,
+compared with the speed distributions computed by the original taxi
+trajectories.
+duct the ablation study by replacing our MWSL-TTE with
+four variations, namely N-Node, N-GRU, N-R-GCN and
+N-PathUpdate, to evaluate the effectiveness of different
+modules in MWSL-TTE (see Fig. 3). In N-Node, we remove
+the node-wise estimation. In N-GRU, we remove the stacked
+GRU and only use the same number of layers of R-GCN.
+In N-R-GCN, we remove the R-GCN and replace it with
+the same number of layers of normal GCN [49]. In N-
+PathUpdate, we only implement the initial path as the in-
+process trajectories of OD orders and do not update the path
+based on the learned travel time and transition probability.
+The result comparison, it shows that R-GCN and node-
+wise estimation are the most critical parts. Regardless of the
+node-wise aspect, the performance becomes worse, and it
+proves the importance of modeling the complex adjacency
+for both the links and nodes. Furthermore, stacked R-GCNs
+with GRU integrate the multiscale information to capture
+both global and local features, and path update is also
+important in improving the travel time estimation. To sum
+up, the key designs of MWSL-TTE are effective.
+Time Cost Analysis. Due to the online travel time esti-
+mation of our proposed MWSL-TTE, we further compare
+the inference time with main GNN-based baselines. As
+Table 2 shows, GNN-based travel time estimation mod-
+els have significantly better performance than common
+deep learning-based methods, with a batch size 32 using a
+RTX2080 GPU card. Table 3 provides the inference time cost
+comparison for main GNN-based models. Although our
+proposed model is slower than DCRNN, the inference time
+of our model can acquire a faster inference time comapred
+with other relatively complex models. And this inference
+time result also proves that our model can process an online
+travel time estimation.
+Performance on Route Recovery. Table 4 shows the per-
+formance of the route recovery task by given OD pairs.
+We can find that our path recovery of MWSL has better
+recovery accuracy and shorter model training time. STRS-
+based methods are very time-consuming due to the long
+iterations of inverse reinforcement learning to acquire the
+transition probability among road segments. Observe that
+the accuracy of Deep-STRS is always worse than that of
+STRS. The reason is that a larger sampling time interval be-
+tween OD pairs leads to a more inaccurate grid-based traffic
+tensor which is the input of DeepGTT to model the traffic
+representation. In particular, a more complex Chengdu road
+network leads that the recovery accuracy of three methods
+dropping as expected.
+Hyper-parameter Analysis. To further show the effective-
+ness of multi-task components of our model, we conduct
+experiments under different combinations of parameters α
+and β based on both of the two datasets. As observed in
+Table 5, on one hand, we find that in terms of the TTE task,
+the overall TTE performance improves as α changes from
+0.6 to 0.8 under the datasets of two cities. However, α = 1
+doesn’t achieve the best TTE performance. This is because
+that the β that controls the loss terms of route recovery also
+has some impacts on TTE prediction. More accurate path
+updates can bring the improvement of TTE prediction. On
+the other hand, the route recovery performance achieves the
+best accuracy when α = 0.8 and β = 0.1. This indicates
+that the DKL(P∥Q) also plays its part in obtaining a higher
+accuracy for route recovery. Furthermore, we compare the
+route recovery performance when α = 0.6. The optimal
+hyper-parameter is the combination of Ltp and DKL(P∥Q)
+as well, but excessive loss weight of DKL(P∥Q) would
+
+Computed distribution
+80
+Learned distribution
+60
+40
+20
+07:00
+09:00
+11:00
+13:00
+15:00
+17:00
+19:00
+21:00
+23:00Computed distribution
+40
+Learned distribution
+30
+20
+10
+07:00
+09:00
+11:00
+13:00
+15:00
+17:00
+19:00
+21:00
+23:00JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, AUGUST 201X
+11
+GT
+05:00
+09:00
+13:00
+17:00
+21:00
+Estimation
+Fig. 7. Comparison between ground truth traffic state and estimated traffic state from MWSL-TTE. Here we use four kinds of colors to represent the
+different road states, which can be defined as 1) red - very congested, 2) orange - congested, 3) yellow - slow, and 4) green - unblocked.
+cause the poor prediction performance. To sum up, we
+conclude that the experimental results demonstrated the
+superiority and generality of the multi-task components of
+our proposed MWSL.
+7
+CASE STUDY
+Our MWSl-TTE not only can conduct path travel time
+estimation and route recovery for OD GPS trips, but also
+learn the travel time distributions of the links and nodes
+based on weakly supervised learning. Thus, in addition to
+the quantified evaluations described in Section 6.4, we also
+conduct a real-world case study in Xi’an, which visualizes
+the learned distributions of the links and nodes from the
+road network. Especially, we conducted the comparison
+with road condition computed by original taxi trajectories.
+7.1
+Learned Distributions for Nodes and Links
+On one hand, we first provide the estimation results for
+different types of nodes, which are depicted in Fig. 5. We
+select three types of nodes in a road network and use the
+0AM
+3AM
+6AM
+9AM
+12NN
+15PM
+18PM
+21PM
+0.8
+Our MWSL
+STRS
+DeepGTT
+(a) Xi’an.
+0AM
+3AM
+6AM
+9AM
+12NN
+15PM
+18PM
+21PM
+0.7
+Our MWSL
+STRS
+DeepGTT
+(b) Chengdu.
+Fig. 8. The 24-h divergence between generated road conditions and
+ground truth under datasets of both Xi’an and Chengdu, compared with
+the temporal components of two baseline methods.
+time-consuming ratio to represent congestion status. The
+time-consuming ratio is calculated by dividing the corre-
+sponding mean value of learned travel time distributions
+by the maximum mean value among all nodes (Noted that
+we filter out top 1% largest node travel time). From Fig.
+5, we can find that nodes with the ”traffic signal” type are
+more time-consuming than the other two types of nodes,
+and most nodes in the morning peak are easy to become
+congested. These indicate that our node travel time estima-
+tion is reasonable in both spatial and temporal aspects.
+On the other hand, we transform the travel time dis-
+tribution into speed distributions in terms of links, and
+this transformation process is based on [41]. The reason
+for this transformation is that most users drive cars with
+a normal speed range (e.g., 10 kmph to 120 kmph), and
+thus we can easily analyze the rationality of learned speed
+distribution compared with link travel time. Two links’
+speed distributions were generated by the proposed MWSL,
+as is shown in Fig. 6, and we compared them with the
+speed distribution that is computed by the original taxi
+trajectories. To test the generalization of our model, we
+select two types of links, where the Xiaozhai east road is
+a busy link (type: ”primary”), and we can find that the
+morning and evening peaks are obvious in both learned
+speed distribution and computed distribution. Another link
+is a highway link, and the commuting pattern only appears
+in the evening for both distributions. Based on the above
+analysis, it is concluded that the learned distributions of
+links can effectively represent different functional types of
+links. Furthermore, the mean values µ of learned speed
+distributions are closer to the ground truth.
+7.2
+A Demo of the Generated Road Conditions
+We generate the road conditions by our MWSL based on taxi
+OD trips, and we compare with the ground truth computed
+by the original taxi trajectories. Especially, we mark it with
+the unblocked state for the road segments without taxi
+trajectories. Since the speed limit for each road is varied,
+which is primarily defined by road type or road length,
+
+JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, AUGUST 201X
+12
+we use four kinds of colors to represent the different road
+states (very congested, congested, slow and unblocked). We
+divide the limiting-velocity for each road type equally, for
+example, the rate-limiting of primary road is 60kph, so
+the interval between very congested is [0, 15), congested
+is [15, 30) ,slow is [30, 45) ,unblocked is [45, 60). The com-
+pared result is shown in Fig. 7. From the comparison of
+the generated road condition and ground truth at several
+time slots, we can acquire the following insights: 1) similar
+traffic state. The road condition generated by our model is
+similar to the ground truth. Most road segments have the
+same road states, and those road segments with different
+road states frequently have a consistent tendency; 2) rational
+adjacency correlation. We can find that the road segments
+with neighbor segments often have the similar road state
+for the generated road condition map. This indicates that
+our model can learn adjacency correlation of road network.
+To better illustrate the model performance for generated
+road conditions, we also conduct quantitative analysis un-
+der Xi’an and Chengdu datasets. As is shown in Fig. 8,
+we computes the 24-h divergence between generated road
+conditions and ground truth. The prediction accuracy is
+relatively worse among these three methods from 22:00 PM
+to 2AM. This is because that a very small number of OD
+pairs in these time slots can not provide efficient model
+training for better prediction performance. However, the
+plots show that the generated conditions achieve the accu-
+racy of around 80% and 70% at ordinary time slots under the
+datasets of both Xi’an and Chengdu, respectively. Compared
+with ground truth, our weak supervision learning method
+can provide believable road conditions only relying on the
+OD pairs.
+8
+CONCLUSION
+For the first time, we consider the OD travel time estimation
+as an inexact supervision problem and propose a multi-task
+framework to infer the optimal route based on transition
+probability and learn the travel time distribution for each
+road segment and intersection through expect MLE frame-
+work [16]. The stacked R-GCN architecture has been em-
+ployed to learn the complex relations of the road network,
+and we generate the travel time distribution for both road
+segments and intersections by 1st-order CP decomposition.
+Finally, we produce the transition probability between road
+segments by multi-layer perception. Moreover, an iterative
+update strategy has been proposed to update the transition
+probability and candidate paths during the training process.
+We evaluate our model on two real-world public datasets
+and verify the effectiveness of our proposed algorithm.
+Future work can be concluded in four parts. Firstly, more
+potential superior distribution can be developed under the
+assumptions of weakly supervised learning, since in this
+paper, only log-normal distribution has been employed.
+Secondly, more advanced route search algorithms could
+be designed based on, for example, transition probability,
+travel time, or route probability. Thirdly, more urban sce-
+narios, such as buses, subways, and people, can be tried to
+extend the applications of weakly supervised learning or
+travel time distribution. Lastly, a federated learning-based
+method can be designed, since the OD types of data save
+abundant storage costs on the client’s mobile phone.
+9
+ACKNOWLEDGMENT
+We are grateful to anonymous reviewers for their help-
+ful comments. This work was partially supported by the
+grants of National Key Research and Development Program
+of China (No. 2018AAA0101100), National Key Research
+and Development Project (2021YFB1714400) of China and
+Guangdong Provincial Key Laboratory (2020B121201001).
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+[49] T. N. Kipf and M. Welling, “Semi-supervised classification with
+graph convolutional networks,” arXiv preprint arXiv:1609.02907,
+2016.
+Hongjun Wang is working toward the M.S. de-
+gree in computer science and technology from
+Southern University of Science and Technology,
+China. He received the B.E. degree from the
+Nanjing University of Posts and Telecommunica-
+tions, China, in 2019. His research interests are
+broadly in machine learning, with urban comput-
+ing, explainable AI, data mining, data visualiza-
+tion.
+Zhiwen Zhang received the B.E. and M.S. de-
+gree in Artificial Intelligence from Nankai Univer-
+sity, China, in 2016 and 2019 respectively. From
+2019, he is currently pursuing a Ph.D. degree at
+the Department of Socio-Cultural Environmental
+Studies, The University of Tokyo. His current
+research interests include urban computing and
+data visualization.
+Zipei Fan received his B.S. degree in Computer
+Science from Beihang University, China, in 2012,
+both M.S. and a Ph.D. degree in Civil Engi-
+neering from The University of Tokyo, Japan, in
+2014 and 2017 respectively. He became Project
+Researcher and Project Assistant Professor in
+2017 and 2019, and he has promoted to Project
+Lecturer at the Center for Spatial Information
+Science, the University of Tokyo in 2020. His
+research interests include ubiquitous computing,
+machine learning, Spatio-temporal data mining,
+and heterogeneous data fusion.
+
+GOLE
+BADLOADJOURNAL OF LATEX CLASS FILES, VOL. XX, NO. X, AUGUST 201X
+14
+Jiyuan Chen is working towards his B.S. de-
+gree in Computer Science and Technology from
+Southern University of Science and Technology,
+China. His major research fields include artifi-
+cial intelligence, deep learning, urban computing
+and data mining.
+Lingyu Zhang joined Baidu in 2012 as a search
+strategy algorithm research and development
+engineer. He joined Didi in 2013 and served as
+senior algorithm engineer, technical director of
+taxi strategy algorithm direction, and technical
+expert of strategy model department. Currently
+a researcher at Didi AI Labs, he used machine
+learning and big data technology to design and
+lead the implementation of multiple company-
+level intelligent system engines during his work
+at Didi, such as the order distribution system
+based on combination optimization, and the capacity based on density
+clustering and global optimization. Scheduling engine, traffic guidance
+and personalized recommendation engine, ”Guess where you are going”
+personalized destination recommendation system, etc. Participated in
+the company’s dozens of international and domestic core technology
+innovation patent research and development, application, good at using
+mathematical modeling, business model abstraction, machine learning,
+etc. to solve practical business problems. He has won honorary titles
+such as Beijing Invention and Innovation Patent Gold Award and QCon
+Star Lecturer, and his research results have been included in top in-
+ternational conferences related to artificial intelligence and data mining
+such as KDD, SIGIR, AAAI, and CIKM.
+Ryosuke Shibasaki was born in Fukuoka,
+Japan. He received his B.S., M.S., and Doctoral
+degrees in Civil Engineering from The Univer-
+sity of Tokyo, Japan, in 1980, 1982, and 1987,
+respectively. From 1982 to 1988, he was with
+the Public Works Research Institute, Ministry
+of Construction. From 1988 to 1991, he was
+an Associate Professor in the Civil Engineering
+Department, The University of Tokyo. In 1991,
+he joined the Institute of Industrial Science, The
+University of Tokyo. In 1998, he was promoted to
+Professor in the Center for Spatial Information Science, The University
+of Tokyo. His research interests cover three-dimensional data acquisi-
+tion for GIS, conceptual modeling for spatial objects, and agent-based
+microsimulation in a GIS environment.
+Prof. Xuan Song received the Ph.D. degree in
+signal and information processing from Peking
+University in 2010. In 2017, he was selected as
+an Excellent Young Researcher of Japan MEXT.
+In the past ten years, he led and participated
+in many important projects as a principal inves-
+tigator or primary actor in Japan, such as the
+DIAS/GRENE Grant of MEXT, Japan; Japan/US
+Big Data and Disaster Project of JST, Japan;
+Young Scientists Grant and Scientific Research
+Grant of MEXT, Japan; Research Grant of MLIT,
+Japan; CORE Project of Microsoft; Grant of JR EAST Company and
+Hitachi Company, Japan. He served as Associate Editor, Guest Editor,
+Area Chair, Program Committee Member or reviewer for many famous
+journals and top-tier conferences, such as IMWUT, IEEE Transactions
+on Multimedia, WWW Journal, Big Data Journal, ISTC, MIPR, ACM
+TIST, IEEE TKDE, UbiComp, ICCV, CVPR, ICRA and etc.
+

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+arXiv:2301.08668v1  [cs.CR]  20 Jan 2023
+1
+Key-and-Signature Compact Multi-Signatures
+for Blockchain: A Compiler with Realizations
+Shaoquan Jiang, Dima Alhadidi, Hamid Fazli Khojir
+Abstract—Multi-signature is a protocol where a
+set of signatures jointly sign a message so that
+the final signature is significantly shorter than con-
+catenating individual signatures together. Recently,
+it finds applications in blockchain, where several
+users want to jointly authorize a payment through a
+multi-signature. However, in this setting, there is no
+centralized authority and it could suffer from a rogue
+key attack where the attacker can generate his own
+keys arbitrarily. Further, to minimize the storage on
+blockchain, it is desired that the aggregated public-
+key and the aggregated signature are both as short
+as possible. In this paper, we find a compiler that
+converts a kind of identification (ID) scheme (which
+we call a linear ID) to a multi-signature so that
+both the aggregated public-key and the aggregated
+signature have a size independent of the number
+of signers. Our compiler is provably secure. The
+advantage of our results is that we reduce a multi-
+party problem to a weakly secure two-party problem.
+We realize our compiler with two ID schemes. The
+first is Schnorr ID. The second is a new lattice-based
+ID scheme, which via our compiler gives the first
+regular lattice-based multi-signature scheme with
+key-and-signature compact without a restart during
+signing process.
+I. INTRODUCTION
+A multi-signature scheme allows a group of
+signers to jointly generate a signature while no
+subset of them can represent all the members to
+generate it. It was first introduced by Itakura and
+Nakamura [26]. A trivial method is to ask each
+signer to generate a signature on the message and
+concatenate their signatures together. However, this
+is not efficient: (1) the signature size is linear in
+All authors are with School of Computer Science, Uni-
+versity of Windsor, Windsor, ON N9B 3P4. Email: jiang-
+shq@uwindsor.ca
+the number of signers n; (2) we need to provide
+n signer public-keys to verifier; (3) the verifica-
+tion needs to verify n signatures; (4) all the n
+public-keys need to be provided to the verifier; (5)
+the communication and storage complexity for the
+signature are both linear in n. With applications
+in blockchain, these problems are crucial as the
+signature will be transmitted, verified and stored
+in the blockchain network. Hence, it is desired to
+find multi-signature that has a signature with these
+efficiency measure independent of n.
+Early multi-signarture schemes [25], [30], [44]
+assumed the signer keys are chosen honestly. In
+Bitcoin [41], every user can choose his own public-
+key. However, this might raise a very serious issue.
+For example, if a user wants to generate a multi-
+signature with users of 3 pubic-keys gx1, gx2, gx3,
+he could choose s randomly and compute his
+public-key as pk = gs(gx1+x2+x3)−1. If the ag-
+gregated public-key (which is the only public-key
+provided to the verifier) is the multiplication of the
+four public-keys, then attacker knows its secret and
+hence can forge a multi-signature. This is called
+a rogue key attack. How to construct a key-and-
+signature compact multi-signature scheme secure
+against a rogue key attack is an important question.
+A. Related Works
+A multi-signature scheme [26] is a special case
+of aggregate signature [12] where each signer of
+the latter can sign a possibly different message.
+In this work, we only discuss a multi-signature
+scheme with a motivation of blockchain application
+where the public-key is arbitrary and the target is to
+minimize the signature and the aggregated public-
+key size. Micali et al. [39] requires an interactive
+key generation among signers and hence is not
+
+suitable. Boldyreva [11] and Lu et al. [32] require
+signers to add proof of possession (PoP) to their
+public-keys, which is typically a signature of the
+user’s public-key. The main disadvantage of this
+assumption is the increase of the public-key size.
+In the signing process, it also requires a signer to
+verify the PoP of all the other signers. In addition,
+this assumption is not compatible with an ordinary
+signature where PoP is not required.
+Bellare and Neven [8] converted the Schnorr
+signature [48] into a multi-signature by linearly
+adding the signature together. Their protocol is of 3-
+round but without the aggregated key aggregation.
+Bagherzandi et al. [3], Ma et al. [36], Syta et al.
+[51] and Maxwell et al. [38] attempted to construct
+a 2-round multi-signature scheme which essentially
+tries to remove the preliminary committing message
+which is a hash of the first message in an ID scheme
+(see [8] for example). However, Drijvers et al. [17]
+pointed out that all these schemes have proof flaws.
+They then proved that a slightly modified scheme
+of Bagherzandi et al. [3] is secure under the PoP
+assumption. Other 2-round proposals that support
+the key-and-signature aggregation are due to Alper
+and Burdges [2] and Nick et al. [42], [43], where
+Nick et al. [43] employed a generic NIZK proof
+while the other two proposals [2], [42] are efficient
+in aggregated key and signature and verification
+cost (similar to the original Schnorr signature).
+Boneh et al. [13] proved the security of a modi-
+fied version of Maxwell et al. [38] via an added
+preliminary committing message and hence it is a
+3-round scheme. Bellare and Dai [4] proposed a 2-
+round multi-signature scheme with a tight reduction
+without the key aggregation.
+The above constructions are all based on variants
+of the discrete logarithm assumption. It is important
+to find out quantum-resistant schemes while this is
+not easy. For instance, lattice-based scheme [28] is
+insecure [31]. Also, the proof for a ring-SIS based
+scheme [27] is invalid. They reduced to find a short
+W for ring-SIS problem AW = 0 with public
+parameter A. However, their obtained W is trivially
+zero which does not contradict the ring-SIS as-
+sumption. Some schemes [21], [22], [37], [14] need
+an exponential number of restarts of the signing
+Key
+Round
+♯
+Assump/
+Compact
+Comp
+Restart
+Remark
+[21]
+No
+3
+exp
+R-LWE
+[22]
+No
+3
+exp
+non-standard
+[14]
+Yes
+2
+exp
+R-MLWE&
+R-MSIS
+[16]
+No
+2
+exp
+R-MLWE&
+R-MSIS
+[20]
+No
+1
+0
+R-SIS
+limited-sign
+ours
+Yes
+3
+0
+R-SIS &
+R-LWE
+Fig. 1.
+Performance of Lattice-based Secure Multi-
+Signature Schemes: compact means the size indepen-
+dent of ♯ signers; all schemes have compact signatures;
+schemes requiring a honest key generations are not in-
+cluded; ♯ restart is ♯ of repeated runs of signing algorithm
+(in case it aborts); exp means exponential in either ♯
+signers or the security parameter; limited-sign restricts
+each user to have a bounded number of signings.
+algorithm, due to a noticeable probability of abort
+event. Some schemes [19], [37] are provably secure
+only when all the user keys are generated honestly
+which is not suitable for blockchain. Damg˚ard et al.
+[16] and Fleischhacker et al. [20] do not support key
+aggregations while the latter can only allow a signer
+to sign a predefined number of signatures. Thus,
+currently no multi-signature scheme can support a
+key-and-signature aggregation without a restart and
+allow an unlimited number of signing. Our work is
+to study this question in details.
+B. Contribution
+In this paper, we consider the key-and-signature
+compact multi-signature. That is, both key and
+signature support aggregation and have a size inde-
+pendent of the number of signers. Toward this, we
+formulate the linear identification scheme (ID) and
+propose a compiler that transforms a linear ID to a
+key-and-signature compact multi-signature scheme,
+where the signature size and the aggregated public-
+key are independent of the number of signers.
+The advantage of our compiler is that we reduce
+the multi-party signature problem to a two-party
+identification problem and hence it is much easier
+
+to deal with and also the security proof for latter
+should be simpler. We formulate the linearity of ID
+via the R-module from algebra. Our compiler is
+provably secure. We realize our compiler with two
+ID schemes. The first is Schnorr ID scheme. The
+second one is a new ID scheme over ring that is
+secure under ring-LWE and ring-SIS assumptions.
+Our ID scheme via the compiler gives the first
+key-and-signature compact multi-signature without
+a restart during the signing process (see Fig. 1 for
+a comparison with other schemes), where a signer
+can do an unlimited number of signing (unlike
+[20], which can only do a predetermined number of
+signings). The security of ID schemes is formulated
+in terms of unforgeability against an aggregated key
+of multi-users with at least one of them honest.
+Our ID schemes are proven secure through a new
+forking lemma (called nested forking lemma). Our
+forking algorithm has a nested rewinding and is
+more effective than the previous algorithms which
+fork at two or more spots sequentially.
+II. PRELIMINARIES
+Notations. We will use the following notations.
+• x ← S samples x uniformly random from a
+set S.
+• For a randomized algorithm A, u = A(x; r)
+denotes the output of A with input x and
+randomness r, while u ← A(x) denotes the
+random output (with unspecified randomness).
+• We use PR(r) to denote the probability
+Pr(R = r); for Boolean variable G, Pr(G)
+means Pr(G = 1).
+• PPT stands for probabilistic polynomial time.
+• Min-entropy
+H∞(X)
+=
+− log(maxx log PX(x)).
+• A|B stands for A concatenating with B.
+• negl(λ)
+is
+negligible:
+limλ→∞ poly(λ)negl(λ)
+=
+0
+for
+any
+polynomial poly(λ).
+• [ν] denotes set {1, · · · , ν}.
+A. Ring and Module
+In this section, we review a math concept: mod-
+ule (for details, see [29]). We start with the concept
+of ring. A ring A is a set, associated with multipli-
+cation and addition operators, respectively written
+as a product and a sum, satisfying the following
+conditions:
+- R-1. A is a commutative group under addition
+operator + with identity element 0.
+- R-2.
+A is associative under multiplication
+operator: for a, b, c ∈ A, (ab)c=a(bc). Also,
+it has a unit element 1: 1a=a.
+- R-3.
+It satisfies the distributive law: for
+a, b, c ∈ A, a(b + c) = ab + ac and (b + c)a =
+ba + ca.
+In this paper, we only consider a commutative ring:
+if a, b ∈ A, then ab = ba. That is, when we say
+ring, it always means a commutative ring. Note that
+a non-zero element in a ring does not necessarily
+have a (multiplicative) inverse, where b is an inverse
+of a if ab = 1. For instance, in Z10, 3 is an inverse
+of 7 while 5 does not have an inverse. If A is a
+commutative ring with 0 ̸= 1 and every non-zero
+element in A has an inverse, then A is a field.
+Now we introduce the concept module.
+Definition 1: Let R be a ring. An Abelian group
+M (with group operator ⊞) is a R-module, if (1) it
+has defined a multiplication operator • between R
+and M: for any r ∈ R, m ∈ M, r•m ∈ M; (2) the
+following conditions are satisfied: for any r, s ∈ R
+and x, y ∈ M,
+1. r • (x ⊞ y) = (r • x) ⊞ (r • y);
+2. (r + s) • x = (r • x) ⊞ (s • x)
+3. (rs) • x = r • (s • x)
+4. 1R • x = x, where 1R is the multiplicative
+identity of R.
+We remark that the group operator ⊞ for M is
+not necessarily the regular number addition (e.g., it
+can be the integer multiplication).
+In the following, we give some R-module exam-
+ples.
+Example 1. Let q be a prime and M is a group of
+order q with generator g (i.e., M = ⟨g⟩). Examples
+of M are a subgroup of Z∗
+p or an elliptic curve
+group. x, y ∈ M, xy denotes its group operation.
+Then, M is a Zq-module with • defined as r•m
+def
+=
+mr, for r ∈ Zq and m ∈ M. It is well-defined:
+
+since mq = 1, any representative r in Zq such as
+r, r + q gives the same result r • m. For r, s ∈ Zq
+and x, y ∈ M, we check the module conditions:
+(1) s • (xy) = (xy)s = xsys = (s • x)(s • y); (2)
+(r + s) • m = mr+s = mrms = (r • m)(s • m);
+(3) (rs) • x = xrs = (xs)r = r • (s • x); (4)
+1 • x = x1 = x.
+Example 2.
+For any integer n > 0, M = Zn
+(as an additive group) is a Zn-module, where • is
+simply the modular multiplication. The verification
+of module properties is straightforward.
+Example 3.
+Let n be a positive integer. Then,
+the polynomial ring M = Zn[x] (as an additive
+group) is a Zn-module with • being the modular n
+multiplication: for s ∈ Zn, m = �t
+i=0 uixi, s•m =
+�t
+i=0 uisxi, where uis is the multiplication over
+Zn. All the other verifications of the properties are
+straightforward.
+III. NESTED FORKING LEMMA
+The original forking lemma was formulated by
+Pointcheval and Stern [46] to analyze Schnorr sig-
+nature [48]. It basically shows that if the attacker
+can forge a Schnorr signature in the random or-
+acle model [7] with a non-negligible probability,
+then it can generate two forgeries when reminding
+to the place where the random oracle value was
+revised. Bellare and Neven [8] generalized the
+forking lemma to a general algorithm A, without
+resorting to a signature scheme. This was further
+generalized by Bagherzandi et al. [3] so that A is
+rewound to many places. However, the algorithm
+needs O(n2q/ǫ) rewindings, where q is the num-
+ber of random values in one run of A (which is
+the number of random oracle queries in typical
+cryptographic applications) and ǫ is the successful
+probability of A while n is the number of rewinding
+spots. However, this is not efficient and can even
+be essentially exponential for a non-negligible ǫ.
+The main issue comes from the fact the rewinding
+for each spot is repeated independently until a
+new success is achieved. But it does not relate
+different rewindings. In this section, we give a
+new forking lemma for two rewinding spots (say
+at index i, j with i < j) while it can be generalized
+to n rewinding spots. The new feature here is
+that the rewinding is nested. To see this, suppose
+that the first run of A uses the list of random
+values: h1, · · · , hi−1, hi, · · · , hj−1, hj, · · · , hq and
+the rewinding spots are chosen at index i and
+j. Then, we execute A for another 3 runs with
+rewindings that respectively the following lists of
+random values:
+h1, · · · , hi−1, hi, · · · , hj−1, h′
+j, · · · , h′
+q;
+(1)
+h1, · · · , hi−1, ¯hi, · · · , ¯hj−1, ¯hj, · · · , ¯hq;
+(2)
+h1, · · · , hi−1, ¯hi, · · · , ¯hj−1, hj, · · · , h′
+q.
+(3)
+That is, execution (1) rewinds the initial execu-
+tion to index j; execution (2) rewinds the initial
+execution to index i while execution (3) rewinds
+the (rewound) execution (2) to index j. With these
+related executions, we are able to claim the outputs
+are all successful with probability at least O(ǫ4),
+which is still non-negligible. The advantage of this
+nested forking is that it can be directly used to
+extract a secret hidden in recursive random oracle
+evaluations. Our algorithm will use the following
+notations.
+h[[1, · · · , q]]
+def
+= h1, · · · , hq (a sequence of ele-
+ments);
+h[[1, · · · , �
+i, · · · , q]]
+def
+= h1, · · · , hi−1, ˆhi, · · · , ˆhq;
+h[[1, · · · , �
+i, · · · , j, j + 1, · · · , q]]
+=h1 · · · , hi−1, ˆhi, · · · , ˆhj, ¯hj+1, · · · , ¯hq.
+Other
+variants
+such
+as
+h[[1, · · · , i, · · · , j, j + 1, · · · , q]]) can be defined
+similarly. Our forking algorithm is in Fig. 2.
+Before introducing our lemma, we give two facts.
+Fact 1.
+For any random variable I, R and any
+function F() on I, R, we have
+Pr(I = i∧F(I, R) = f) = Pr(I = i∧F(i, R) = f).
+Proof. For any function G and any random variable
+W, Pr(G(W) = g) = �
+w:G(w)=g PW (w). Apply-
+ing this to W = (I, R) and G = (I, F), a simple
+calculation gives the result as (I, F) = (i, f) is
+I = i ∧ F = f.
+□
+
+—————————————————————-
+Algorithm FA(x)
+—————————————————————-
+pick coin ρ for A at random
+h1, · · · , hq ← H
+(I0, J0, σ0) ← A(x, h[[1, · · · , q]]; ρ)
+If I0 = 0 or J0 = 0 or I0 ≥ J0, return Fail
+ˆhJ0, · · · , ˆhq ← H
+(I1, J1, σ1) ← A(x, h[[1, · · · ,
+�
+J0, · · · , q]]; ρ)
+If I1 = 0 or J1 = 0, return Fail
+¯hI0, · · · , ¯hq ← H
+(I2, J2, σ2) ← A(x, h[[1, · · · , I0, · · · , q]]; ρ)
+If I2 = 0 or J2 = 0, return Fail
+hJ0, · · · , hq ← H
+(I3, J3, σ3) ← A(x, h[[1, · · ·, I0, · · · , J0 − 1, J0, · · · , q]]; ρ)
+If I3 = 0 or J3 = 0, return Fail
+Let Flag1 = (I0 = I1 = I2 = I3) ∧ (J0 = J1 =
+J2 = J3)
+Let Flag2 = (hI0 ̸= ¯hI0)∧(hJ0 ̸= ˆhJ0)∧(¯hJ0 ̸= hJ0)
+If Flag1 ∧ Flag2, return (I0, J0, {σi}3
+i=0)
+else return Fail.
+—————————————————————-
+Fig. 2. Forking Algorithm FA
+Fact 2.
+Let B′, B, R be independent random
+variables with B′, B identically distributed. Let G
+be a fixed boolean function. Then,
+Pr(G(R, B) ∧ G(R, B′)) =
+�
+r
+PR(r) · Pr2(G(r, B)).
+Proof. Notice Pr(X = x) = �
+r Pr(R = r, X =
+x) for variable R, X. Together with Fact 1, we have
+Pr(G(R, B) ∧ G(R, B′))
+=
+�
+r
+Pr(R = r, {G(R, B) ∧ G(R, B′)} = 1)
+=
+�
+r
+Pr(R = r, {G(r, B) ∧ G(r, B′)} = 1)
+=
+�
+r
+PR(r) · Pr(G(r, B)) · Pr(G(r, B′))
+=
+�
+r
+PR(r) · Pr2(G(r, B)),
+where the third equality uses the independence of
+R, B, B′ and the last equality uses the fact that B′
+and B are identically distributed.
+□
+Now we are ready to present our forking lemma.
+Lemma 1: Let q ≥ 2 be a fixed integer and H
+be a set of size N ≥ 2. Let A be a randomized
+algorithm that on input x, h1, · · · , hq returns a
+triple, the first two elements of which are integers
+from {0, 1, · · · , q} and the last element of which
+is a side output. Let IG be a randomized algorithm
+(called input generator). The accepting probability
+of A, denoted by acc, is defined as the probability
+that I, J ≥ 1 in the experiment
+x ← IG; h1, · · · , hq ← H;
+(I, J, σ) ← A(x, h[[1, · · · , q]]).
+The forking algorithm FA associated with A is a
+randomized algorithm that takes x as input and
+proceeds as in Fig. 2.
+Let frk = Pr[FA(x) ̸= Fail : x ← IG]. Then,
+frk ≥
+8 · acc4
+q3(q − 1)3 − 3
+N .
+(4)
+Proof. With respect to Flag1, we define Flag∗
+1 as
+event
+(I0 = · · · = I3 ≥ 1)∧(J0 = · · · = J3 ≥ 1)∧(J0 > I0).
+Then, it is easy to check that FA(x) ̸= Fail is
+equivalent to Flag∗
+1 ∧ Flag2 = 1. Since hI0 = ¯hI0
+(resp. hJ0 = ˆhJ0, or, ¯hJ0 = hJ0) in ¬Flag2 holds
+with probability 1/N. It follows that
+frk = Pr(Flag∗
+1 ∧ Flag2 = 1)
+≥ Pr(Flag∗
+1 = 1) − 3/N.
+(5)
+Notice that
+Pr(Flag∗
+1 = 1)
+=
+q
+�
+i=1
+q
+�
+j=i+1
+Pr(∧3
+b=0{Ib = i ∧ Jb = j}).
+(6)
+Let A1 (resp. A2, A12) be three variants of
+algorithm A with the only difference in the output
+which is the first element (resp. the second element,
+the first two elements) of A’s output. For instance,
+J1 =A2(x, h[[1, · · · , J0 − 1,
+�
+J0, · · · , q]]; ρ),
+(7)
+I2 =A1(x, h[[1, · · · , I0 − 1, I0, · · · , q]]; ρ).
+(8)
+Assigning I0 = i and J0 = j, we denote
+J′
+1 =A2(x, h[[1, · · · , j − 1, �
+j, · · · , q]]; ρ),
+(9)
+I′
+2 =A1(x, h[[1, · · · , i − 1, i, · · · , q]]; ρ).
+(10)
+
+We can similarly define I′
+1, J′
+2, I′
+3, J′
+3. So Ib, Jb for
+b ≥ 1 are functions (of A’s inputs and randomness)
+and when assigning I0 = i and J0 = j, they
+become I′
+b, J′
+b. Hence, we can apply fact 1 to
+evaluate Eq. (6). Then, assigning I0 = i and J0 = j,
+applying Fact 1 to realize I0 = i and J0 = j in
+A1 (for Ib) and A2 (for Jb), Ib and Jb respectively
+become I′
+b and J′
+b. Hence, we have
+Pr(Flag∗
+1 = 1)
+(11)
+=
+q
+�
+i=1
+q
+�
+j=i+1
+Pr(∧3
+b=0{I′
+b = i ∧ J′
+b = j}),
+(12)
+where I0, J0 is rewritten as I′
+0, J′
+0 for brevity (so the
+term {I0 = i ∧ J0 = j} becomes {I′
+0 = i ∧ J′
+0 =
+j}). Notice ∧1
+b=0(I′
+b = i ∧ J′
+b = j) is a random
+variable, with randomness R = (x, ρ, h1, · · · , hi−1)
+and B
+= (hi, · · · , hq, ˆhj, · · · , ˆhq). So we can
+define
+∧1
+b=0 (I′
+b = i ∧ J′
+b = j) = G(R, B)
+(13)
+for some boolean function G.
+Besides, by verifying the definition of I′
+b, J′
+b, we
+can see that
+∧3
+b=2 (I′
+b = i ∧ J′
+b = j) = G(R, B′)
+(14)
+with B′ = (¯hi, · · · , ¯hq, hj, · · · , hq).
+Hence, applying Fact 2 to Eq. (12), we have
+Pr(Flag∗
+1 = 1)
+=
+�
+r
+1≤i<j≤q
+PR(r)Pr2(∧1
+b=0(I′
+br = i ∧ J′
+br = j))
+=
+�
+r
+1≤i<j≤q
+PR(r)Pr2(∧1
+b=0(I′
+br, J′
+br) = (i, j)).
+(15)
+where I′
+br (resp. J′
+br) is I′
+b (resp. J′
+b) with R = r.
+Notice that (I′
+0r, J′
+0r)
+= (i, j) is a boolean
+random variable (i.e., the result is true only if the
+equality holds), determined by hi, · · · , hq. We can
+define
+G′(S, C)
+def
+= {(I′
+0r, J′
+0r) = (i, j)}
+(16)
+for some function G′, where S = hi, · · · , hj−1 and
+C = hj, · · · , hq.
+Checking the definition of (I′
+1r, J′
+1r), we can see
+{(I′
+1r, J′
+1r) = (i, j)} = G′(S, C′)
+(17)
+with C′ = ˆhj, · · · , ˆhq.
+Thus, Eq. (15) is
+Pr(Flag∗
+1 = 1)
+=
+�
+r
+PR(r)Pr2�
+G′(S, C) ∧ G′(S, C′)
+�
+.
+(18)
+Hence, we can apply Fact 2 to Eq. (18) and
+obtain
+Pr(Flag∗
+1 = 1)
+=
+�
+r
+1≤i<j≤q
+PR(r)[
+�
+s
+PS(s)Pr2((I′
+0rs, J′
+0rs) = (i, j))]2
+≥
+�
+1≤i<j≤q
+[
+�
+r,s
+PR(r)PS(s)Pr2((I′
+0rs, J′
+0rs) = (i, j))]2
+≥
+�
+1≤i<j≤q
+[
+�
+r,s
+PR(r)PS(s)Pr((I′
+0rs, J′
+0rs) = (i, j))]4
+=
+�
+1≤i<j≤q
+[Pr((I′
+0, J′
+0) = (i, j))]4,
+≥
+
+
+�
+1≤i<j≤q
+Pr((I′
+0, J′
+0) = (i, j))
+
+
+4
+/(q3(q − 1)3/23)
+where (I′
+0rs, J′
+0rs) is (I′
+0r, J′
+0r) with S = s, the
+first two inequalities follow from Cauchy-Schwarz
+inequality1 (the first one is over distribution PR(·)
+and the second one is over distribution PR(·)PS(·));
+the last inequality is to apply Cauchy-Schwarz in-
+equality �n
+i=1 x2
+i ≥ (�
+i xi)2/n twice by noticing
+that y4
+i
+= (y2
+i )2 so that the first time we use
+xi = y2
+i for Cauchy-Schwarz inequality. Finally,
+notice that I′
+0 = I0 and J′
+0 = J0 by definition.
+Also, �
+1≤i<j≤q Pr((I0, J0) = (i, j)) is exactly
+acc by definition. It follows that Pr(Flag∗
+1
+=
+1) ≥
+acc4
+q3(q−1)3/23 . From Eq. (5), we have frk ≥
+8·acc4
+q3(q−1)3 − 3/N.
+□
+IV. MODEL OF MULTI-SIGNATURE
+In this section, we introduce the model of multi-
+signature. It consists of the multi-signature defini-
+tion and the security formalization.
+1�
+i pix2
+i ≥ (�
+i pixi)2, if pi ≥ 0 and �
+i pi = 1
+
+A. Syntax
+Mult-signature is a signature with a group of
+signers, where each of them has a public-key and
+a private key. They jointly generate a signature.
+The interaction between them proceeds in rounds.
+Signers are pair-wise connected but the channel is
+not secure. The signing protocol is to generate a
+signature so that the successful verification would
+indicate that all signers have agreed to sign the
+message. The target is to generate a compact signa-
+ture that is shorter than concatenating all signers’
+individual signatures together.
+Definition 2: A multi-signature is a tuple of al-
+gorithms (Setup, KeyGen, Sign, Verify), described
+as follows.
+Setup. Given security parameter λ, it generates a
+system parameter param that serves as part of the
+input for KeyGen, Sign, Verify (but for brevity, we
+omit it).
+KeyGen. It takes param as input and outputs for
+a user a private key sk and a public-key pk.
+Sign.
+Assume
+n
+users
+with
+public-keys
+(pk1, · · · , pkn) want to jointly sign a message
+M. Then, each user i takes its private key ski as
+input and interacts with other signers. Finally, each
+of them outputs a signature σ (note: this is for
+simplicity only; in literature, usually a designated
+leader outputs σ). Besides, there is a function F
+that aggregates (pk1, · · · , pkn) into a compact
+public-key pk = F(pk1, · · · , pkn).
+Verify. Upon (σ, M) with the aggregated public-
+key pk = F(pk1, · · · , pkn), verifier takes σ, M and
+pk as input, outputs 1 (for accept) or 0 (for reject).
+Remark.
+The verify algorithm only uses the
+aggregated key pk to verify the signature. This
+is important for blockchain, where the recipient
+only uses pk as the public-key. Also, the redeem
+signature only uses the multi-signature σ. It is
+desired that both pk and σ are independent of
+n while no attacker can forge a valid signature
+w.r.t. this short pk. Even though, our definition
+generally does not make any restriction on pk and
+it especially can be (pk1, · · · , pkn).
+B. Security Model
+In this section, we introduce the security model
+[13] of a multi-signature. It formulates the exis-
+tential unforgeability. Essentially, it says that no at-
+tacker can forge a valid signature on a new message
+as long as the signing group contains an honest
+member. Toward this, the attacker can access to a
+signing oracle and create fake public-keys at will.
+The security is defined through a game between a
+challenger CHAL and an attacker A.
+Initially, CHAL runs Setup(1λ) to generate sys-
+tem parameter param and executes KeyGen to
+generate a public-key pk∗ and a private key sk∗. It
+then provides pk∗|param to A who interacts with
+CHAL through signing oracle below.
+Sign Os(PK, M).
+Here PK is a set of distinct
+public-keys with pk∗ ∈ PK. Upon this query,
+CHAL represents pk∗ and A represents PK −
+{pk∗} to run the signing protocol on message M.
+Finally, Os outputs the multi-signature σ (if it
+succeeds) or ⊥ (if it fails).
+Forgery.
+Finally, A outputs a signature σ∗
+for
+a
+message
+M∗,
+w.r.t.
+a
+set
+of
+distinct
+public-keys
+(pk∗
+1, · · · , pk∗
+N)
+s.t.
+pk∗
+=
+pk∗
+i
+for
+some
+i.
+A
+succeeds
+if
+two
+conditions
+are met: (a) Verify(pk∗, σ∗, M∗)
+=
+1 (where
+pk∗
+=
+F(pk∗
+1, · · · , pk∗
+N));
+(b)
+no
+query
+((pk∗
+1, · · · , pk∗
+N), M∗) was issued to Os. Denote a
+success forgery event by succ.
+Now we can define the security of a multi-
+signature.
+Definition
+3:
+A
+multi-signature
+scheme
+(Setup, KeyGen, Sign, Verify) is existentially
+unforgeable against chosen message attack
+(or EU-CMA for short), if satisfies the correctness
+and existential unforgeability below.
+• Correctness.
+For (sk1, pk1), · · · , (skn, pkn)
+generated by KeyGen, the signature generated
+by signing algorithm on a message M will
+pass the verification, except for a negligible
+probability.
+• Existential Unforgeability. For any PPT ad-
+versary A, Pr(succ(A)) is negligible.
+The multi-signature scheme is said t-EU-CMA, if it
+is EU-CMA w.r.t. adversary A who always restricts
+
+the number of signers in each signing query and the
+final forgery to be at most t.
+V. MODEL OF CANONICAL LINEAR
+IDENTIFICATION
+In this section, we introduce a variant model
+of canonical identification (ID) scheme and extend
+it with linearity. We label the ID scheme with a
+parameter τ. This is needed in order to include
+our lattice-based ID scheme as a realization for our
+multi-signature compiler.
+Definition 4: A canonical identification scheme
+with parameter τ ∈ N is a tuple of algorithms
+ID = (Setup, KeyGen, P, Vτ, Θ), where Setup
+takes security parameter λ as input and generates
+a system parameter param; KeyGen is a key
+generation algorithm that takes param as input and
+outputs a public key pk and a private key sk;
+P is an algorithm, executed by prover; Vτ is a
+verification algorithm parameterized by τ, executed
+by Verifier; Θ is a set. ID scheme is a three-
+round protocol depicted in Fig. 3, where Prover
+first generates a committing message CMT with
+H∞(CMT) = ω(log λ), and then Verifier replies
+with a challenge CH ← Θ and finally Prover
+finishes with a response Rsp which will be either
+rejected or accepted by Vτ.
+Denote the domain of sk, pk, CMT, Rsp respec-
+tively by SK, PK, CMT , RSP. In the following,
+we define linearity and simutability for an ID
+scheme. Simulatbility appeared before (e.g., [1])
+while the linearity is new.
+Linearity.
+A
+canonical
+ID
+scheme
+ID
+=
+(Setup, KeyGen, P, Vτ, Θ) is linear if it satisfies
+the following conditions.
+i. SK, PK, CMT , RSP
+are
+R-modules
+for
+some ring R with Θ ⊆ R (as a set);
+ii. For any λ1, · · · , λt ∈ Θ and public/private
+pairs (ski, pki) (i = 1, · · · , t), we have that
+sk
+=
+�t
+i=1 λi • ski is a private key of
+pk = �t
+i=1 λi • pki.
+Note:
+Operator • between R and SK (resp.
+PK, CMT , RSP) might be different (as long
+as it is clear from the context), even though
+we use the same symbol •.
+iii. Let λi ← Θ and (pki, ski) ← KeyGen(1κ),
+for i = 1, · · · , t. If CMTi|CH|Rspi is a faith-
+fully generated transcript of the ID scheme
+w.r.t. pki, then
+Vτ(pk, CMT|CH|Rsp) = 1,
+(19)
+where pk = �t
+i=1 λi • pki, CMT = �t
+i=1 λi •
+CMTi and Rsp = �t
+i=1 λi • Rspi.
+Simulability.
+ID
+is
+simulatable
+if
+there
+ex-
+ists a PPT algorithm SIM s.t. for (sk, pk) ←
+KeyGen(1λ), CH ← Θ and (CMT, Rsp) ←
+SIM(CH, pk, param), it holds that CMT|CH|Rsp
+is indistinguishable from a real transcript, even
+if the distinguisher is given pk|param and has
+access to oracle Oid(sk, pk), where Oid(sk, pk) is
+as follows: (st, CMT) ← P(param); CH ← Θ;
+Rsp ← P(st|sk|pk, CH); output CMT|CH|Rsp.
+Now we define the security for an ID scheme.
+Essentially, it is desired that an attacker is unable
+to impersonate a prover w.r.t. an aggregated public-
+key, where at least one of the participating public-
+keys is not generated by attacker. Later we will
+use this definition to convert an ID scheme into a
+secure multi-signature. In our definition, the prover
+does not access to additional information. He is not
+given extra capability, either. Thus, our definition is
+rather weak.
+Definition 5: A canonical identification scheme
+ID = (Setup, KeyGen, P, Vτ, Θ) with linearity
+and τ ∈ N is secure if it satisfies correctness and
+security below.
+Correctness. When no attack presents, Prover will
+convince Verifier, except for a negligible probabil-
+ity.
+Security.
+For
+any
+PPT
+adversary
+A,
+Pr(EXPID,A = 1) is negligible, where EXPID,A
+is defined as follows, where pki ∈ PK for i ∈ [t]
+and pk = �t
+i=1 λi • pki.
+Experiment EXPID,A(λ)
+param← Setup(1λ);
+(pk1, sk1) ← KeyGen(param);
+(st0, pk2, · · · , pkt) ← A(param, pk1)
+λ1, · · · , λt ← Θ
+st1|CMT ← A(st0, λ1, · · · , λt);
+
+Prover (sk, pk|τ)
+Verifier (pk|τ)
+(st, CMT) ← P(param)
+CMT
+�
+CH ← Θ
+CH
+�
+Rsp ← P(st|sk|pk, CH)
+Rsp
+�
+Vτ(pk, CMT|CH|Rsp) ?= 1
+Fig. 3. Canonical Identification Protocol
+CH ← Θ; Rsp ← A(st1, CH);
+b ← Vt(pk, CMT|CH|Rsp);
+output b.
+ID is said t∗-secure if the security holds for any
+t ≤ t∗.
+VI. FROM CANONICAL LINEAR ID SCHEME TO
+KEY-AND-SIGNATURE COMPACT
+MULTI-SIGNATURE
+In this section, we show how to convert a linear
+ID scheme into a multi-sinagure so that the aggre-
+gated public-key and signature are both compact.
+The idea is to linearly add the member signatures
+(resp. public-keys) together with weights while the
+weight depends on all public-keys and is different
+for each user.
+A. Construction
+Let
+ID = (Setupid, KeyGenid, P, Vτ, Θ)
+be a canonical linear ID with parameter τ ∈ N.
+H0, H1 are two random oracles from {0, 1}∗ to
+Θ with Θ ⊆ R, where R is the ring defined for
+the linearity property of ID. Our multi-signature
+scheme Π = (Setup, KeyGen, Sign, Verify) is
+as follows.
+Setup.
+Sample
+and
+output
+param
+←
+Setupid(1λ).
+Note:
+param
+should
+be
+part
+of the input to the algorithms below. But for
+brevity, we omit it in the future.
+KeyGen.
+Sample
+(pk, sk)
+←
+KeyGenid(param);
+output
+a
+public-key
+pk
+and private key sk.
+Sign.
+Suppose
+that
+users
+with
+public-keys
+pki, i = 1, · · · , t want to jointly sign a message
+M. Let λi = H0(pki, PK) and pk = �t
+i=1 λi•pki,
+where PK = {pk1, · · · , pkt}. They run the follow-
+ing procedure.
+• R-1.
+User
+i
+takes
+(sti, CMTi)
+←
+P(param) and sends ri := H0(CMTi|pki) to
+other users.
+• R-2.
+Upon rj for all j (we do not restrict
+j ̸= i for simplicity), user i verifies if rj =
+H0(CMTj|pkj). If no, it aborts; otherwise, it
+sends CMTi to other users.
+• R-3.
+Upon CMTj, j = 1, · · · , t, user i com-
+putes CMT = �t
+j=1 λj • CMTj. It computes
+CH = H1(pk|CMT|M). Finally, it computes
+Rspi = P(sti|ski|pki, CH) and sends it to
+other signers.
+• Output. Upon Rspj, j = 1, · · · , t, user i com-
+putes Rsp = �t
+j=1 λj • Rspj, and outputs the
+aggregated public-key pk|t and multi-signature
+CMT|Rsp.
+Verify.
+Upon signature (CMT, Rsp) on mes-
+sage M with the aggregated public key pk|t,
+it outputs Vt(pk, CMT|CH|RSP), where CH =
+H1(pk|CMT|M).
+Remark. (1) Since pk|t is the aggregated public-
+key, we assume that it will be correctly computed
+and available to verifier, which is true for the
+Bitcoin application.
+(2) The most damaging attack to a multi-signature
+is the rogue key attack, where an attacker chooses
+his public-key after seeing other signers’ public-
+keys. By doing this, the attacker could man-
+age to reach an aggregated key for which he
+knows the private key. In our construction, attacker
+can not achieve this. Indeed, notice that pk =
+
+H0(pkn, PK) • pkn + �n−1
+i=1 H0(pki, PK) • pki,
+where PK
+= {pk1, · · · , pkn}. The hash-value
+weights can be computed only after PK has been
+determined. Also, if pkn is the honest user’s key,
+then it is quite random. So, H0(pkn, PK) (hence
+H0(pkn, PK) • pkn and also pk) will be random,
+given other variables in pk. So it is unlikely that
+attacker can predetermined pk and so the rogue key
+attack can not succeed.
+B. Security Theorem
+In this section, we prove the security of our
+scheme. The idea is as follows. We notice that
+the multi-signature is (CMT, RSP) that satis-
+fies Vt(pk, CMT|CH|RSP) = 1, where CH =
+H0(pk|CMT|M). Assume PK = {pk1, · · · , pkt},
+where pk1 is an honest user’s key and other keys are
+created by attacker. We want to reduce the multi-
+signature security to the security of ID scheme.
+In this case, pk will be the aggregated key with
+weights λi = H0(pki, PK). If an attacker can
+forge a multi-signature with respect to pk, we want
+to convert it into an impersonate attack to the ID
+scheme w.r.t. pk. There are two difficulties for
+this task. First, we need to simulate the signing
+oracle without sk1, where we have to compute the
+response Rsp for user of pk1 without sk1. Our idea
+is to use the simulability of the ID scheme to help:
+take a random CH and simulate an ID transcript
+CMT′|CH|Rsp′. Then, we send CMT1 = CMT′
+as the committing message. The simulation will
+be well done if we can manage to define CH as
+H1(pk|CMT|M). This will be fine if pk|CMT|M
+was never queried to H1 oracle. Fortunately, this
+is true with high probability: due to the initial
+registration message at round R-1, attacker can
+not know CMT1 before registering CMTj using
+rj (hence CMTj is known to us through oracle
+H0). Hence, CMT will have a min-entropy of
+H∞(CMT1), which is super-logarithmic and hence
+can not be guessed. That is, pk|CMT|M was
+unlikely to be queried to H1 before. Hence, the
+signing oracle will be simulated without difficulty.
+The second difficulty is how to convert the forgery
+into an impersonating attack. In the ID attack, CH
+is provided by challenger while in the forgery, CH
+is the hash value from H1. The problem is the
+attacker could make a query pk|CMT|M to H1
+oracle (we maintain) while we do not know whether
+this query is toward his final forgery output or not
+and so we do not know which CMT should be
+sent to our challenger and consequently we do not
+know which of such queries should be answered
+with our challenger’s CH. Fortunately, this is not a
+big issue as we can guess which query will be used
+for the forgery. There are a polynomial number of
+such queries. Our random guess only degrades the
+success probability by a polynomial fraction. This
+completes our idea. Now we give full details below.
+Theorem 1: Assume that h ← Θ is invertible
+in R with probability 1 − negl(λ). Let ID =
+(Setupid, KeyGenid, P, Vτ, Θ) be a secure iden-
+tification scheme with linearity and simulability.
+Then, our multi-signature scheme is EU-CMA se-
+cure.
+Proof.
+We show that if the multi-signature is
+broken by D with non-negligible probability ǫ,
+then we can construct an attacker B to break ID
+scheme with a non-negligible probability ǫ′. Given
+the challenge public-key pk∗
+1, B needs to come
+up with some other public-keys pk∗
+2, · · · , pk∗
+ν for
+some ν of his choice and receives a list of random
+numbers λ∗
+i ← Θ for i = 1, · · · , ν. Then, he needs
+to play as a prover in the ID protocol for public-
+key pk∗ = �ν
+i=1 λ∗
+i • pk∗
+i to convince the verifier
+(his challenger). Toward this, B will simulate an
+environment for D and use the responses from D to
+help complete his attack activity. The details follow.
+Upon receiving the challenge public-key pk∗
+1
+and system parameter param, B samples ℓ∗
+H0 ←
+{1, · · · , q∗
+H0}, where q∗
+H0 is the upper bound on
+the number of new queries (i.e., not queried be-
+fore) of form (pk, PK) to random oracle H0 s.t.
+pk, pk∗
+1 ∈ PK (call it a Type-I irregular query).
+In addition, a new query of format CMT|pk∗|∗ to
+oracle H1 after the ℓ∗
+H0th Type-I irregular query
+will be called a Type-II irregular query, where
+CMT ∈ CMT , pk∗ = �ν
+i=1 H0(pk∗
+i , PK∗) • pk∗
+i
+and PK∗ = {pk∗
+1, · · · , pk∗
+ν} is the public-key set
+for the ℓ∗
+H0th Type-I irregular query. Let q∗
+ch be the
+upper bound on the number of the Type-II irregu-
+
+lar queries. It then samples ℓ∗
+ch ← {1, · · · , q∗
+ch}.
+B invokes D with pk∗
+1 and param and answers
+his random oracle queries and signing queries as
+follows.
+Random Oracle H(·).
+For simplicity, we main-
+tain one random oracle H with H0(x) = H(0, x)
+and H1(x) = H(1, x). The query x to Hb is
+automatically interpreted as query b|x to H. With
+this in mind, it maintains a hash list LH (initially
+empty), consisting of records of form (u, y), where
+y = H(u). Upon a query b|x, it first checks if
+there was a record (b|x, y) in LH for some y. If
+yes, it returns y; otherwise, there are three cases
+(all irregular queries will be in these cases as they
+are unrecorded by definition).
+•
+x is not a (Type-I or Type-II) irregular query
+to Hb.
+In this case, it takes y ← Θ and adds
+(b|x, y) into LH.
+•
+x is a Type-I irregular query to Hb (thus b =
+0).
+In this setting, there are two cases.
+- x is not the ℓ∗
+H0th irregular query.
+In this
+case, for each pk′ ∈ PK, it takes h ← Θ
+and adds (0|(pk′, PK), h) into LH. Note for
+convenience, we treat each new record in LH
+as created due to a hash query (from either
+simulator B or D). For the technical reason,
+for given PK with pk∗
+1
+∈ PK, we treat
+(0|(pk∗
+1, PK), ∗) as the last record created in
+LH among all records of (0|(pk′, PK), ∗) with
+pk′ ∈ PK. Our treatment is well-defined and
+perfectly consistent with random oracle, as
+by our convention, all records of (pk′, PK)
+with pk′, pk∗
+1 ∈ PK will be recorded in LH
+simultaneously whenever it receives a Type-I
+irregular query (which is 0|x in our case).
+- x is the ℓ∗
+H0th irregular query.
+In this
+case, let 0|x = 0|(pk, PK∗) with PK∗ =
+{pk∗
+1, · · · , pk∗
+ν} for some ν ≥ 2. B sends
+{pk∗
+2, · · · , pk∗
+ν} to his challenger and re-
+ceives λ∗
+1, · · · , λ∗
+ν
+(each of which is uni-
+formly random over Θ). Then, B inserts
+(0|(pk∗
+i , PK∗), λ∗
+i ) into LH for i = 1, · · · , ν.
+This treatment is perfectly consistent with ran-
+dom oracles: a Type-I irregular query by defi-
+nition is an unrecorded query (i.e., not queried
+before) and 0|(pk′, PK∗) for each pk′ ∈ PK∗
+will be recorded in LH within one hash query
+(thus none of them was queried before).
+•
+x is a Type-II irregular query to Hb (thus b =
+1).
+In this setting, there are two cases.
+- x is not the ℓ∗
+chth Type-II irregular query. In
+this case, it takes y ← Θ and adds (1|x, y)
+into LH.
+- x is the ℓ∗
+chth Type-II irregular query.
+In
+this case, it parses x = CMT∗|pk∗|M∗ with
+CMT∗ ∈ CMT . Then, it sends CMT∗ to
+its challenger and receive CH∗. Then, it adds
+(1|x, CH∗) to LH.
+After our treatment above, x now has been recorded
+in LH. Then, the oracle returns y for (b|x, y) ∈ LH.
+Sign Os (pk1, · · · , pkn, M).
+By our security
+model, it is assumed that pk∗
+1 = pkt for some t.
+Then, B plays the role of user pkt while D plays
+users of pkj for j ̸= t in the signing algorithm. The
+action of B is as follows.
+• R-1.
+B generates rt ← Θ and sends to other
+signers (played by D).
+• R-2.
+Upon {rj}j̸=t from D, B first is-
+sues hash queries (pki, PK) for each pki ∈
+PK to compute λi = H0(pki, PK), where
+PK = {pk1, · · · , pkn}. Then, it computes pk,
+takes h ← Θ and runs SIM(h, pk∗, param)
+to simulate an ID transcript (CMT′, h, Rsp′).
+Then,
+he
+defines
+CMTt
+=
+CMT′.
+He
+also adds (0|CMTt|pkt, rt) into LH (in case
+(0|CMTt|pkt, ∗) not in LH) and otherwise
+aborts with ⊥ (denoted by event Bad0).
+Next, for each j ̸= t, it searches a record
+(0|CMTj|pkj, rj) in LH
+for some CMTj
+which results in two cases.
+(i)
+If
+(0|CMTj|pkj, rj)
+for
+all
+j
+̸=
+t are found in LH, it computes
+CMT = �n
+i=1 λi • CMTi and checks whether
+(1|pk|CMT|M, y) ∈ LH for some y. If this y
+does not exist, it records (1|pk|CMT|M, h)
+into LH and defines CH = h and sends CMTt
+to D; otherwise (denote this event by Bad1),
+B aborts with ⊥ .
+(ii)
+If (0|CMTj∗|pkj∗, rj∗) does not ex-
+ist in LH
+for some j∗, it sends CMTt
+
+to D (normally). However, we remark that
+CMTj∗ later in Step R-3 (from j∗) satis-
+fies H0(CMTj∗|pkj∗) = rj∗ (which will be
+checked there) only negligibly (so this case
+will not raise a simulation difficulty), as the
+hash value is even undefined yet and hence
+equals rj with probability 1/|Θ| only, which
+we ignore it now.
+• R-3.
+Upon
+{CMTj}j̸=t,
+B
+checks
+if
+H0(CMTj|pkj) = rj for each j. If it does
+not hold for some j, Os outputs ⊥ (normally);
+otherwise, it sends Rspt := Rsp′ to D. We
+clarify two events: (1) some CMTj found in
+R-2(i) is different from that received in the
+current step. In this case, the check in the
+current step is consistent with a negligible
+probability only as H for two different inputs
+are independent. (2) R-2(ii) occurs to some j∗
+(so CMTj∗ is not found there) while CMTj∗
+received in the current step is consistent with
+rj. As seen above, this holds with proba-
+bility 1/|Θ| only. Ignoring these events, CH
+and {CMTj}j are determined in R-2(i) and
+{CMTj}j are consistent with those received
+in the current step.
+• Output.
+Upon Rspj for j ̸= t, it computes
+RSP = �n
+j=1 λj • Rspj. The final signature is
+(CMT, RSP) with the aggregated key pk|t.
+Finally, D outputs a forgery (α, β) for mes-
+sage M′ and public keys PK′. If α|PK′|M′ ̸=
+CMT∗|PK∗|M∗ or α|β is invalid (when verified
+using V (·)), B exits with ⊥; otherwise, he verifies
+(α, β). If invalid, he outputs ⊥; otherwise, he de-
+fines Rsp∗ = β and sends it back to his challenger.
+This completes the description of B.
+We now analyze the success probability of B.
+First, the view of D is identical to the real game,
+except for the following events.
+a. In step R-2 of Os, (CMT′, h, Rsp′) is sim-
+ulated by SIM (instead of being computed
+using sk∗
+1). However, by hybrid reduction to
+simulability of ID, the view of D is statistical
+close from his view when this transcript is
+generated using sk∗
+1 (with the same h).
+b. In
+step
+R-2
+of
+oracle
+Os,
+when
+(0|CMTt|pkt, y)
+∈
+LH, Bad0 occurs for
+some y (hence the view of D is inconsistent
+if y ̸= rt). However, since CMT′ (i.e., CMTt)
+is just simulated in this oracle query and
+H∞(CMT′) = ω(log λ), CMT′ is independent
+of current records in LH. Hence, Bad0 occurs
+with
+probability
+at
+most
+Q/2H∞(CMT
+′)
+(negligible),
+where
+Q
+is
+the
+number
+of
+records in LH. We ignore this negligible
+probability from now on.
+c. In step R-2 (i), if (1|pk|CMT|M, y) ∈ LH
+for some y, then event Bad1 occurs. In this
+case, A can not define CH = h and the
+simulation can not continue. However, since
+CMT = λt • CMT′ + �
+j̸=t λj • CMTj and
+CMT′ is simulated in the current oracle and
+hence independent of the rest variables in this
+equation. Hence, as long as λt is invertible
+(which is violated only negligibly), CMT has
+a min-entropy at least H∞(CMT) = ω(log λ).
+Thus, similar to Bad0 event, Bad1 occurs
+negligibly only.
+d. Finally, when D outputs (α, β) for mes-
+sage M′ and public-key set PK′, it has
+α|PK′|M′ ̸= CMT∗|PK∗|M∗. Since (α, β)
+has been verified, a Type-I irregular query
+(pk, PK′)
+and
+a
+Type-II irregular
+query
+α|pk′|M must have been issued (the first query
+(pk, PK′) for some pk ∈ PK′ is the Type-I ir-
+regular query while the first query of α|pk′|M
+is the Type-II query; the existence of such
+queries are guaranteed as the verification of
+(α, β) by B will certainly issue these queries).
+Since ℓ∗
+H and ℓ∗
+ch are chosen uniformly ran-
+dom, they happened to the foregoing two
+queries with probability
+1
+q∗
+Hq∗
+ch ≥
+1
+q0q1 , where
+q0 (resp. q1) is the upper bound on ♯ queries
+to H0 (resp. H1).
+From the analysis of (a)(b)(c), their occurrence
+changes the adversary view negligibly. Ignoring
+this, from item d, when ℓ∗
+H and ℓ∗
+ch is chosen
+correctly, the view of D is indistinguishable from its
+view in the real game. On the other hand, it is easy
+to verify that conditional on this correct choice, a
+valid forgery indicates a successful attack by B.
+Hence, B can break the ID security with probability
+at least ǫ/q0q1, non-negligible. This contradicts the
+
+security of our ID scheme.
+□
+If the adversary always restricts the number of
+signers in the signing query and the forgery to be
+at most T, then Theorem immediately implies the
+following corollary.
+Corollary 1: Let T ≥ 2. Assume that h ← Θ
+is invertible in R with probability 1 − negl(λ).
+Let ID = (Setupid, KeyGenid, P, Vτ, Θ) be a
+T-secure identification scheme with linearity and
+simulability. Then, our multi-signature scheme is
+T-EU-CMA secure.
+VII. REALIZATIONS
+In this section, we will realize our compiler with
+ID schemes: Schnorr ID scheme and a lattice-based
+ID scheme. The first scheme is similar to Boneh
+et al. [13]. But we keep it as it is very simple
+and efficient and can demonstrate the usage of our
+compiler. The second one is new and breaks a
+barrier that the previous schemes can not overcome.
+A. Realization I: Schnorr Identification
+In this section, we apply our compiler to the well-
+known Schnorr ID scheme [48]. Toward this, we
+only need to show that it is linear with simula-
+bility and security. For clarity, we first review this
+scheme.
+Let q be a large prime. Consider a prime group of
+order q with a random generator g (e.g., the group
+on elliptic curve secp256k1 of y2 = x3 + 7 for
+Bitcoin). The Schnorr identification is depicted in
+Fig. 4. This scheme can be regarded as a realization
+of the parameterized ID scheme with parameter τ
+never used. In the following, we show that Schnorr
+ID scheme satisfies the three properties.
+Linearity.
+Notice that SK = RSP = R =
+Θ = Zq, CMT = PK = ⟨g⟩. We now verify the
+linearity property.
+i. As seen in Section II-A, Zq and ⟨g⟩ are
+both Zq-modules, where the multiplication •
+between R = Zq and Zq is the multiplica-
+tion of Zq, while • between R = Zq and
+⟨g⟩ is exponentiation: s • m = ms. Hence,
+SK, PK, CMT , RSP are R-modules.
+ii. Let pki = gsi with ski = si, i = 1, · · · , n. Let
+λ1, · · · , λn ∈ R. Then, sk = �n
+i=1 λi • ski =
+�n
+i=1 λisi, where the addition is the group
+operation for SK (i.e., addition in Zq). Note
+the group operation for PK is the multipli-
+cation in ⟨g⟩. Hence, pk = �n
+i=1 λi • pki =
+�n
+i=1 pkλi
+i
+= g
+�n
+i=1 λisi. Thus, sk ∈ SK is the
+private key of pk ∈ PK.
+iii. Let Xi|c|zi be a transcript of ID w.r.t., pki =
+gsi and ski = si, i = 1, · · · , n. For λi ∈ R,
+X = �n
+i=1 λi • Xi = �n
+i=1 Xλi
+i
+and z =
+�n
+i=1 λi • zi = �n
+i=1 λizi. If gzi = pkc
+iXi,
+then �n
+i=1 gλizi = �n
+i=1(pkc
+i Xi)λi. Hence,
+gz = pk
+cX, desired!
+Simulability.
+Let pk = gs be the public-key
+and sk = s be the private key. For c ← Zq, we
+define SIM by taking z ← Zq and X = gzpk−c.
+The simulated ID transcript is X|c|z. Obviously,
+this transcript is valid (i.e., it passes the verifica-
+tion). Now we show that for any (even unbounded)
+distinguisher D that has oracle access to Oid can
+not distinguish the output of SIM from the real
+ID transcript. Notice for both simulated and real
+transcripts X|c|z, it satisfies gz = pkcX. Hence,
+X = gx for some x ∈ Zq and z = cs + x. In
+the real transcript, x ← Zq while the simulated
+transcript z ← Zq. Hence, given c, (x, z) (hence
+X, z) in both transcripts has the same distribution.
+Since c is uniformly random in Zq in the simulation,
+the simulated and real transcripts have the same
+distribution (independent of adversary view before
+the challenge which includes the responses from
+Oid). Thus, the adversary view, given oracle access
+to Oid, in both cases has the same distribution. The
+simulability follows.
+Security.
+We now prove the security of Schnorr
+ID scheme under Definition 5.
+Lemma 2: Under discrete logarithm assumption,
+Schnorr ID scheme is secure w.r.t. Definition 5.
+Proof. If there exists an adversary D that breaks the
+Schnorr ID scheme with non-negligible probability
+ǫ, then we construct an adversary A that breaks
+discrete logarithm in ⟨g⟩ with a non-negligible
+probability ǫ′. The idea is to make use of D to con-
+struct an algorithm A for the nested forking lemma
+
+Prover (s, A = gs)
+Verifier (A = gs)
+x ← Zq, X = gx
+X
+�
+c ← Zq
+c
+�
+z = sc + x mod q
+z
+�
+gz ?= AcX
+Fig. 4. Schnorr Identification Scheme
+and then use the output of the forking algorithm
+to derive the discrete logarithm for the challenge.
+Upon a challenge A1 = gx and parameters q, g,
+A constructs A((A1, g, q), λ1, c; ρ) as follows (so
+h1|h2 = λ1|c with q = 2 in the forking algorithm),
+where A = �t
+i=1 Aλi
+i .
+Algorithm A((A1, g, q), λ1, c; ρ)
+Parse ρ as two parts: ρ = ρ0|ρ1
+(st0, A2, · · · , At) ← D(q, g, A1; ρ0)
+λ2, · · · , λt ← Zq using randomness ρ1
+st1|X ← D(st0, λ1, · · · , λt);
+z ← D(st1, c);
+If gz = A
+c · X, then b = 1;
+else b = 0;
+output (b, 2b, {Ai|λi}t
+1|X|z|c|g|q).
+From the description of A and the forking algorithm
+FA (for the forking lemma), the rewinding in the
+forking algorithm FA only changes λ1 and/or c as
+well as those affected by (λ1, c). In terms of forking
+lemma terminology, we have (h1, h2) = (λ1, c)
+and I0 = 1, J0 = 2 (for a successful execution;
+otherwise, A will abort when I0 ≤ J0). Let us now
+analyze algorithm forking algorithm FA. When four
+executions are executed successfully (i.e., b = 1 for
+all cases), then the output for each execution will be
+described as follows. Let Ai = gai for i = 1, · · · , t.
+- Execution
+0.
+It
+outputs
+(1, 2, {Ai|λi}t
+1|X|z|c|g|q). As the verification
+passes,
+z = (
+t
+�
+i=1
+λiai)c + x,
+(20)
+where X = gx.
+- Execution 1. Compared with execution 0, the
+input only changes c to ˆc. From the code of
+A, the output is (1, 2, {Ai|λi}t
+1|X|ˆz|ˆc|g|q). As
+the verification passes,
+ˆz = (
+t
+�
+i=1
+λiai)ˆc + x.
+(21)
+- Execution 2.
+Compared with execution 0,
+the
+input
+changes
+λ1
+to
+¯λ1
+and
+c
+to
+¯c.
+From
+the
+code
+of
+A,
+the
+output
+is
+(1, 2, {Ai|λi}t
+2|A1|¯λ1|X′|¯z|¯c|g|q). As the ver-
+ification passes,
+¯z = (¯λ1a1 +
+t
+�
+i=2
+λiai)¯c + x′,
+(22)
+where X′ = gx′.
+- Execution 3.
+Compared with execution 0,
+the
+input
+changes
+λ1
+to
+¯λ1
+and
+c
+to
+c.
+From
+the
+code
+of
+A,
+the
+output
+is
+(1, 2, {Ai|λi}t
+2|A1|¯λ1|X′|z|c|g|q). As the ver-
+ification passes,
+z = (¯λ1a1 +
+t
+�
+i=2
+λiai)c + x′.
+(23)
+From Eqs. (23)(22), A can derive ¯λ1a1+�t
+i=2 λiai,
+as long as c ̸= c′ in Zq. Similarly, from Eqs.
+(21)(20), A can derive λ1a1 + �t
+i=2 λiai, as long
+as ¯c ̸= c. This can further give a1, as long as
+λ1 ̸= ¯λ1 in Zq. Finally, if the forking algorithm
+does not fail, then the four executions succeeds
+and (c ̸= c′) ∧ (¯c ̸= c) ∧ (λ1 ̸= ¯λ1) =True. By
+forking lemma, it does not fail with probability at
+least ǫ4/(1 · 1) − 3/|Θ| = ǫ4 − 3/q. Hence, A can
+obtain a1 with probability at least ǫ4 − 3/q, non-
+negligible. This contradicts to the discrete logarithm
+assumption.
+□
+Key-and-Signature
+Compact
+Multi-Signature
+from Schnorr ID Scheme.
+Since Schnorr ID
+
+scheme satisfies the linearity, simulability and
+special soundness, the multi-signature from this
+scheme using our compiler is obtained. For clarity,
+we give the complete signature in the following.
+Let pki = gsi be the public-key with private key
+ski = si for i = 1, · · · , n. When users PK =
+{pk1, · · · , pkn} want to jointly sign a message M,
+they act as follows.
+• R-1.
+User i generates Xi = gxi for xi ← Zq
+and sends H0(Xi|pki) to other users.
+• R-2.
+Upon {rj}n
+j=1, user i sends Xi to other
+users.
+• R-3. Upon {Xj}n
+j=1, user i checks rj
+?=
+H0(Xj|pkj) for all j. If not, he rejects; other-
+wise, he computes
+pk =
+n
+�
+i=1
+pkH0(pki,P K)
+i
+(24)
+X =
+n
+�
+i=1
+XH0(pki,P K)
+i
+.
+(25)
+Then, he computes
+c = H1(pk|X|M), zi = sic + xi
+(26)
+and sends zi to leader.
+• Output.
+Receiving all zj’s, user i computes
+z =
+n
+�
+j=1
+H0(pkj, PK)zj.
+Finally, it outputs (X, z) as the multi-signature
+of M with the aggregated public-key pk (note:
+the compiler protocol includes n in the aggre-
+gated key; we omit it here as it is not used in
+the verification).
+• Verification.
+To verify signature (X, z) for
+M with the aggregated public-key pk, it com-
+putes c = H1(pk|X|M). It accepts only if
+gz = pk
+c · X.
+We denote this signature scheme by Schnorr-
+MultiSig. Notice that c ← Zq is invertible in R
+with probability 1 − 1/q. As it satisfies linearity,
+simulability and security, by Theorem 1, we have
+the following.
+Corollary 2: If Discrete logarithm assumption in
+⟨g⟩ holds, then Schnorr-MultiSig is EU-CMA.
+Remark. Boneh et al. [13] proposed a method
+that transforms Schnorr ID to a key-and-signature
+compact multi-signature. Their protocol is an im-
+provement of Maxwell et al. [38] to overcome
+a simulation flaw. Their protocol is also 3-round
+but computationally more efficient in the signing
+process than ours. However, our sizes of aggregated
+(public-key, signature) as well as the verification
+cost are all the same as theirs (also identical to
+the original Schnorr signature case). Aggregated
+public-key and signature have impacts on the stor-
+age at a large number of blockchain nodes and
+the verification cost has the impact on the power
+consumption on these nodes. The signing cost is
+relatively not so important as it only has impact on
+the involved signers. Boneh et al. [13] uses λisi as
+a secret for public-key pkλi
+i
+to generate a member
+signature Xi|c|zi and the final multi-signature �
+X =
+�
+i Xi and ˜z = �
+i zi. Their main saving (over
+us) is to avoid n exponentiations in computing our
+X. One might be motivated to modify our general
+compiler so that it uses λi•pki (whose private key is
+λi•ski) to generate a member signature CMTi|Rspi
+so that the final multi-signature is �
+CMT|�
+RSP with
+�
+CMT = �
+i CMTi and �
+RSP = �
+i Rspi. However,
+this looking secure scheme has a simulation issue
+in general when we prove Theorem 1: it is required
+that {SIM(CH, λ•pk)}λ is indistinguishable from
+the list of real transcripts for a fixed but random
+pk while it is not clear how this can be proven
+generally.
+B. Realization II: a new lattice-based ID scheme
+In this section, we propose a new ID scheme
+from lattice and then apply our compiler to obtain
+a lattice-based multi-signature scheme. This is the
+first lattice-based multi-signature that has both a
+compact public-key and a compact signature with-
+out a restart during the signing process.
+Notations.
+The following notations are specific
+for this section (see Section II for more).
+• As a convention for lattice over ring, this
+section uses security parameter n (a power of
+2), instead of λ;
+• q is a prime with q ≡ 3 mod 8;
+
+• R = Z[x]/(xn + 1); Rq = Zq[x]/(xn + 1); R∗
+q
+is the set of invertible elements in Rq;
+• for a vector w, we implicitly assume it is a
+column vector and the ith component is wi or
+w[i];
+• for a matrix or vector X, XT is its transpose;
+• 1 denotes the all-1 vector (1, · · · , 1)T of di-
+mension that is clear from the context;
+• for u = �n−1
+i=0 uixi ∈ R, ||u||∞ = maxi |ui|;
+• α ∈ Zq always uses the default representative
+with −(q−1)/2 ≤ α ≤ (q−1)/2 and similarly,
+for u ∈ Rq, each coefficient of u by default
+belongs to this range;
+• e = 2.71828 · · · is the Euler’s number;
+• C = {c ∈ R | ||c||∞ ≤ log n, deg(c) < n/2}
+• Y = {y ∈ R | ||y||∞ ≤ n1.5σ log3 n}
+• Z = {z ∈ R | ||z||∞ ≤ (n − 1)n1/2σ log3 n}.
+1) Ring-LWE and Ring-SIS: In this section, we
+introduce the ring-LWE amd ring-SIS assumptions
+(see [35], [45], [33] for details). For σ > 0, distri-
+bution DZn,σ assigns the probability proportional to
+e−π||y||2/σ2 for any y ∈ Zn and 0 for other cases.
+As in [1], y ← DR,σ samples y = �n−1
+i=0 yixi from
+R with yi ← DZ,σ.
+The
+Ring
+Learning
+With
+Error
+(Ring-
+LWEq,σ,2n) problem over R with standard deviation
+σ is defined as follows. Initially, it takes s ← DR,σ
+as secret. It then takes a ← Rq, e ← DR,σ and
+outputs (a, as + e). The problem is to distinguish
+(a, as + e) from a tuple (a, b) for a, b ← Rq. The
+Ring-LWEq,σ,2n assumption is to say that no PPT
+algorithm can solve Ring-LWEq,σ,2n problem with
+a non-negligible advantage. According to [34],
+[18], ring-LWE assumption with σ =
+˜Ω(n3/4)
+is provably hard and so it is safe to assume
+σ = Ω(n).
+The Small Integer Solution problem with pa-
+rameters q, m, β over ring R (ring-SISq,m,β) is
+as follows: given m uniformly random elements
+a1, · · · , am over Rq, find (t1, · · · , tm) so that
+||ti||∞ ≤ β and a1t1 + · · · + amtm = 0 (note:
+here we use || · ||∞ norm while the literature
+regularly uses square-root norm || · ||. However, the
+gap is only a factor n on β and does not affect
+the validity of the assumption according to the
+current research status for ring-SIS). We consider
+the case m = 3. As we use q = 3 mod 8, by
+[9, Theorem 1], xn + 1 = Φ1(x)Φ2(x) for irre-
+ducible polynomials Φ1(x), Φ2(x) of degree n/2.
+So by Chinese remainder theorem, ai is invertible,
+except for probability 2q−n/2. Hence, ring-SIS is
+equivalent to the case of invertible a2 which is
+further equivalent to problem a1t1 + t2 + a3t3 = 0,
+as we can multiply it by a−1
+2 . By [33], [15], the
+best quantum polynomial algorithm for ring-SIS
+problem with q, m can only solve β = 2 ˜O(√n) case.
+Thus, it is safe to assume Ring-SISq,m,β for any
+polynomial β or even β = 2
+4√n.
+2) Construction: We now describe our new ID
+scheme from ring R. Initially, take s1, s2
+←
+DR,σ, a ← R∗
+q and compute u = as1 + s2. The
+system parameter is a; the public key is u and the
+private key is (s1, s2). Our ID scheme is as follows;
+also see Fig. 5.
+1. Prover generates y1, y2 ← Yµ and computes
+v = ay1 + y2 and sends v to Verifier, where
+µ ≥ log2 n.
+2. Receiver samples c ← C and sends it to Prover.
+3. Upon c, Prover does the following:
+a. Compute z1 = s1c · 1 + y1, z2 = s2c ·
+1 + y2;
+b. Let A = {j | z1j, z2j ∈ Z}. If A = ∅,
+abort; otherwise, take j∗ ← A and com-
+pute
+z1 = z1j∗ +
+�
+j̸=j∗
+y1j, z2 = z2j∗ +
+�
+j̸=j∗
+y2j.
+4. Upon z1, z2, Verifier checks
+µ
+�
+i=1
+vi
+?= az1 + z2 − uc, ||zb||∞
+?
+≤ η, b = 1, 2,
+where ηt = 5σn2√tµ log6 n and t is a positive
+integer (see the remark below). If all are valid,
+it accepts; otherwise, it rejects.
+Remark. We give two clarifications.
+(1)
+The correctness does not need ηt to vary with
+t as it is defined so. Actually, η1 = 3σn1.5√µ log4 n
+suffices for all t. However, we need the dependency
+on t for the linearity and later for the multi-
+signature. Especially, for linearity with t transcripts,
+ηt is needed to depend on t. Further, for the
+
+multi-signature scenario, t stands for the number
+of signers.
+(2)
+It should be pointed out that the choice of j∗
+(if it exists) does not affect z1, z2 at all as zb = sbc+
+�t
+j=1 ybj for b = 1, 2. In addition, the probability
+that j∗ does not exist is exponentially small in n and
+so defining j∗ is unnecessary. However, we keep it
+for ease of analysis later.
+Correctness. We now prove the correctness with ηt
+replaced by a smaller value η1 = 3σn1.5√µ log4 n.
+When all signers are honest, the protocol is easily
+seen to be correct if we can show A = ∅ or
+||zb||∞ > η1 has a negligible probability. The
+former is shown in Lemma 5 below. For the latter,
+notice that z1 = s1c + y11 + · · · + y1µ. If we
+use w ∈ R to denote the coefficient vector of the
+polynomial w. Then,
+y11 + · · · + y1µ = y11 + · · · + y1µ.
+(27)
+Notice each component of y1j is uniformly random
+in {−σn1.5 log3 n, · · · , σn1.5 log3 n}. By Hoeffd-
+ing inequality on each of the vector component
+in Eq. (27), || �
+i y1i||∞ > 2σn1.5√µ log4 n only
+has a probability at most 2ne− log2 n. By Lemma
+3 below, ||sc||∞ > σn1/2 log3 n with probability
+at most e−Ω(log2 n). Hence, correctness holds for
+bound η1, except for probability at most e−Ω(log2 n)
+(note: for brevity, this quantity should be under-
+stood as there exists constant C so that the excep-
+tion probability is at most e−C log2 n; we will later
+keep this convention without a mention).
+3) Analysis: In this section, we analyze our
+ID scheme. We start with some preparations. The
+following lemma is adapted from [1, Lemma 4],
+where our restriction that the element c of C has a
+degree at most n/2, does not affect the proof.
+Lemma 3: [1] If s ← DR,σ and c ← C, then
+Pr(||sc||∞ ≤ σn1/2 log3 n) ≥ 1 − e−Ω(log2 n),
+where e = 2.71828 · · · is the Euler’s number
+The lemma below was in the proof of [1, Lemma
+3].
+Lemma 4: [1] Fix γ
+∈
+R with ||γ||∞
+≤
+σn1/2 log3 n. Then, for y ← Y, we have
+Pr(γ + y ∈ Z) ≥ 1
+e − 1
+en
+Pr(γ + y = g | γ + y ∈ Z) =
+1
+|Z|, ∀g ∈ Z.
+Lemma 5: Let A be the index set in our ID
+scheme. Then, Pr(A
+=
+∅)
+<
+e−Ω(log2 n) for
+µ ≥ Ω(log2 n).
+Proof.
+Notice zbj = sc + ybj for b = 1, 2. By
+Lemma 3, ||sc||∞ ≤ σn1/2 log3 n with probabil-
+ity 1 − e−Ω(log2 n). Fixing sc (that satisfies this
+condition), zbj for b = 1, 2, j = 1, · · · , µ are
+independent and thus by Lemma 4, A = ∅ with
+probability at most (1− 1
+e2 (1− 1
+n)2)µ < (1−
+1
+4e2 )µ,
+exponentially small. Together with the probability
+for ||sc||∞ ≤ σn1/2 log3 n, we conclude the lemma.
+□
+Lemma 6: If u ← C, then u is invertible in Rq
+with probability 1 − (1 + 2 log n)−n/2.
+Proof. Recall that q ≡ 3 mod 8 in this section. By
+Blake et al. [9, Theorem 1], xn + 1 = Φ1(x)Φ2(x)
+mod q, where Φ1(x), Φ2(x) have degree n/2 and
+are irreducible over Zq. By Chinese remainder
+theorem, u is invertible in Rq if and only if it is
+non-zero mod Φb(x) for both b = 1, 2. Since u has
+a degree at most n/2 − 1, u remains unchanged
+after mod Φb(x). Hence, it is invertible in Rq if
+and only if u is non-zero. This has a probability
+1 − (1 + 2 log n)−n/2.
+□
+Simulability. We now show the simulability of our
+ID scheme. Given the public-key u and c ← C, we
+define the simulator SIM as follows.
+- Sample j∗ ← [µ] and z1j∗, z2j∗ ← Z; compute
+vj∗ = az1j∗ + z2j∗ − uc;
+- For j ∈ [µ] − {j∗}, sample y1j, y2j ← Y and
+compute vj = ay1j + y2j.
+- Compute zb = zbj∗ + �
+j̸=j∗ ybj, b = 1, 2.
+- Output v = (v1, · · · , vµ)T and z1, z2.
+This simulation is valid by the following lemma.
+Lemma 7: The output of SIM is statistically
+close to the real transcript, even if the distin-
+guisher has oracle access to O((s1, s2), u), where
+
+Prover ((s1, s2), u|t)
+Verifier (u|t)
+y1, y2 ← Yµ, v = ay1 + y2
+v
+�
+b = 1, 2 :
+zb = sbc·1+yb
+A = {j | z1j, z2j ∈
+Z}, j∗ ← A
+b = 1, 2 : zb = zbj∗+�
+j̸=j∗ ybj
+c ← C
+c
+�
+z1,z2
+�
+b = 1, 2 :
+||zb||∞ < ηt?
+�µ
+j=1 vj
+?= az1 + z2 − uc
+Fig. 5. Our Lattice-based ID Scheme (Note: Membership checks c ∈ C at Prover is important but omitted in the figure; 1 is the vector of all 1 of length µ.)
+(s1, s2) ← D2
+R,σ is the private key and u = as1+s2
+is the public-key.
+Proof.
+First, we can assume A ̸= ∅ for the
+real transcript as by Lemma 5 this is violated
+negligibly only. Then, by symmetry, j∗ for the real
+transcript is uniformly random over {1, · · · , µ}.
+By the definition of j∗, we know that z1j∗, z2j∗
+both belong to Z. In this case, by Lemma 4,
+sc + y1j∗, sc + y2j∗ for the real transcript with
+given sc satisfying ||sc||∞ < σn1/2 log3 n, are
+independent and uniformly random over Z. By
+lemma 3, we conclude that z1j∗ and z2j∗ are
+statistically close to uniform over Z if they belong
+to Z. On the other hand, when z1j∗ and z2j∗ are
+given, vj∗ is fixed as vj∗ = az1j∗ + z2j∗ − uc.
+Thus, our simulation of z1j∗, z2j∗, vj∗ is statistically
+close to that in the real transcript. On the other
+hand, our simulation of y1j, y2j, vj for j ̸= j∗ is
+exactly according to the real distribution. Thus, our
+simulation is statistically close to the real transcript.
+This closeness holds (even given adversary view,
+which includes the responses from Oid). Hence, the
+simulability follows.
+□
+Security.
+Now we prove the security of our ID
+scheme, where the attacker needs to generate z1, z2
+(given challenge c) to pass the verification w.r.t.
+an aggregated public-key u. We show that this is
+unlikely by the ring-SIS assumption.
+Lemma 8: Under ring-LWEq,σ,2n
+and ring-
+SIS3,q,βt∗ assumptions, our scheme is t∗-secure
+(with respect to Definition 5), where βt∗
+=
+16ηt∗√n log2 n and σ = Ω(n).
+Proof.
+If there exists an adversary D that breaks
+our ring-based ID scheme with non-negligible prob-
+ability ǫ, then we construct an adversary A that
+breaks ring-SIS assumption with a non-negligible
+probability ǫ′. The idea is to make use of D to
+construct an algorithm A for the nested forking
+lemma and then uses the output of the forking
+algorithm to obtain a solution for ring-SIS problem.
+Upon a challenge u1 and a (both uniformly over
+Rq), A needs to find short α1, α2, α3 ∈ R so that
+aα1 +α2 +u1α3 = 0. Toward this, A constructs an
+algorithm A((u1, a), λ1, c; ρ) as follows (so q = 2
+in the forking algorithm), where λi, c ← C and
+u = �t
+i=1 λi · ui with ui ∈ Rq (in the description
+of A).
+Algorithm A((u1, a), λ1, c; ρ)
+Parse ρ as two parts: ρ = ρ0|ρ1
+(st0, u2, · · · , ut) ← D(u1, a; ρ0)
+λ2, · · · , λt ← C using randomness ρ1
+st1|v ← D(st0, λ1, · · · , λt);
+(z1, z2) ← D(st1, c);
+If ||zb||∞ < ηt and �µ
+j=1 vj = az1 + z2 − uc,
+then
+b = 1;
+else
+b = 0;
+Output (b, 2b, {ui|λi}t
+1|v|z1|z2|c|a).
+From the description of A and the forking al-
+gorithm FA (for the forking lemma), the rewind-
+ing in FA only updates λ1 and/or c as well as
+variables affected by (λ1, c). In terms of forking
+
+lemma terminology, we have (h1, h2) = (λ1, c)
+and I0 = 1, J0 = 2 (for a successful execution;
+otherwise, A will abort when I0 ≤ J0). Let us now
+analyze algorithm forking algorithm FA. When four
+executions are executed successfully (i.e., b = 1 for
+all cases), then the output for each execution will
+be described as follows.
+- Execution
+0.
+It
+outputs
+(1, 2, {ui|λi}t
+1|v|z1|z2|c|a). Since it succeeds,
+||zb||∞ ≤ ηt (b = 1, 2) and
+µ
+�
+i=1
+vi = az1 + z2 − uc.
+(28)
+- Execution 1. Compared with execution 0, the
+input only changes c to ˆc. From the code of A,
+the output is (1, 2, {ui|λi}t
+1|v|ˆz1|ˆz2|ˆc|a). Since
+it succeeds, ||ˆzb||∞ ≤ ηt (b = 1, 2) and
+µ
+�
+i=1
+vi = aˆz1 + ˆz2 − uˆc.
+(29)
+- Execution 2.
+Compared with execution 0,
+the input changes λ1 to ¯λ1 and changes
+c to ¯c. From the code of A, the output
+is (1, 2, {ui|λi}t
+2|u1|¯λ1|v′|¯z1|¯z2|¯c|a). Since it
+succeeds, ||¯zb||∞ ≤ ηt (b = 1, 2) and
+µ
+�
+i=1
+v′
+i = a¯z1 + ¯z2 − u′¯c,
+(30)
+where u′ = ¯λ1u1 + �t
+i=2 λiui.
+- Execution 3.
+Compared with execution 0,
+the input changes λ1 to ¯λ1 and changes
+c to c. From the code of A, the output
+is (1, 2, {ui|λi}t
+2|u1|¯λ1|v′|z1|z2|c|a). Since it
+succeeds, ||zb||∞ ≤ ηt (b = 1, 2) and
+µ
+�
+i=1
+v′
+i = az1 + z2 − u′c.
+(31)
+From Eqs. (31)(30), A can derive
+a(z1 − ¯z1) + (z2 − ¯z2) − u′(c − c) = 0.
+(32)
+From Eqs. (29)(28),
+a(ˆz1 − z1) + (ˆz2 − z2) − u(ˆc − c) = 0.
+(33)
+Notice that Eq. (32)×(ˆc−c)-Eq. (33)×(c−c) gives
+aα1 + α2 − u1α3 = 0,
+(34)
+where
+α1 =(z1 − ¯z1)(ˆc − c) − (ˆz1 − z1)(c − c)
+(35)
+α2 =(z2 − ¯z2)(ˆc − c) − (ˆz2 − z2)(c − c)
+(36)
+α3 =(λ1 − ¯λ1)(ˆc − c)(c − c).
+(37)
+Hence, (α1, α2, −α3) forms a solution to ring-
+SIS problem with parameter (a, 1, u). It suffices to
+verify that each αi is short and also at least one
+of them is non-zero. For the second condition, it
+suffices to make sure that the probability for α3 = 0
+is small. Notice that by Chinese remainder theorem,
+α3 = 0 implies λ1 = ¯λ1 mod Φ1(x) or c = c
+mod Φ1(x) or ˆc = c mod Φ1(x). Similarly, this
+must also hold for modular Φ2(x) but it suffices
+to consider Φ1(x) only. Since λ1, ¯λ1, c, c, ˆc is uni-
+formly random over C, each of the equality holds
+with probability (1 + 2 log n)−n/2 only and hence
+Pr(α3 = 0) ≤ 3(1 + 2 log n)−n/2, negligible! For
+the first condition, notice that ||ˆc − c||∞ ≤ 2 log n
+and ||z1 − z1||∞ ≤ 2ηt. Further, the constant term
+of (ˆc − c)(z1 − z1) is
+(ˆc−c)[0]·(z1−z1)[0]−
+n
+2 −1
+�
+k=1
+(ˆc−c)[k]·(z1−z1)[n−k]
+which, by Heoffding inequality, has an abso-
+lute value at most
+�
+n/2 log n · 8ηt log n
+≤
+8ηt
+√n log2 n,
+with
+probability
+at
+least
+1 −
+e−Ω(log2 n). The constant term of (ˆz1 − z1)(c −
+c) is similar. Hence, |α1[0]| ≤ 16ηt
+√n log2 n,
+with probability at least 1 − e−Ω(log2 n). The gen-
+eral case of α1[i] is similar. Hence, ||α1||∞ ≤
+16ηt
+√n log2 n with probability 1−e−Ω(log2 n). Sim-
+ilarly, ||α2||∞ has the same property. We can use
+the above proof technique to show that ||(ˆc −
+c)(c − c)||∞ ≤ 8 log n · √n log2 n with proba-
+bility 1 − e−Ω(log2 n). Since λ1, ¯λ1 is uniformly
+random over C, using the same technique, we have
+||α3||∞ ≤ √n log n · 32√n log4 n = 32n log5 n,
+with probability 1 − e−Ω(log2 n). Thus, we find a
+ring-SIS solution (α1, α2, −α3) of length at most
+16ηt
+√n log2 n. Assume that the probability that D
+succeeds in one execution is ˆǫ. Then, by forking
+lemma, it succeeds in four executions with proba-
+bility ˆǫ4 − 3(1 + 2 log n)−n/2. This implies that A
+breaks the ring-SIS assumption with probability at
+least ˆǫ4 − 3(1 + 2 log n)−n/2 − e−Ω(log2 n).
+
+Finally, notice that the input u1 is uniformly ran-
+dom over Rq while in our ID scheme u1 = as1+s2
+for s1, s2 ← DR,σ. However, under ring-LWE
+assumption, it is immediate that ˆǫ ≥ ǫ − negl(n).
+Hence, A can succeed with probability at least
+ǫ4 − negl(n), this contradicts the assumption of
+ring-SIS.
+□
+Linearity. Let SK = RSP = (Rq, Rq), CMT =
+Rµ
+q , PK = Rq, R = Rq. We now verifies the
+linearity.
+i. Obviously, SK is a R-module under the op-
+eration •: for (s1, s2) ∈ SK and c ∈ R,
+c•(s1, s2) = (cs1, cs2), where cs1 and cs2 are
+multiplications in Rq. Other cases are similar.
+ii. If
+(s1i, s2i)
+∈
+SK
+and
+λi
+∈
+C
+for
+i
+=
+1, · · · , t, then �t
+i=1(λis1i, λis2i)
+=
+(�t
+i=1 λis1i, �t
+i=1 λis2i) is obviously the pri-
+vate
+key
+of
+�t
+i=1 λi · (as1i + s2i)
+=
+a(�t
+i=1 λis1i) + (�t
+i=1 λis2i). However, we
+emphasize
+that
+this
+key
+is
+not
+neces-
+sarily
+short.
+But for
+randomly generated
+(pki, ski, λi)’s, Lemma 9 implicitly implies
+that the aggregated private key has length
+at most 2
+√
+ntσ log3 n (except for probability
+e−Ω(log2 n)); see maxv |Sv| with |Sv| given in
+the proof of Lemma 9).
+iii. If {(vi, c, z1i, z2i)}t
+i=1 are honestly generated
+accepting transcripts w.r.t the honestly pub-
+lic/private key pairs {(ui, (s1i, s2i))}i, then
+µ
+�
+j=1
+vij = az1i + z2i − uic.
+(38)
+Together
+with
+Lemma
+9
+be-
+low,
+for
+h1, · · · , ht
+←
+C,
+(�t
+i=1 hivi, c, �t
+i=1 hiz1i, �t
+i=1 hiz2i)
+satisfies (except for probability e−Ω(log2 n))
+||
+t
+�
+i=1
+hiz1i||∞ ≤ηt,
+||
+t
+�
+i=1
+hiz1i||∞ ≤ ηt,
+µ
+�
+j=1
+(
+t
+�
+i=1
+hivij) =a(
+t
+�
+i=1
+hiz1i) + (
+t
+�
+i=1
+hiz2i) − (
+t
+�
+i=1
+hiui)c,
+where ηt = 5σn2√tµ log6 n. The linearity
+follows.
+Lemma 9: Fix integer t ≥ 2 and σ ≥ ω(log n).
+Assume si ← DR,σ, hi ← C, yij ← Y for i ∈
+[t], j ∈ [µ], c ← C. Let
+Z =
+t
+�
+i=1
+hi(sic +
+µ
+�
+j=1
+yij).
+(39)
+Then, ||Z||∞ ≤ ηt with probability 1 − e−Ω(log2 n).
+Proof. Notice
+Z[0] =
+n−1
+�
+v=0
+Sv · c[v] −
+t
+�
+i=1
+n−1
+�
+k=0
+hi[n − k] · Yik,
+where Yik = �µ
+j=1 yij[k], hi[n]
+def
+= −hi[0] and
+Sv =
+t
+�
+i=1
+n−1
+�
+k=0
+hi[n − k]si[k − v].
+By [24, Lemma 4.2], ||si||∞ ≤ σ log n, except
+for probability e−Ω(log2 n). When this is satisfied,
+terms hi[n − k]si[k − v] in Sv are independent
+random variables in the range [−σ log2 n, σ log2 n].
+By Heoffding inequality, |Sv| ≤ 2
+√
+ntσ log3 n, ex-
+cept for probability e−Ω(log2 n). Since yij[k] is uni-
+formly random over [−σn1.5 log3 n, σn1.5 log3 n],
+by Heoffding inequality, |Yik| ≤ 2σ√µn1.5 log4 n,
+except for a probability e− log2 n. Assuming these
+inequalities for Sv and Yik, we know that from
+Heoffding inequality again,
+|
+n−1
+�
+v=1
+Sv · c[v]| ≤ 4σn
+√
+t log5 n
+|
+�
+i,k
+hi[n − k] · Yik| ≤
+√
+nt log n · 4σ√µn1.5 log5 n,
+except for probability e−Ω(log2 n). Hence, we con-
+clude that |Z[0]| ≤ 5σn2√tµ log6 n, except for
+e−Ω(log2 n). We can similarly bound Z[i] for i ≥ 1
+and so ||Z||∞ ≤ 5σn2√tµ log6 n, except for prob-
+ability e−Ω(log2 n).
+□
+4) Key-and-Signature Compact Multi-signature
+Scheme from our ID scheme.:
+With the simu-
+lability, linearity and security for our ID, we can
+use our compiler to convert it into a secure multi-
+signature. We now describe this scheme as follows.
+Let (s1i, s2i) be the private key of public-key
+ui = as1i + s2i for i = 1, · · · , t. If the users
+
+of u1, · · · , ut want to jointly sign M, they com-
+pute the aggregated public-key u|t and execute the
+protocol as follows, where H0, H1 : {0, 1}∗ → C
+and we define w = �t
+i=1 H0(ui, U)wi for any
+list of variables w1, · · · , wt in the description and
+U = (u1, · · · , ut) (e.g., v = �t
+i=1 H0(ui, U) · vi).
+• R-1.
+User generates y1i, y2i ← Yµ, com-
+putes vi = ay1i + y2i and sends H0(vi|ui) to
+other users.
+• R-2.
+Upon receiving all rj, j = 1, · · · , t,
+user i sends vi to other users.
+• R-3.
+Upon all vj, user i verifies its consis-
+tency with rj. If verification fails, it rejects;
+otherwise, it computes v and c = H1(u|v|M)
+as well as the response (z1i, z2i) for challenge
+c in the ID scheme with committing message
+vi.
+• output.
+After receiving (z1j, z2j) for j ∈ [t],
+user i computes multi-signature (z1, z2, v).
+The aggregated public-key is u|t.
+• Verify.
+Upon (z1, z2, v), it verifies the fol-
+lowing with u|t and accepts only if it is valid:
+||z1||∞ ≤ ηt,
+||z2||∞ ≤ ηt,
+(40)
+µ
+�
+j=1
+vj = az1 + z2 − uc,
+(41)
+where ηt = 5σn2√tµ log6 n. Denote this multi-
+signature scheme by RLWE-MultiSig. From our
+compiler and the properties of our ID scheme, we
+obtain the following.
+Corollary 3: Let ηt∗
+=
+5σn2√t∗µ log6 n,
+σ = Ω(n) and βt∗
+= 16ηt∗√n log2 n. Under
+Ring-LWEq,σ,2n and Ring-SIS3,q,βt∗ assumptions,
+RLWE-MultiSig is t∗-EU-CMA secure. Especially,
+if the assumptions hold for t∗ = 2
+4√n, then RLWE-
+MultiSig is EU-CMA secure.
+Remark.
+As the best algorithm [33], [15] can
+only solve ring-SISq,3,β with β = 2 ˜O(√n), it is safe
+to assume ring-SISq,3,β with any polynomial β. If
+the assumption is sound for β = 2
+4√n, our multi-
+signature scheme is EU-CMA secure, as t∗ can only
+be polynomial for a PPT adversary.
+VIII. CONCLUSION
+In this paper, we proposed a compiler that con-
+verts a type of identification scheme to a key-
+and-signature compact multi-signature. This special
+type of ID owns a linear property. The aggregated
+public-key and multi-signature are of size both in-
+dependent of the number of signers. We formulated
+this compiler through linear ID via the language of
+R-module and proved the security through a new
+forking lemma called nested forking lemma. Under
+our compiler, the compact multi-signature problem
+has been reduced from a multi-party problem to a
+two-party problem. We realized our compiler with
+Schnorr ID scheme and a new lattice-based scheme.
+Our lattice multi-signature is the first of its kind
+that is key-and-signature compact without a restart
+in the signing process.
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+

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@@ -0,0 +1,2861 @@
+EmbedDistill: A Geometric Knowledge Distillation
+for Information Retrieval
+Seungyeon Kim, Ankit Singh Rawat, Manzil Zaheer,
+Sadeep Jayasumana, Veeranjaneyulu Sadhanala, Wittawat Jitkrittum,
+Aditya Krishna Menon, Rob Fergus, Sanjiv Kumar
+Google LLC, USA
+{seungyeonk,ankitsrawat,manzilzaheer}@google.com
+{sadeep,veerus,wittawat,adityakmenon,robfergus,sanjivk}@google.com
+Abstract
+Large neural models (such as Transformers) achieve state-of-the-art performance for information
+retrieval (IR). In this paper, we aim to improve distillation methods that pave the way for the deployment
+of such models in practice. The proposed distillation approach supports both retrieval and re-ranking
+stages and crucially leverages the relative geometry among queries and documents learned by the large
+teacher model. It goes beyond existing distillation methods in the IR literature, which simply rely on
+the teacher’s scalar scores over the training data, on two fronts: providing stronger signals about local
+geometry via embedding matching and attaining better coverage of data manifold globally via query
+generation. Embedding matching provides a stronger signal to align the representations of the teacher and
+student models. At the same time, query generation explores the data manifold to reduce the discrepancies
+between the student and teacher where training data is sparse. Our distillation approach is theoretically
+justified and applies to both dual encoder (DE) and cross-encoder (CE) models. Furthermore, for distilling
+a CE model to a DE model via embedding matching, we propose a novel dual pooling-based scorer for the
+CE model that facilitates a distillation-friendly embedding geometry, especially for DE student models.
+1
+Introduction
+Neural models for information retrieval (IR) are increasingly used to model the true ranking function in
+various applications, including web search [Mitra and Craswell, 2018], recommendation [Zhang et al., 2019],
+and question-answering (QA; Chen et al. 2017). Notably, the recent success of Transformers [Vaswani et al.,
+2017]-based pre-trained language models [Devlin et al., 2019, Liu et al., 2019, Raffel et al., 2020] on a
+wide range of natural language understanding tasks has also prompted their utilization in IR to capture
+query-document relevance [see, e.g., Dai and Callan, 2019b, MacAvaney et al., 2019a, Nogueira and Cho,
+2019, Lee et al., 2019, Karpukhin et al., 2020a].
+A typical IR system comprises two stages: (1) A retriever first selects a small subset of potentially relevant
+candidate documents (out of a large collection) for a given query; and (2) A re-ranker then identifies a
+precise ranking among the candidates provided by the retriever. Dual-encoder (DE) models are the de-facto
+architecture for retrievers [Lee et al., 2019, Karpukhin et al., 2020a]. Such models independently embed
+queries and documents into a common space, and capture their relevance by simple operations on these
+embeddings such as the inner product. This enables offline creation of a document index and supports fast
+retrieval during inference via efficient maximum inner product search (MIPS) implementations [Guo et al.,
+1
+arXiv:2301.12005v1  [cs.LG]  27 Jan 2023
+
+2020, Johnson et al., 2021], with online query embedding generation primarily dictating the inference latency.
+Cross-encoder (CE) models, on the other hand, are preferred as re-rankers, owing to their excellent perfor-
+mance [Nogueira and Cho, 2019, Dai and Callan, 2019a, Yilmaz et al., 2019]. A CE model jointly encodes a
+query-document pair while enabling early interaction among query and document features. Employing a CE
+model for retrieval is often infeasible, as it would require processing a given query with every document in
+the collection at inference time. In fact, even in the re-ranking stage, the inference cost of CE models is high
+enough [Khattab and Zaharia, 2020] to warrant exploration of efficient alternatives [Hofst¨atter et al., 2020,
+Khattab and Zaharia, 2020, Menon et al., 2022].
+Knowledge distillation [Bucilˇa et al., 2006, Hinton et al., 2015] provides a general strategy to address the
+prohibitive inference cost associated with high-quality large neural models. In the IR literature, most existing
+distillation methods only rely on the teacher’s query-document relevance scores [see, e.g., Lu et al., 2020,
+Hofst¨atter et al., 2020, Chen et al., 2021, Ren et al., 2021, Santhanam et al., 2021] or their proxies [Izacard
+and Grave, 2021]. However, given that neural IR models are inherently embedding-based, it is natural to ask:
+Is it useful to go beyond matching of the teacher and student models’ scores, and directly aim to align their
+embedding spaces?
+With this in mind, we propose a novel distillation method for IR models that utilizes an embedding matching
+task to train the student. The proposed method supports cross-architecture distillation and improves upon
+existing distillation methods for both retriever and re-ranker models. When distilling a large DE model
+into a smaller DE model, for a given query (document), one can minimize the distance between the query
+(document) embeddings of the teacher and student after compatible projection layers to account for any
+dimensionality mismatch. In contrast, defining an embedding matching task for distilling a CE model into
+a DE model is not as straightforward. For Transformers-based CE models, it is common to use the final
+embedding of a special token, e.g., [CLS] in BERT [Devlin et al., 2019], to compute query-document
+relevance [Nogueira and Cho, 2019]. However, as we note in Sec. 4.2, this token embedding does not capture
+semantic similarity between the query and document. To make CE models more amenable to distillation via
+embedding matching, we propose a modified CE scoring approach by utilizing a novel dual-pooling strategy:
+this separately pools the final query and document token embeddings, and then computes the inner product
+between the pooled embeddings as the relevance score. Our key contributions toward improving IR models
+via distillation are:
+• We propose a novel distillation approach for neural IR models, namely EmbedDistill, that goes beyond
+score matching and aligns the embedding spaces of the teacher and student models.
+• We consider a novel DE to DE distillation setup to showcase the effectiveness of EmbedDistill (Sec. 4.1).
+Specifically, we consider a student DE model with an asymmetric configuration, consisting of a small
+query encoder and a frozen document encoder inherited from the teacher. This configuration significantly
+reduces inference latency of query embedding generation, while leveraging the teachers’ high-quality
+document index.
+• We show that EmbedDistill can leverage synthetic data to improve a student by further aligning the
+embedding spaces of the teacher and student (Sec. 4.3).
+• We theoretically justify both embedding matching and query generation components of our proposed
+method (Sec. 5). Further, we provide a comprehensive empirical evaluation of the method (Sec. 6) on two
+standard IR benchmarks – Natural Questions [Kwiatkowski et al., 2019a] and MSMARCO [Nguyen et al.,
+2016]. Additionally, we also evaluate EmbedDistill on BEIR benchmark [Thakur et al., 2021] which is
+used to measure the zero-shot performance of an IR model.
+• Finally, we demonstrate the utility of embedding matching for CE to DE distillation on MSMARCO
+by employing a novel pooling strategy, namely dual pooling (Sec. 4.2), which may be of independent
+interest.
+2
+
+2
+Related work
+Here, we review some existing Transformers-based IR models, and discuss prior work on distillation and data
+augmentation for such models. We also cover prior efforts on aligning representations during distillation for
+non-IR settings. Unlike our problem setting where the DE student is factorized, these works mainly consider
+distilling a single large Transformer into a smaller one.
+Transformers-based architectures for IR. Besides DE and CE models described in Sec. 1, intermediate
+configurations [MacAvaney et al., 2020, Khattab and Zaharia, 2020, Nie et al., 2020, Luan et al., 2021] have
+been proposed. Such models independently encode query and document before applying a more complex late
+interaction between the two. Interestingly, Nogueira et al. [2020] explore generative encoder-decoder style
+model for re-ranking, where a T5 [Raffel et al., 2020] model takes a query-document pair as input and its
+score for certain target tokens (e.g., True/False) defines the relevance score for the pair. In this paper, we
+focus on basic DE and CE models to showcase the benefits of our proposed geometric distillation approach.
+Exploring embedding matching for other aforementioned architectures is an interesting avenue for future
+exploration.
+Distillation for IR. Traditional distillation techniques have been widely applied in the IR literature, often to
+distill a teacher CE model to a student DE model [Li et al., 2020, Chen et al., 2021]. Recently, distillation from
+a DE model (with complex late interaction) to another DE model (with inner-product scoring) has also been
+considered [Lin et al., 2021, Hofst¨atter et al., 2021]. As for distilling across different model architectures, Lu
+et al. [2020], Izacard and Grave [2021] consider distillation from a teacher CE model to a student DE model.
+Hofst¨atter et al. [2020] conduct an extensive study of knowledge distillation across a wide-range of model
+architectures. Most existing distillation schemes for IR rely on only teacher scores; by contrast, we propose a
+geometric approach that also utilizes the teacher embeddings. Many recent efforts [Qu et al., 2021, Ren et al.,
+2021, Santhanam et al., 2021] show that iterative multi-stage (self-)distillation improves upon single-stage
+distillation [Qu et al., 2021, Ren et al., 2021, Santhanam et al., 2021]. These approaches use a model from
+the previous stage to obtain labels [Santhanam et al., 2021] as well as mine harder-negatives [Xiong et al.,
+2021]. We only focus on the single-stage distillation in this paper. Multi-stage procedures are complementary
+to our work, as one can employ our proposed embedding-matching approach in various stages of such a
+procedure. Interestingly, we demonstrate in Sec. 6 that our proposed EmbedDistill can successfully benefit
+from high quality models trained with such complex procedures [Reimers et al., 2019, Zhang et al., 2022].
+In particular, our single-stage distillation method can transfer almost all of their performance gains to even
+smaller models. Also to showcase that our method brings gain orthogonal to how teacher was trained, we
+conduct experiments with single-stage trained teacher in Appendix F.5.
+Distillation with representation alignments. Outside of the IR context, a few prior works proposed to utilize
+alignment between hidden layers during distillation [Romero et al., 2014, Sanh et al., 2019, Jiao et al., 2020,
+Aguilar et al., 2020, Zhang and Ma, 2020]. Chen et al. [2022] utilize the representation alignment to re-use
+teacher’s classification layer. Our work differs from these as it needs to address multiple challenges presented
+by an IR setting: 1) cross-architecture distillation such as CE to DE distillation; 2) partial representation
+alignment of query or document representations as opposed to aligning for the entire input, i.e., a query-
+documents pair; 3) catering representation alignment approach to novel IR setups such as asymmetric DE
+configuration; and 4) dealing with negative sampling due to a large number of classes (documents). To the
+best of our knowledge, our work is first in the IR literature that goes beyond simply matching scores (or its
+proxies).
+Semi-supervised learning for IR. Data augmentation or semi-supervised learning has been previously
+used to ensure data efficiency in IR [see, e.g., MacAvaney et al., 2019b, Zhao et al., 2021]. More inter-
+estingly, data augmentation via large pre-trained models have enabled performance improvements as well.
+3
+
+Teacher
+Cross Encoder
+Student Query 
+Encoder
+Student Doc
+Encoder
+…
+…
+Score
+Score
+Special tokens
+Query & doc tokens
+Score-based 
+distillation
+…
+…
+Doc tokens
+Query tokens
+Doc tokens
+Query tokens
+Pooling
+Pooling
+(a) Score-based CE to DE distillation
+Special tokens
+Query & doc tokens
+Student Query 
+Encoder
+…
+…
+Score
+Teacher Doc
+Encoder
+Score
+Teacher Query
+Encoder
+Student Doc
+Encoder
+Pooling
+…
+…
+Doc tokens
+Query tokens
+Doc tokens
+Query tokens
+Score-based distillation
+Pooling
+Pooling
+Pooling
+(b) Score-based DE to DE distillation
+Figure 1: Illustration of score-based distillation for IR (cf. Section 3.2). Fig. 1a describes distillation from
+a teacher [CLS]-pooled CE model to a student DE model. Fig. 1b depicts distillation from a teacher DE
+model to a student DE model. Here, both distillation setup employ symmetric DE configurations where query
+and document encoders of the student model are of the same size.
+Doc2query [Nogueira et al., 2019b,a] performs document expansion by generating queries that are relevant to
+the document and appending those queries to the document. Query expansion has also been considered, e.g.,
+for document re-ranking [Zheng et al., 2020]. Notably, generating synthetic (query, passage, answer) triples
+from a text corpus to augment existing training data for QA systems also leads to significant gains [Alberti
+et al., 2019, O˘guz et al., 2021]. Furthermore, even zero-shot approaches, where no labeled query-document
+pairs are used, can also perform competitively to supervised methods. Such methods train a DE model by
+relying on inverse cloze task [Lee et al., 2019, Izacard et al., 2021], synthetic query-document pairs given a
+target text corpus [Ma et al., 2021], or relevance scores from large pretrained models [Sachan et al., 2022].
+Unlike these works, we utilize query-generation capability to ensure tighter alignment between the embedding
+spaces of the teacher and student.
+3
+Background
+Let Q and D denote the query and document spaces, respectively. An IR model is equivalent to a scorer
+s : Q × D → R, i.e., it assigns a (relevance) score s(q, d) for a query-document pair (q, d) ∈ Q × D.
+Ideally, we want to learn an IR model or scorer that is faithful to the true query-document relevance, i.e.,
+s(q, d) > s(q, d′) iff the document d is more relevant to the query q than document d′. We assume access to
+n labeled training examples Sn = {(qi, di, yi)}i∈[n]. Here, di = (di,1, . . . , di,L) ∈ DL, ∀i ∈ [n], denotes a
+list of L documents and yi = (yi,1, . . . , yi,L) ∈ {0, 1}L denotes the corresponding labels such that yi,j = 1
+iff the document di,j is relevant to the query qi. Given Sn, we learn an IR model by minimizing
+R(s; Sn) := 1
+n
+�
+i∈[n] ℓ
+�
+sqi,di, yi
+�
+,
+(1)
+where sqi,di := (s(qi, d1,i), . . . , s(qi, d1,L)) and, accordingly, ℓ
+�
+sqi,di, yi
+�
+denotes the loss s incurs on
+(qi, di, yi). We defer concrete choices for the loss function ℓ to Appendix A.
+While this learning framework is general enough to work with any IR models as the scorers, next, we formally
+introduce two families of Transformer-based IR models that are prevalent in the recent literature.
+4
+
+3.1
+Transformer-based IR models: Cross-encoders and Dual-encoders
+Let query q = (q1, . . . , qm1) and document d = (d1, . . . , dm2) consist of m1 and m2 tokens, respectively. We
+now discuss how Transformers-based CE and DE models process a the (q, d) pair.
+Cross-encoder model. Let p = [q; d] be the sequence obtained by concatenating q and d. Further, let ˜p
+be the sequence obtained by adding special tokens such [CLS] and [SEP] to p. Given an encoder-only
+Transformer model Enc, the relevance score for the (q, d) pair is
+s(q, d) = ⟨w, pool
+�
+Enc(˜p)
+�
+⟩ = ⟨w, embq,d⟩,
+(2)
+where w is a d-dimensional classification vector, and pool(·) denotes a pooling operation that transforms the
+contextualized token embeddings Enc(˜p) to a joint embedding vector embt
+q,d. [CLS]-pooling is a common
+operation that simply outputs the embedding of the [CLS] token as embt
+q,d.
+Dual-encoder model. Let ˜q and ˜d be the sequences obtained by adding appropriate special tokens to q and
+d, respectively. A DE model comprises two (encoder-only) Transformers EncQ and EncD, which we call
+query and document encoders, respectively.1 Let embq = pool
+�
+EncQ(˜q)
+�
+and embd = pool
+�
+EncD( ˜d)
+�
+denote
+the query and document embeddings, respectively. Now, one can define s(q, d) = ⟨embq, embd⟩ to be the
+relevance score assigned to the (q, d) pair by the DE model.
+3.2
+Score-based distillation for IR models
+Most distillation schemes for IR [e.g., Lu et al., 2020, Hofst¨atter et al., 2020, Chen et al., 2021] rely on
+teacher relevance scores (cf. Fig. 1). In particular, given a training set Sn and a teacher with scorer st, one
+learns a student with scorer ss by minimizing
+R(ss, st; Sn) = 1
+n
+�
+i∈[n] ℓd
+�
+ss
+q,di, st
+q,di
+�
+,
+(3)
+where ℓd captures the discrepancy between ss and st. See Appendix A for common choices for ℓd.
+4
+Embedding-matching based distillation
+Since modern neural IR models are inherently embedding-based, we propose to explicitly align the embedding
+spaces of the teacher and student via a novel distillation method, namely EmbedDistill. Our proposal goes
+beyond existing distillation methods in the IR literature that only use the teacher scores. Next, we introduce
+EmbedDistill for two prevalent settings: (1) distilling a large DE model to a smaller DE model;2 and (2)
+distilling a CE model to a DE model.
+4.1
+DE to DE distillation
+Given a (q, d) pair, let embt
+q and embt
+d be the query and document embeddings produced by the query encoder
+Enct
+Q and document encoder Enct
+D of the teacher DE model, respectively. Similarly, let embs
+q and embs
+d
+1It is common to employ dual-encoder models where query and document encoders are shared.
+2We focus on DE to DE distillation setup as the CE to CE distillation is special case of the former with the classification vector w
+(cf. Eq. 2) being the trivial second encoder.
+5
+
+Special tokens
+Query & doc tokens
+Student Query 
+Encoder
+Teacher Doc
+Encoder
+…
+…
+Doc tokens
+Query tokens
+Score
+Teacher Doc
+Encoder
+…
+…
+Doc tokens
+Query tokens
+Score
+Teacher Query
+Encoder
+≈
+Embedding matching
+Pooling
+Pooling
+Pooling
+Pooling
+Score-based distillation
+Figure 2: Proposed DE to DE distillation with query embedding matching. The figure describes a setting
+where student employs an asymmetric DE configuration with a small query encoder and a large (non-trainable)
+document encoder inherited from the teacher. The smaller query encoder ensures small latency for encoding
+query during inference, and large document encoder leads to a good quality document index.
+denote the query and document embeddings produced by a student DE model with (Encs
+Q, Encs
+D) as its
+query and document encoders. Now, EmbedDistill optimizes the following embedding alignment loss in
+addition to the score-matching loss from Sec. 3.2:
+REmb(t, s; Sn)= 1
+n
+�
+(q,d)∈Sn
+�
+∥embt
+q−proj
+�
+embs
+q
+�
+∥ + ∥embt
+d−proj
+�
+embs
+d)∥
+�
+,
+(4)
+where proj is an optional trainable layer that is required if the teacher and student produce different sized
+embeddings. Alternatively, one can employ other variants of EmbedDistill, e.g., focusing on only aligning the
+query embeddings takes the following form (cf. Fig. 2).
+REmb,Q(t, s; Sn) = 1
+n
+�
+q∈Sn
+∥embt
+q − proj
+�
+embs
+q
+�
+∥.
+(5)
+Asymmetric DE. We also propose a novel student DE configuration where the student employs the teacher’s
+document encoder (i.e., Encs
+D = Enct
+D) and only train its query encoder, which is much smaller compared
+to the teacher’s query encoder. For such a setting, it is natural to only employ the embedding matching loss in
+Eq. 5 as the document embeddings are aligned by design (cf. Fig. 2).
+Note that this asymmetric student DE does not incur an increase in latency despite the use of a large teacher
+document encoder. This is because the large document encoder is only needed to create a good quality
+document index offline, and only the query encoder is evaluated at inference time. Thus, for DE to DE
+distillation, we prescribe the asymmetric DE configuration universally. Our theoretical analysis (cf. Sec. 5)
+and experimental results (cf. Sec. 6) suggest that the ability to inherit the document tower from the teacher
+DE model can drastically improve the final performance, especially when combined with EmbedDistill.
+4.2
+CE to DE distillation
+Let Enct be the (single) teacher CE model encoder, and (Encs
+Q, Encs
+D) denote the student DE model’s query
+and document encoders. When distilling from a CE to DE model, defining an effective embedding matching
+task is not as straightforward as in Sec. 4.1: since CE models jointly encode query-document pairs, it is not
+obvious how to extract individual query and document embeddings for matching.
+6
+
+Teacher
+Cross Encoder
+Student Query 
+Encoder
+Student Doc
+Encoder
+…
+…
+Doc tokens
+Query tokens
+Score
+Score
+Special tokens
+Query & doc tokens
+…
+…
+Doc tokens
+Query tokens
+Pooling
+Pooling
+Score-based 
+distillation
+≈
+Embedding matching
+Figure 3: Illustration of CE to DE distillation using EmbedDistill, with CE model employing dual pooling.
+As a na¨ıve solution, for a (q, d) pair, one can simply match a joint transformation of the student’s query
+embedding embs
+q and document embedding embs
+d to the teacher’s joint embedding embt
+q,d. However, we
+observed that including such an embedding matching task often leads to severe over-fitting, and results in
+a poor student. Since st(q, d) = ⟨w, embt
+q,d⟩, during CE model training, the joint embeddings embt
+q,d for
+relevant and irrelevant (q, d) pairs are encouraged to be aligned with w and −w, respectively. This produces
+degenerate embeddings that do not capture semantic query-to-document relationships. We notice that even
+the final query and document token embeddings lose such semantic structure (cf. Appendix G.2). Thus, a
+teacher CE model with st(q, d) = ⟨w, embt
+q,d⟩ does not add value for distillation beyond score-matching; in
+fact, it hurts to include na¨ıve embedding matching. Next, we propose a modified CE model training strategy
+that facilitates EmbedDistill.
+CE models with dual pooling. We propose dual pooling to produce two embeddings embt
+q←(q,d) and
+embt
+d←(q,d) from a CE model that serve as the proxy query and document embeddings, respectively. Ac-
+cordingly, we define the relevance score as st(q, d) = ⟨embt
+q←(q,d), embt
+d←(q,d)⟩. We explore two variants of
+dual pooling: (1) special token-based pooling that pools from [CLS] and [SEP]; and (2) segment-based
+weighted mean pooling that separately performs weighted averaging on the query and document segments of
+the final token embeddings. See Appendix B for details.
+In addition to employing the dual pooling, we also utilize a reconstruction loss during the CE training, which
+measures the likelihood of predicting each token of the original input from the final token embeddings. This
+loss encourages reconstruction of query and document tokens based on the final token embeddings and
+prevents the degeneration of the token embeddings during training. Given proxy embeddings from the teacher
+CE, we can perform EmbedDistill with the embedding matching loss defined in Eq. 4 or Eq. 5 (cf. Fig. 3).
+4.3
+Task-specific online data generation
+Data augmentation as a general technique has been previously considered in the IR literature [see, e.g.,
+Nogueira et al., 2019b, O˘guz et al., 2021, Izacard et al., 2021], especially in data-limited, out-of-domain,
+or zero-shot settings. As EmbedDistill aims to align the embeddings spaces of the teacher and student, the
+ability to generate similar queries or documents can naturally help enforce such an alignment globally on the
+task-specific manifold. Given a set of unlabeled task-specific query and document pairs Um, we can further
+add the embedding-alignment loss REmb(t, s; Um) to our training objective (cf. Eq.4). Interestingly, for DE
+to DE distillation setting, our approach can even benefit from a large collection of task-specific queries Q′ or
+documents D′. Here, we can independently employ embedding matching losses REmb,Q(t, s; Q′) (cf. Eq. 5)
+7
+
+and REmb,D(t, s; D′) that focus on queries and documents, respectively. Please refer to Appendix E for
+additional details.
+5
+Improvements in the student generalization
+Note that we motivate EmbedDistill as well as asymmetric DE configuration where the student DE model
+inherits the teacher’s document encoder from their potential ability to ensure a better alignment between the
+teacher and student embedding spaces. In this section, we provide a theoretical justification for both of these
+proposals by showing that they indeed result in a better generalization (test-time) performance for the student
+and reduce the gap between the teacher and the student.
+Let R(s) = E
+�
+ℓ
+�
+sq,d, y
+��
+be the population version of the empirical risk in Eq. 1. Note that R(s) is a
+measure of the test time performance of the IR model defined by the scorer s. Since we want the student
+model to (at least) closely approximate the teacher model’s performance, we are interested in bounding
+R(ss) − R(st), i.e., the gap between the test time performance of the student and teacher models. The
+following result provides a bound on this quantity (see Appendix C.1 for a formal statement and proof). For
+simplicity, we focus on L = 1 (cf. Sec. 3). The result can be extended to L > 1 with more complex notation.
+Theorem 5.1 (Teacher-student performance gap (informal)). Let F and G denote the function classes for the
+query and document encoders for the student model, respectively. Suppose that the score-based distillation
+loss ℓd in Eq. 3 is based on binary cross entropy loss (Eq. 11 in Appendix A). Let one-hot (label-dependent)
+loss ℓ in Eq. 1 be the binary cross entropy loss (Eq. 9 in Appendix A). Further, assume that all encoders have
+the same output dimension and embeddings have their ℓ2-norm bounded by K. Then, we have
+R(ss) − R(st) ≤ En(F, G ) + 2KREmb,Q(t, s; Sn) + 2KREmb,D(t, s; Sn)
+�
++
+∆(st; Sn) + K2�
+E
+���σ(st
+q,d) − y
+���
++ 1
+n
+�
+i∈[n]
+��σ(st
+qi,di) − yi
+�� �
+,
+where σ denotes the sigmoid function,
+En(F, G ) :=
+sup
+ss∈F×G
+��R(ss, st; Sn) − Eℓd
+�
+ss
+q,d, st
+q,d
+���
+and ∆(st; Sn) denotes the deviation between the empirical risk (on Sn) and population risk of the teacher
+model st.
+The last three quantities in our bound on the teacher-student performance gap, namely ∆(st; Sn), E[|σ(st
+q,d)−
+y|], and 1
+n
+�
+i∈[n] |σ(st
+qi,di)−yi|, are independent of the underlying student model. These terms solely depend
+on the quality of the underlying teacher model st.
+More importantly, the teacher-student gap can be made small by reducing the following three terms: 1)
+uniform deviation of the student’s empirical distillation risk from its population version En(F, G ); 2) query
+embedding matching loss for the student REmb,Q(t, s; Sn); and 3) doc embedding matching loss for the
+student REmb,D(t, s; Sn). The last two terms suggest that the teacher-student gap naturally goes down as
+the embeddings of the student and teacher models become aligned. In particular, when the student inherits
+the document encoder from the teacher, the third term REmb,D(t, s; Sn) vanishes. In general, these last two
+terms justify EmbedDistill which either employ Eq. 4 or Eq. 5 (when student inherits teacher’s document
+encoder).
+8
+
+Table 1: Full recall performance of various student DE models on NQ dev set, including symmetric DE
+student model (67.5M or 11.3M transformer for both encoders), and asymmetric DE student model (67.5M
+or 11.3M transformer as query encoder and document embeddings inherited from the teacher). All distilled
+students used the same teacher (110.1M parameter BERT-base models as both encoders), with the full
+Recall@5 = 72.3, Recall@20 = 86.1, and Recall@100 = 93.6.
+Method
+6-Layer (67.5M)
+4-Layer (11.3M)
+R@5 R@20 R@100
+R@5 R@20 R@100
+Train student directly
+36.2
+59.7
+80.0
+24.8
+44.7
+67.5
++ Distill from teacher
+65.3
+81.6
+91.2
+44.3
+64.9
+81.0
++ Inherit doc embeddings
+69.9
+83.9
+92.3
+56.3
+70.9
+82.5
++ Query embedding matching
+72.7
+86.5
+93.9
+61.2
+75.2
+85.1
++ Query generation
+73.4
+86.3
+93.8
+64.3
+77.8
+87.9
+Train student using only
+embedding matching and
+inherit doc embeddings
+71.4
+84.9
+92.6
+64.6
+50.2
+76.8
++ Query generation
+71.8
+85.0
+93.0
+54.2
+68.9
+80.8
+Next, we focus on the first term En(F, G ) which captures the uniform deviation between the training and test
+time performance of the student, as measured by the distillation loss. Again, we restrict ourselves to the setting
+of Theorem 5.1 which assumes a binary cross-entropy loss with L = 1 for simplicity. (See Appendix C.2 for
+a more precise statement and proof.)
+Proposition 5.2. Let ℓd be a distillation loss which is Lℓd-Lipschitz in its first argument. Let F and G
+denote the function classes for the query and document encoders, respectively. Further assume that, for each
+query and document encoder in our function class, the query and document embeddings have their ℓ2-norm
+bounded by K. Then,
+En(F, G ) ≤ ESn
+48KLℓd
+√n
+� ∞
+0
+�
+log
+�
+N(u, F)N(u, G )
+�
+du.
+(6)
+Furthermore, with a fixed document encoder, i.e., G = {g∗},
+En(F, {g∗}) ≤ ESn
+48KLℓd
+√n
+� ∞
+0
+�
+log N(u, F) du.
+(7)
+Here, N(u, ·) is the u-covering number of a function class.
+Note that Eq. 6 and Eq. 7 correspond to uniform deviation when we train without and with a frozen document
+encoder, respectively. It is clear that the bound in Eq. 7 is less than or equal to that in Eq. 6 (because
+N(u, G ) ≥ 1 for any u), which alludes to desirable impact of employing a frozen document encoder (in
+terms of deviation in train and test performance). When we further employ EmbedDistill (e.g., with the loss in
+Eq. 5), it regularizes the function class of query encoders; effectively, reducing it to F ′ with |F ′| ≤ |F|.
+This has further desirable implication for reducing the uniform deviation bounds as N(u, F ′) ≤ N(u, F).
+6
+Experiments
+We now conduct a comprehensive evaluation of the proposed distillation approach. Specifically, we highlight
+the utility of the approach for both DE to DE and CE to DE distillation. We also showcase the benefits of
+combining our distillation approach with query generation methods.
+9
+
+Table 2: Performance of EmbedDistill for DE to DE distillation on NQ test set. Note that the prior work
+mentioned in the table rely on techniques such as negative mining and multi-stage training. In contrast, we
+explore the orthogonal direction of embedding-matching that improves single-stage distillation, which can be
+combined with the aforementioned techniques.
+Method
+#Layers
+R@20
+R@100
+DPR [Karpukhin et al., 2020a]
+12
+78.4
+85.4
+DPR + PAQ [O˘guz et al., 2021]
+12
+84.0
+89.2
+DPR + PAQ [O˘guz et al., 2021]
+24
+84.7
+89.2
+ACNE [Xiong et al., 2021]
+12
+81.9
+87.5
+RocketQA [Qu et al., 2021]
+12
+82.7
+88.5
+MSS-DPR [Sachan et al., 2021]
+12
+84.0
+89.2
+MSS-DPR [Sachan et al., 2021]
+24
+84.8
+89.8
+Our teacher [Zhang et al., 2022]
+12 (220.2M)
+85.4
+90.0
+Student w/ proposed method
+6 (67.5M)
+85.1
+89.8
+Student w/ proposed method
+4 (11.3M)
+81.2
+87.4
+6.1
+Setup
+Benchmarks and evaluation metrics. We consider two popular IR benchmarks — Natural Questions
+(NQ; Kwiatkowski et al. 2019b) and MSMARCO [Nguyen et al., 2016], which focus on finding the most
+relevant passage/document given a question and a search query, respectively. NQ provides both standard
+test and dev sets, whereas MSMARCO has only standard dev set publicly available. In what follows, we
+use the terms query (document) and question (passages) interchangeably. For NQ, we use the standard
+full recall (strict) as well as the relaxed recall metric [Karpukhin et al., 2020a] to evaluate the retrieval
+performance. For MSMARCO, we focus on the standard metrics Mean Reciprocal Rank (MRR)@10, and
+normalized Discounted Cumulative Gain (nDCG)@10 to evaluate both re-ranking and retrieval performance.
+See Appendix D for a detailed discussion on these evaluation metrics. Finally, we also evaluate EmbedDistill on
+the BEIR benchmark [Thakur et al., 2021] – a zero-shot retrieval benchmark – in terms of nDCG@10 and
+recall@100 metrics.
+Model architectures. We follow the standard Transformers-based IR model architectures similar
+to Karpukhin et al. [2020a], Qu et al. [2021], O˘guz et al. [2021]. Our CE model is based on a RoBERTa-base
+model (Liu et al. [2019]; 12-layer, 768 dim, 124M parameters). We utilized various sizes of DE models based
+on BERT-base ([Devlin et al., 2019], 12-layer, 768 dim, 110M parameters), DistilBERT (Sanh et al. [2019];
+6-layer, 768 dim, 67.5M parameters – ∼ 2/3 of base), or BERT-mini (Turc et al. [2019]; 4-layer, 256 dim,
+11.3M parameters – ∼ 1/10 of base). For query generation (cf. Sec. 4.3), we employ BART-base [Lewis
+et al., 2020], an encoder-decoder model, to generate similar questions from each training example’s input
+question (query). We randomly mask 10% of tokens and inject zero mean Gaussian noise with σ = {0.1, 0.2}
+between the encoder and decoder. See Appendix E for more details on query generation and Appendix F.1 for
+hyperparameters.
+10
+
+Table 3: Performance of various DE models on MSMARCO dev set for re-ranking task. A teacher model
+(110.1M parameter BERT-base models as both encoders) achieving MRR@10 of 36.8 and nDCG@10 of
+42.7 is used. The table shows performance of the symmetric DE student model (67.5M or 11.3M transformer
+as both encoders), and asymmetric DE student model (67.5M or 11.3M transformer as query encoder and
+document embeddings inherited from the teacher).
+Method
+MRR@10
+nDCG@10
+67.5M
+11.3M
+67.5M
+11.3M
+Train student directly
+27.0
+23.0
+32.2
+29.7
++ Distill from teacher
+34.6
+30.4
+40.2
+35.8
++ Inherit doc embeddings
+35.2
+32.1
+41.0
+37.7
++ Query embedding matching
+36.2
+35.0
+42.0
+40.8
++ Query generation
+36.2
+34.4
+42.0
+40.1
+Train student using only
+embedding matching and
+inherit doc embeddings
+36.5
+33.5
+42.3
+39.3
++ Query generation
+36.4
+34.1
+42.3
+39.9
+6.2
+DE to DE distillation
+We employ AR2 [Zhang et al., 2022]3 and SentenceBERT-v5 [Reimers et al., 2019]4 as teacher DE models
+for NQ and MSMARCO. Note that both models are based on BERT-base. For DE to DE distillation, we
+consider two kinds of configurations for the student DE model: (1) Symmetric: We use identical question and
+document encoders. We evaluate DistilBERT and BERT-mini on both datasets. (2) Asymmetric: The student
+DE model inherits its document embeddings from the teacher DE model, which are not trained during the
+distillation. For query encoder, we use DistilBERT or BERT-mini which are smaller than document encoder.
+Student DE model training. We train student DE models using a combination of (i) one-hot loss (cf. Eq. 8
+in Appendix A) on training data; (ii) distillation loss in (cf. Eq. 10 in Appendix A); and (iii) embedding
+matching loss in Eq. 5. We used [CLS]-pooling for all student encoders. Unlike DPR [Karpukhin et al.,
+2020a] or AR2, we do not use hard negatives from BM25 or other models, which greatly simplifies our
+distillation procedure.
+Results and discussion. To understand the impact of various proposed configurations and losses, we train
+models by sequentially adding components and evaluate their retrieval performance on NQ and MSMARCO
+dev set as shown in Table 1 and 4, respectively. (See Table 7 in Appendix F.2 for performance on NQ in terms
+of the relaxed recall.) We also evaluate re-ranking task on MSMARCO dev set (Table 3).
+We begin by training a symmetric DE without distillation. As expected, moving to distillation brings in
+considerable gains. Next, we swap the student document encoder with document embeddings from the teacher
+(non-trainable), which leads to a good jump in the performance. Now we can introduce EmbedDistill with
+Eq. 5 for aligning query representations between student and teacher. The two losses are combined with
+weight of 1.0 (except for BERT-mini models in the presence of query generation with 5.0). This improves
+performance significantly, e.g.,it provides ∼3 and ∼5 points increase in recall@5 on NQ with students based
+on DistilBERT and BERT-mini, respectively (Table 1). We further explore the utility of EmbedDistill in
+aligning the teacher and student embedding spaces in Appendix G.1.
+3https://github.com/microsoft/AR2/tree/main/AR2
+4https://huggingface.co/sentence-transformers/msmarco-bert-base-dot-v5
+11
+
+Table 4: Performance of various DE models on MSMARCO dev set for retrieval task. A teacher model
+(110.1M parameter RoBERTa-base models as both encoders) achieving MRR@10 of 37.2 and nDCG@10 of
+44.2 is used. The table shows performance of the symmetric DE student model (67.5M or 11.3M transformer
+as both encoders), and asymmetric DE student model (67.5M or 11.3M transformer as query encoder and
+document embeddings inherited from the teacher).
+Method
+MRR@10
+nDCG@10
+67.5M
+11.3M
+67.5M
+11.3M
+Train student directly
+22.6
+18.6
+27.2
+22.5
++ Distill from teacher
+35.0
+28.6
+41.3
+34.1
++ Inherit doc embeddings
+35.7
+30.3
+42.2
+36.2
++ Query embedding matching
+37.1
+35.4
+43.8
+41.9
++ Query generation
+37.2
+34.8
+43.8
+41.2
+Train student using only
+embedding matching and
+inherit doc embeddings
+36.6
+31.4
+43.3
+37.6
++ Query generation
+36.7
+32.8
+43.4
+39.2
+Table 5: Average BEIR performance of our DE teacher and EmbedDistill student models and their numbers of
+trainable parameters. Both models are trained on MSMARCO and evaluated on 14 other datasets (the average
+does not include MSMARCO). The full table is at Appendix F.4. With EmbedDistill, student materializes
+most of the performance of the teacher on the unforeseen datasets.
+Method
+#Layers
+nDCG@10
+R@100
+DPR [Karpukhin et al., 2020b]
+12
+22.5
+47.7
+ANCE [Xiong et al., 2021]
+12
+40.5
+60.0
+TAS-B [Hofst¨atter et al., 2021]
+6
+42.8
+64.8
+GenQ [Thakur et al., 2021]
+6
+42.5
+64.2
+Our teacher [Reimers et al., 2019]
+12 (220.2M)
+45.7
+65.1
+Student w/ EmbedDistill
+6 (67.5M)
+44.0
+63.5
+On top of the two losses (standard distillation and embedding matching), we also use REmb,Q(t, s; Q′) from
+Sec. 4.3 on 2 additional questions (per input question) generated from BART. We also try a variant where we
+eliminate the standard distillation loss and only employ the embedding matching loss in Eq. 5 along with
+inheriting teacher’s document embeddings. This configuration without the standard distillation loss leads to
+excellent performance (with query generation again providing additional gains in most cases.)
+It is worth highlighting that DE models trained with the proposed methods (e.g., asymmetric DE with
+embedding matching and generation) achieve 99% of the performance in both NQ/MSMARCO tasks with a
+query encoder that is 2/3rd the size of that of the teacher. Furthermore, even with 1/10th size of the query
+encoder, our proposal can achieve 95-97% of the performance. This is particularly useful for latency critical
+applications with minimal impact on the final performance.
+Finally, we take our best student models, i.e., one trained using with additional embedding matching loss and
+using data augmentation from query generation, and evaluate on test sets. We compare with various prior
+work and note that most prior work used considerably bigger models in terms of parameters, depth (12 or
+24 layers), or width (upto 1024 dims). For NQ test set results are reported in Table 2, but as MSMARCO
+does not have any public test set, we instead present results for the BEIR benchmark in Table 5. Note we
+12
+
+Table 6: Performance of DE models distilled from [CLS]-pooled and Dual-pooled CE models on MS-
+MARCO re-ranking task. While both teacher models perform similarly, embedding matching-based distilla-
+tion only works with the Dual-pooled teacher. See Appendix F for nDCG@10 metric.
+Method
+MRR@10
+[CLS]-pooled teacher
+37.1
+Dual-pooled teacher
+37.0
+Standard distillation from [CLS]-pooled teacher
+33.0
++Joint matching
+32.4
+Standard distillation from Dual-pooled teacher
+33.3
++Query matching
+33.7
+also provide evaluation of our SentenceBERT teacher achieving very high performance on the benchmark
+which can be of independent interest (please refer to Appendix F.4 for details). For both NQ and BEIR, our
+approach obtains competitive student model with fewer than 50% of the parameters: even with 6 layers, our
+student model is very close (98-99%) to its teacher.
+6.3
+CE to DE distillation
+We consider two CE teachers for MSMARCO re-ranking task5: a standard [CLS]-pooled CE teacher, and
+the Dual-pooled CE teacher (cf. Sec. 4.2). Both teachers are based on RoBERTa-base and trained on triples
+in the training set for 300K steps with cross-entropy loss.
+Student DE model training. We considered the following distillation variants: standard score-based distil-
+lation from the [CLS]-pooled teacher, and our novel Dual-pooled CE teacher (with and without embedding
+matching loss). For each variant, we initialize encoders of the student DE model with two RoBERTa-base
+models and train for 500K steps We performed the na¨ıve joint embedding matching for the [CLS]-pooled
+teacher (cf. Sec. 4.2) and employed the query embedding matching (cf. Eq.5) for the Dual-pooled CE teacher.
+In either case, embedding-matching loss is added on top of the standard cross entropy loss with the weight of
+1.0 (when used).
+Results and discussion. Table 6 evaluates the effectiveness of the dual pooling and the embedding matching
+for CE to DE distillation. As described in Sec. 4.2, the traditional [CLS]-pooled teacher did not provide any
+useful embedding for the embedding matching (see Appendix G.2 for the further analysis of the resulting
+embedding space). However, with the Dual-pooled teacher, embedding matching does boost student’s
+performance.
+7
+Conclusion
+We propose EmbedDistill — a novel distillation method for IR that goes beyond simple score matching. We
+specialize it to distill a DE model into another DE model by (a) reusing the teacher’s document encoder in
+the student and (b) aligning query embeddings of the teacher and student. This simple approach delivers im-
+mediate quality and computational gains in practical deployments and we demonstrate them on MSMARCO,
+NQ, and BEIR benchmarks. We show that query generation technique further improves the performance of
+the distilled student in most cases. We generalize the proposed approach to distill a CE model to a DE model
+5Note: Full retrieval is prohibitively expensive with CE models.
+13
+
+and show the benefits on MSMARCO. Finally, our theoretical analysis alludes to the favorable implications
+of both embedding matching and inheriting document encoder in DE to DE distillation setting. A more
+comprehensive and systematic analysis of embedding matching-based distillation for IR is an exciting avenue
+for future research.
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+findings-emnlp.424.
+19
+
+A
+Loss functions
+Here, we state various (per-example) loss functions that most commonly define training objectives for IR
+models. Typically, one hot training with original label is performed using softmax-based cross-entropy loss
+functions:
+ℓ
+�
+sq,di, yi
+�
+= −
+�
+j∈[L]
+yi,j · log
+�
+exp(s(qi, di,j))
+�
+j′∈[L]
+exp(s(qi, di,j′))
+�
+.
+(8)
+In our experiments, we use in-batch negatives for obtaining the irrelevant documents di,j′. Alternatively, it is
+also common to employ a one-vs-all loss function based on binary cross-entropy loss as follows:
+ℓ
+�
+sq,di, yi
+�
+= −
+�
+j∈[L]
+�
+yi,j · log
+�
+1
+1 + exp(−s(qi, di,j))
+�
++
+(1 − yi,j) · log
+�
+1
+1 + exp(s(qi, di,j))
+��
+.
+(9)
+As for distillation, one can define a distillation objective based on the softmax-based cross-entropy loss as6:
+ℓd
+�
+ss
+q,di, st
+q,di
+�
+= −
+�
+j∈[L]
+�
+exp(st
+i,j)
+�
+j′∈[L] exp(st
+i,j′) · log
+�
+exp(ss
+i,j)
+�
+j′∈[L] exp(ss
+i,j′)
+��
+,
+(10)
+where st
+i,j := st(qi, di,j) and ss
+i,j := ss(qi, di,j) denote the teacher and student scores, respectively. On the
+other hand, the distillation objective with the binary cross-entropy takes the form:
+ℓd
+�
+ss
+q,di, st
+q,di
+�
+= −
+�
+j∈[L]
+�
+1
+1 + exp(−st
+i,j) · log
+�
+1
+1 + exp(−ss
+i,j)
+�
++
+1
+1 + exp(st
+i,j) · log
+�
+1
+1 + exp(ss
+i,j)
+��
+.
+(11)
+Finally, distillation based on the meas square error (MSE) loss (aka. logit matching) employs the following
+loss function:
+ℓd
+�
+ss
+q,di, st
+q,di
+�
+=
+�
+j∈[L]
+�
+st(qi, di,j) − ss(qi, di,j)
+�2.
+(12)
+B
+Dual pooling details
+In this work, we focus on two kinds of dual pooling strategies:
+• Special tokens-based dual pooling. Let poolCLS and poolSEP denote the pooling operations that
+return the embeddings of the [CLS] and [SEP] tokens, respectively. We define
+embt
+q←(q,d) = poolCLS
+�
+Enct(˜o)
+�
+,
+embt
+d←(q,d) = poolSEP
+�
+Enct(˜o)
+�
+,
+(13)
+where ˜o denotes the input token sequence to the Transformers-based encoder, which consists of { query,
+document, special } tokens.
+6It is common to employ temperature scaling with softmax operation. We do not explicitly show the temperature parameter for
+ease of exposition.
+20
+
+• Segment-based weighted-mean dual pooling. Let Enct(˜o)|Q and Enct(˜o)|D denote the final query
+token embeddings and document token embeddings produced by the encoder, respectively. We define
+the proxy query and document embeddings
+embt
+q←(q,d) = meanwt
+�
+Enct(˜o)|Q
+�
+,
+embt
+d←(q,d) = meanwt
+�
+Enct(˜o)|D
+�
+,
+(14)
+where meanwt(·) denotes the weighted mean operation. We employ the specific weighting scheme
+where each token receives a weight equal to the inverse of the square root of the token-sequence length.
+C
+Deferred details and proofs from Section 5
+In this section we present more precise statements and proofs of Theorem 5.1 and Proposition 5.2 (stated
+informally in Section 5 of the main text) along with the necessary background. First, for the ease of exposition,
+we define new notation which will facilitate theoretical analysis in this section.
+Notation. Denote the query and document encoders as f : Q → Rk and g: D → Rk for the student, and
+F : Q → Rk, G: D → Rk for the teacher (in the dual-encoder setting). With q denoting a query and d
+denoting a document, f(q) and g(d) then denote query and document embeddings, respectively, generated by
+the student. We define F(q) and G(d) similarly for embeddings by the teacher.7
+Theorem C.1 (Formal statement of Theorem 5.1). Let F and G denote the function classes for the query
+and document encoders for the student model, respectively. Given n examples Sn = {(qi, di, yi)}i∈[n] ⊂
+Q × D × {0, 1}, let ss(q, d) := sf,g(qi, di) = f(qi)T g(di) be the scores assigned to the (qi, di) pair by a
+dual-encoder model with f ∈ F and g ∈ G as query and document encoders, respectively. Let ℓ and ℓd be
+the binary cross-entropy loss (cf. Eq. 9 with L = 1) and the distillation-specific loss based on it (cf. Eq. 11
+with L = 1), respectively. In particular,
+ℓ(sF,G(qi, di), yi) := −yi log σ
+�
+F(qi)⊤G(di)
+�
+− (1 − yi) log
+�
+1 − σ
+�
+F(qi)⊤G(di)
+��
+ℓd(sf,g(qi, di), sF,G(qi, di)) := −σ
+�
+F(qi)⊤G(di)
+�
+· log σ
+�
+f(qi)⊤g(di)
+�
+−
+[1 − σ
+�
+F(qi)⊤G(di)
+�
+] · log
+�
+1 − σ
+�
+f(qi)⊤g(di)
+��
+,
+where σ is the sigmoid function and st := sF,G denotes the teacher dual-encoder model with F and Q as its
+query and document encoders, respectively. Assume that
+1. All encoders f, g, F, and G have the same output dimension.
+2. ∃ K ∈ (0, ∞) such that
+sup
+q∈Q
+max {∥f(q)∥2, ∥F(q)∥2} ≤ K
+and
+sup
+d∈D
+max {∥g(d)∥2, ∥G(d)∥2} ≤ K.
+7Note that, as per the notations in the main text, we have (f, g) = (Encs
+Q, Encs
+D) and (F, G) = (Enct
+Q, Enct
+D). Similarly, we
+have (embt
+q, embt
+d) = (f(q), g(d)) and (embt
+q, embt
+d) = (F(q), G(d)).
+21
+
+Then, we have
+E
+�
+sf,g(q, d)
+�
+�
+��
+�
+:=R(ss)=R(sf,g)
+− E
+�
+sF,G(q, d)
+�
+�
+��
+�
+:=R(st)=R(sF,G)
+≤
+sup
+(f,g)∈F×G
+���R(sf,g, sF,G; Sn) − E
+�
+ℓd
+�
+sf,g(q, d), sF,G(q, d)
+�����
+�
+��
+�
+:=En(F,G )
++ 2K
+� 1
+n
+�
+i∈[n]
+∥g(di) − G(di)∥2
+�
+��
+�
+:=REmb,D(t,s;Sn)
++ 1
+n
+�
+i∈[n]
+∥f(qi) − F(qi)∥2
+�
+�
+��
+�
+:=REmb,Q(t,s;Sn)
++ R(sF,G; Sn) − R(sF,G)
+�
+��
+�
+:=∆(st;Sn)
++ K2�
+E
+����σ(F(q)⊤G(d)) − y
+���
+�
++ 1
+n
+�
+i∈[n]
+���σ
+�
+F(qi)⊤G(di)
+�
+− yi
+���
+�
+.
+(15)
+Proof. Note that
+R(sf,g) − R(sF,G) = R(sf,g) − R(sf,g, sF,G) + R(sf,g, sF,G) − R(sF,G)
+(a)
+≤ K2E
+����σ(F(q)⊤G(d)) − y
+���
+�
++ R(sf,g, sF,G) − R(sF,G)
+= K2E
+����σ(F(q)⊤G(d)) − y
+���
+�
++ R(sf,g, sF,G) − R(sf,g, sF,G; Sn) + R(sf,g, sF,G; Sn) − R(sF,G)
+(b)
+≤ K2E
+����σ(F(q)⊤G(d)) − y
+���
+�
++ En(F, G ) + R(sf,g, sF,G; Sn) − R(sF,G)
+= K2E
+����σ(F(q)⊤G(d)) − y
+���
+�
++ En(F, G ) + R(sf,g, sF,G; Sn) − R(sF,G; Sn)+
+R(sF,G; Sn) − R(sF,G)
+(c)
+≤ K2E
+����σ(F(q)⊤G(d)) − y
+���
+�
++ En(F, G ) + R(sF,G; Sn) − R(sF,G)
+�
+��
+�
+:=∆(st;Sn)
++
+2K
+n
+�
+i∈[n]
+∥g(di) − G(di)∥2 + 2K
+n
+�
+i∈[n]
+∥f(qi) − F(qi)∥2 + K2
+n
+�
+i∈[n]
+���σ
+�
+F(qi)⊤G(di)
+�
+− yi
+���
+(16)
+where (a) follows from Lemma C.3, (b) follows from the definition of En(F, G ), and (c) follows from
+Proposition C.2.
+C.1
+Bounding the difference between student’s empirical distillation risk and teacher’s em-
+pirical risk
+Lemma C.2. Given n examples Sn = {(qi, di, yi)}i∈[n] ⊂ Q×D×{0, 1}, let sf,g(qi, di) = f(qi)T g(di) be
+the scores assigned to the (qi, di) pair by a dual-encoder model with f and g as query and document encoders,
+respectively. Let ℓ and ℓd be the binary cross-entropy loss (cf. Eq. 9 with L = 1) and the distillation-specific
+loss based on it (cf. Eq. 11 with L = 1), respectively. In particular,
+ℓ(sF,G(qi, di), yi) := −yi log σ
+�
+F(qi)⊤G(di)
+�
+− (1 − yi) log
+�
+1 − σ
+�
+F(qi)⊤G(di)
+��
+ℓd(sf,g(qi, di), sF,G(qi, di)) := −σ
+�
+F(qi)⊤G(di)
+�
+· log σ
+�
+f(qi)⊤g(di)
+�
+−
+[1 − σ
+�
+F(qi)⊤G(di)
+�
+] · log
+�
+1 − σ
+�
+f(qi)⊤g(di)
+��
+,
+22
+
+where σ is the sigmoid function and sF,G denotes the teacher dual-encoder model with F and Q as its query
+and document encoders, respectively. Assume that
+1. All encoders f, g, F, and G have the same output dimension k ≥ 1.
+2. ∃ K ∈ (0, ∞) such that
+sup
+q∈Q
+max {∥f(q)∥2, ∥F(q)∥2} ≤ K
+and
+sup
+d∈D
+max {∥g(d)∥2, ∥G(d)∥2} ≤ K.
+Then, we have
+1
+n
+�
+i∈[n]
+ℓd
+�
+sf,g(qi, di), sF,G(qi, di)
+�
+− 1
+n
+�
+i∈[n]
+ℓ
+�
+sF,G(qi, di), yi
+�
+≤
+2K
+n
+�
+i∈[n]
+∥g(di) − G(di)∥2 + 2K
+n
+�
+i∈[n]
+∥f(qi) − F(qi)∥2 + K2
+n
+�
+i∈[n]
+���σ
+�
+F(qi)⊤G(di)
+�
+− yi
+��� .
+(17)
+Proof. We first note that the distillation loss can be rewritten as
+ℓd
+�
+sf,g(q, d), sF,G(q, d)
+�
+=
+�
+1 − σ(F(q)⊤G(d)
+�
+f(q)⊤g(d) + γ(−f(q)⊤g(d)),
+where γ(v) := log[1 + ev] is the softplus function. Similarly, the one-hot (label-dependent) loss can be
+rewritten as
+ℓ
+�
+sF,G(q, d), y
+�
+= (1 − y)F(q)⊤G(d) + γ(−F(q)⊤G(d)).
+Recall from our notation in Section 3 that
+R(sf,g, sF,G; Sn) := 1
+n
+�
+i∈[n]
+ℓd
+�
+sf,g(qi, di), sF,G(qi, di)
+�
+,
+(18)
+R(sF,G; Sn) := 1
+n
+�
+i∈[n]
+ℓ
+�
+sF,G(qi, di), yi
+�
+,
+(19)
+as the empirical risk based on the distillation loss, and the empirical risk based on the label-dependent loss,
+respectively. With this notation, the quantity to upper bound can be rewritten as
+R(sf,g, sF,G; Sn) − R(sF,G; Sn) = R(sf,g, , sF,G; Sn) − R(sf,G, sF,G; Sn)
+�
+��
+�
+:=□1
++
+R(sf,G, sF,G; Sn) − R(sF,G, sF,G; Sn)
+�
+��
+�
+:=□2
++
+R(sF,G, sF,G; Sn) − R(sF,G; Sn)
+�
+��
+�
+:=□3
+.
+(20)
+We start by bounding □1 as
+□1 = 1
+n
+�
+i∈[n]
+�
+ℓd
+�
+sf,g(qi, di), sF,G(qi, di)
+�
+− ℓd
+�
+sf,G(qi, di), sF,G(qi, di)
+��
+23
+
+= 1
+n
+�
+i∈[n]
+� �
+1 − σ(F(qi)⊤G(di))
+�
+f(qi)⊤g(di) + γ(−f(qi)⊤g(di))
+−
+�
+1 − σ(F(qi)⊤G(di))
+�
+f(qi)⊤G(di) − γ(−f(qi)⊤G(di))
+�
+= 1
+n
+�
+i∈[n]
+�
+f(qi)⊤�
+g(di) − G(di)
+� �
+1 − σ(F(qi)⊤G(di))
+�
++ γ(−f(qi)⊤g(di)) − γ(−f(qi)⊤G(di))
+�
+(a)
+≤ 1
+n
+�
+i∈[n]
+�
+f(qi)⊤�
+g(di) − G(di)
+� �
+1 − σ(F(qi)⊤G(di))
+�
++
+���f(qi)⊤g(di) − f(qi)⊤G(di)
+���
+�
+(b)
+≤ 1
+n
+�
+i∈[n]
+�
+∥f(qi)∥∥g(di) − G(di)∥
+�
+1 − σ(F(qi)⊤G(di))
+�
++ ∥f(qi)∥∥g(di) − G(di)∥
+�
+≤ K
+n
+�
+i∈[n]
+∥g(di) − G(di)∥2
+�
+2 − σ(F(qi)⊤G(di))
+� �
+≤ 2K
+n
+�
+i∈[n]
+∥g(di) − G(di)∥2,
+(21)
+where at (a) we use the fact that γ is a Lipschitz continuous function with Lipschitz constant 1, and at (b) we
+use Cauchy-Schwarz inequality.
+Similarly for □2, we proceed as
+□2 = 1
+n
+�
+i∈[n]
+�
+ℓd
+�
+sf,G(qi, di), sF,G(qi, di)
+�
+− ℓd
+�
+sF,G(qi, di), sF,G(qi, di)
+��
+= 1
+n
+�
+i∈[n]
+� �
+1 − σ(F(qi)⊤G(di))
+�
+f(qi)⊤G(di) + γ(−f(qi)⊤G(di))
+−
+�
+1 − σ(F(qi)⊤G(di))
+�
+F(qi)⊤G(di) − γ(−F(qi)⊤G(di))
+�
+= 1
+n
+�
+i∈[n]
+�
+G(di)⊤(f(qi) − F(qi))
+�
+1 − σ(F(qi)⊤G(di))
+�
++ γ(−f(qi)⊤G(di)) − γ(−F(qi)⊤G(di))
+�
+≤ 1
+n
+�
+i∈[n]
+�
+∥G(di)∥∥f(qi) − F(qi)∥ +
+���f(qi)⊤G(di) − F(qi)⊤G(di)
+���
+�
+≤ 2K
+n
+�
+i∈[n]
+∥f(qi) − F(qi)∥2.
+(22)
+□3 can be bounded as
+□3 = R(sF,G, sF,G; Sn) − R(sF,G; Sn)
+= 1
+n
+�
+i∈[n]
+�
+ℓd
+�
+sF,G(qi, di), sF,G(qi, di)
+�
+− ℓ
+�
+sF,G(qi, di), yi
+��
+= 1
+n
+�
+i∈[n]
+� �
+1 − σ(F(qi)⊤G(di))
+�
+F(qi)⊤G(di) + γ(−F(qi)⊤G(di))
+− (1 − yi)F(qi)⊤G(di) − γ(−F(qi)⊤G(di))
+�
+24
+
+= 1
+n
+�
+i∈[n]
+��
+1 − σ(F(qi)⊤G(di)) − (1 − yi)
+�
+F(qi)⊤G(di)
+�
+≤ K2
+n
+�
+i∈[n]
+���σ(F(qi)⊤G(di)) − yi
+��� .
+(23)
+Combining Eq. 20, 21, 22, and 23 establishes the bound in Eq. 17.
+Lemma C.3. Given an example (q, d, y) ∈ Q×D×{0, 1}, let sf,g(q, d) = f(q)T g(d) be the scores assigned
+to the (q, d) pair by a dual-encoder model with f and g as query and document encoders, respectively. Let ℓ
+and ℓd be the binary cross-entropy loss (cf. Eq. 9 with L = 1) and the distillation-specific loss based on it
+(cf. Eq. 11 with L = 1), respectively. In particular,
+ℓ(sf,g(q, d), y) := −y log σ
+�
+f(q)⊤g(d)
+�
+− (1 − y) log
+�
+1 − σ
+�
+f(q)⊤g(d)
+��
+ℓd(sf,g(q, d), sF,G(q, d)) := −σ
+�
+F(q)⊤G(d)
+�
+· log σ
+�
+f(q)⊤g(d)
+�
+−
+[1 − σ
+�
+F(q)⊤G(d)
+�
+] · log
+�
+1 − σ
+�
+f(q)⊤g(d)
+��
+,
+where σ is the sigmoid function and sF,G denotes the teacher dual-encoder model with F and Q as its query
+and document encoders, respectively. Assume that
+1. All encoders f, g, F, and G have the same output dimension k ≥ 1.
+2. ∃ K ∈ (0, ∞) such that
+sup
+q∈Q
+max {∥f(q)∥2, ∥F(q)∥2} ≤ K
+and
+sup
+d∈D
+max {∥g(d)∥2, ∥G(d)∥2} ≤ K.
+Then, we have
+E
+�
+ℓ
+�
+sf,g(q, d), y
+��
+�
+��
+�
+:=R(sf,g)
+− E
+�
+ℓd
+�
+sf,g(q, d), sF,G(q, d)
+��
+�
+��
+�
+:=R(sf,g,sF,G)
+≤ KQKDE
+����σ(F(q)⊤G(d)) − y
+���
+�
+(24)
+where expectation are defined by a joint distribution P(q, d, y) over Q × D × {0, 1}
+Proof. Similar to the proof of Proposition C.2, we utilize the fact that
+ℓ
+�
+sF,G(q, d), y
+�
+= (1 − y)F(q)⊤G(d) + γ(−F(q)⊤G(d)),
+ℓd
+�
+sf,g(q, d), sF,G(q, d)
+�
+=
+�
+1 − σ(F(q)⊤G(d)
+�
+f(q)⊤g(d) + γ(−f(q)⊤g(d)),
+where γ(v) := log[1 + ev] is the softplus function. Now,
+E
+�
+ℓ
+�
+sf,g(q, d), y
+�
+− ℓd
+�
+sf,g(q, d), sF,G(q, d)
+��
+(25)
+=E
+�
+(1 − y)f(q)⊤g(d) + γ(−f(q)⊤g(d))
+�
+− E
+��
+1 − σ(F(q)⊤G(d))
+�
+f(q)⊤g(d) + γ(−f(q)⊤g(d))
+�
+25
+
+= E
+��
+1 − y −
+�
+1 − σ(F(q)⊤G(d))
+��
+F(q)⊤G(d)
+�
+≤ K2E
+����σ(F(q)⊤G(d)) − y
+���
+�
+,
+(26)
+which completes the proof.
+C.2
+Uniform deviation bound
+Let F denote the class of functions that map queries in Q to their embeddings in Rk via the query encoder.
+Define G analogously for the doc encoder, which consists of functions that map documents in D to their
+embeddings in Rk. To simplify exposition, we assume that each training example consists of a single relevant
+or irrelevant document for each query, i.e., L = 1 in Section 3. Let
+FG = {(q, d) �→ f(q)⊤g(d) | f ∈ F, g ∈ G }
+Given Sn = {(qi, di, yi) : i ∈ [n]}, let N(ϵ, H ) denote the ϵ-covering number of a function class H with
+respect to L2(Pn) norm, where ∥h∥2
+L2(Pn) := ∥h∥2
+n := 1
+n
+�n
+i=1 ∥h(qi, di)∥2
+2. Depending on the context, the
+functions in H may map to R or Rd.
+Proposition C.4. Let st be scorer of a teacher model and ℓd be a distillation loss function which is Lℓd-
+Lipschitz in its first argument. Let the embedding functions in F and G output vectors with ℓ2 norms at most
+K. Define the uniform deviation
+En(F, G ) =
+sup
+f∈F,g∈G
+����
+1
+n
+�
+i∈[n] ℓd
+�
+f(qi)⊤g(di), st
+qi,di
+�
+− Eq,dℓd
+�
+f(q)⊤g(d), st
+q,d
+����� .
+For any g∗ ∈ G , we have
+ESnEn(F, G ) ≤ ESn
+48KLℓd
+√n
+� ∞
+0
+�
+log N(u, F) + log N(u, G ) du,
+ESnEn(F, {g∗}) ≤ ESn
+48KLℓd
+√n
+� ∞
+0
+�
+log N(u, F) du.
+Proof of Proposition C.4. We first symmetrize excess risk to get Rademacher complexity, then bound the
+Rademacher complexity with Dudley’s entropy integral.
+For a training set Sn, the empirical Rademacher complexity of a class of functions H that maps Q × D to
+R is defined by
+Radn(H ) = Eσ sup
+h∈H
+1
+n
+n
+�
+i=1
+εih(qi, di),
+where {εi} denote i.i.d. Rademacher random variables taking the value in {+1, −1} with equal probability.
+By symmetrization [Bousquet et al., 2004] and the fact that ℓd is Lℓd-Lipschitz in its first argument, we get
+ESnEn(F, G ) ≤ 2LℓdESnRadn(FG ).
+Then, Dudley’s entropy integral [see, e.g., Ledoux and Talagrand, 1991] gives
+Radn(FG ) ≤ 12
+√n
+� ∞
+0
+�
+log N(u, FG ) du.
+26
+
+From Lemma C.5 with KQ = KD = K, for any u > 0,
+N(u, FG ) ≤ N
+� u
+2K , F
+�
+N
+� u
+2K , G
+�
+.
+Putting these together,
+ESnEn(F, G ) ≤ 24Lℓd
+√n
+� ∞
+0
+�
+log N(u/2K, F) + log N(u/2K, G ) du.
+(27)
+Following the same steps with G replaced by {g∗}, we get
+ESnEn(F, {g∗}) ≤ 24Lℓd
+√n
+� ∞
+0
+�
+log N(u/2K, F) du
+(28)
+By changing variable in Eq. 27 and Eq. 28, we get the stated bounds.
+For f : Q → Rk, g : D → Rk, define fg : Q × D → R by fg(q, d) = f(q)⊤g(d).
+Lemma C.5. Let f1, . . . , fN be an ϵ-cover of F and g1, . . . , gM be an ϵ-cover of G in L2(Pn) norm. Let
+supf∈F supq∈Q ∥f(q)∥2 ≤ KQ and supg∈G supd∈D ∥g(d)∥2 ≤ KD. Then,
+{figj | i ∈ [N], j ∈ [M]}
+is a (KQ + KD)ϵ-cover of FG .
+Proof of Lemma C.5. For arbitrary f ∈ F, g ∈ G , there exist ˜f ∈ {f1, . . . , fN}, ˜g ∈ {g1, . . . , gM} such
+that ∥f − ˜f∥n ≤ ϵ, ∥g − ˜g∥n ≤ ϵ. It is sufficient to show that ∥fg − ˜f˜g∥n ≤ (KQ + KD)ϵ. Decomposing
+using triangle inequality,
+∥fg − ˜f˜g∥n = ∥fg − f˜g + f˜g − ˜f˜g∥n
+≤ ∥fg − f˜g∥n + ∥f˜g − ˜f˜g∥n.
+(29)
+To bound the first term, using Cauchy-Schwartz inequality, we can write
+1
+n
+n
+�
+i=1
+�
+f(qi)⊤g(di) − ˜f(qi)⊤˜g(di)
+�2
+≤ sup
+q∈Q
+∥f(q)∥2
+2 · 1
+n
+n
+�
+i=1
+∥(g − ˜g)(di)∥2
+2.
+Therefore
+∥fg − f˜g∥n ≤ KQ∥g − ˜g∥n ≤ KQϵ.
+Similarly
+∥f˜g − ˜f˜g∥n ≤ KD∥f − ˜f∥n ≤ KDϵ
+Plugging these in Eq. 29, we get
+∥fg − ˜f˜g∥n ≤ (KQ + KD)ϵ.
+This completes the proof.
+27
+
+D
+Evaluation metric details
+For NQ, we evaluate models with full strict recall metric, meaning that the model is required to find a golden
+passage from the whole set of candidates (21M). Specifically, for k ≥ 1, recall@k or R@k denotes the
+percentage of questions for which the associated golden passage is among the k passages that receive the
+highest relevance scores by the model. In addition, we also present results for relaxed recall metric considered
+by Karpukhin et al. [2020a], where R@k denotes the percentage of questions where the corresponding answer
+string is present in at least one of the k passages with the highest model (relevance) scores.
+For both MSMARCO retrieval and re-ranking tasks, we follow the standard evaluation metrics Mean
+Reciprocal Rank(MRR)@10 and normalized Discounted Cumulative Gain (nDCG)@10. For retrieval tasks,
+these metrics are computed with respect to the whole set of candidates passages (8.8M). On the other hand,
+for re-ranking task, the metrics are computed with respect to BM25 generated 1000 candidate passages for
+each query. We report 100 × MRR@10 and 100 × nDCG@10, as per the convention followed in the prior
+works.
+E
+Query generation details
+We introduced query generation to encourage geometric matching in local regions, which can aid in trans-
+ferring more knowledge in confusing neighborhoods. As expected, this further improves the distillation
+effectiveness on top of the embedding matching in most cases. To focus on the local regions, we generate
+queries from the observed examples by adding local perturbation in the data manifold (embedding space).
+Specifically, we employ an off-the-shelf encoder-decoder model – BART-base Lewis et al. [2020]. First,
+we embed an observed query in the corresponding dataset. Second, we add a small perturbation to the
+query embedding. Finally, we decode the perturbed embedding to generate a new query in the input space.
+Formally, the generated query x′ given an original query x takes the form x′ = Dec(Enc(x) + ϵ), where
+Enc() and Dec() correspond to the encoder and the decoder from the off-the-shelf model, respectively, and ϵ
+is an isotropic Gaussian noise. Furthermore, we also randomly mask the original query tokens with a small
+probability. We generate two new queries from an observed query and use them as additional data points
+during our distillation procedure.
+As a comparison, we tried adding the same size of random sampled queries instead of the ones generated via
+the method described above. That did not show any benefit, which justifies the use of our query/question
+generation method.
+F
+Experimental details and additional results
+F.1
+Additional training details
+Optimization. For all of our experiments, we use ADAM weight decay optimizer with a short warm up
+period and a linear decay schedule. We use the initial learning rate of 10−5 and 2.8 × 10−5 for experiments
+on NQ and MSMARCO, respectively. The batch sizes is 128.
+28
+
+F.2
+Additional results on NQ
+See Table 7 for the performance of various DE models on NQ, as measured by the relaxed recall metric.
+Table 7: Relaxed recall performance of various student DE models on NQ dev set, including symmetric DE
+student model (67.5M or 11.3M transformer for both encoders), and asymmetric DE student model (67.5M
+or 11.3M transformer as query encoder and document embeddings inherited from the teacher). All distilled
+students used the same teacher (110M parameter BERT-base models as both encoders), with the performance
+(in terms of relaxed recall) of Recall@5 = 87.2, Recall@20 = 94.7, Recall@100 = 98.1. Note: the proposed
+method can achieve 100% of teacher’s performance even with 2/3rd size of the query encoder, and 92-97%
+with even 1/8th size.
+Method
+Recall@5
+Recall@20
+Recall@100
+67.5M
+11.3M
+67.5M
+11.3M
+67.5M
+11.3M
+Train student directly
+62.5
+49.7
+82.5
+73.0
+93.7
+88.2
++ Distill from teacher
+82.7
+66.1
+92.9
+84.0
+97.3
+93.1
++ Inherit document embeddings
+84.7
+73.0
+93.7
+85.4
+97.6
+93.3
++ Query embedding matching
+87.2
+77.6
+95.0
+88.0
+97.9
+94.3
++ Query generation
+87.8
+80.3
+94.8
+89.9
+98.0
+95.6
+Train student only using embedding
+matching and inherit doc embeddings
+86.4
+69.1
+94.2
+81.6
+97.7
+89.9
++ Query generation
+86.7
+72.9
+94.4
+84.9
+97.8
+92.2
+F.3
+Additional results on MSMARCO
+See Table 8 for CE to DE distillation results on MSMARCO re-ranking task, as measured by the nDCG@10
+metric.
+Table 8: Performance of CE to DE distillation on MSMARCO re-ranking task, as measured by the nDCG@10
+metric. As for the teacher CE models, we consider two kinds of CE models based on two different pooling
+mechanism.
+Method
+nDCG@10
+[CLS]-pooled teacher
+43.0
+Dual-pooled teacher
+42.8
+Standard distillation from [CLS]-teacher
+38.8
++Joint matching
+38.0
+Standard distillation from Dual-pooling teacher
+39.2
++Query matching
+39.4
+29
+
+F.4
+Additional results on BEIR benchmark
+See Table 9 (NDCG@10) and Table 10 (Recall@100) for BEIR benchmark results. All numbers are from
+BEIR benchmark paper [Thakur et al., 2021]. As common practice, non-public benchmark sets8, {BioASQ,
+Signal-1M(RT), TREC-NEWS, Robust04}, are removed from the table. Following the original BEIR pa-
+per [Thakur et al., 2021] (Table 9 and Appendix G from the original paper), we utilized Capped Recall@100
+for TREC-COVID dataset.
+Table 9: In-domain and zero-shot retrieval performance on BEIR benchmark [Thakur et al., 2021], as
+measured by nDCG@10. All the baseline number in the table are taken from Thakur et al. [2021]. We
+exclude (in-domain) MSMARCO from average computation as common practice.
+Model (→)
+Lexical
+Sparse
+Dense
+Dataset (↓)
+BM25
+DeepCT
+SPARTA
+docT5query
+DPR
+ANCE
+TAS-B
+GenQ
+SentenceBERT
+(our teacher)
+EmbedDistill
+(ours)
+MS MARCO
+22.8
+29.6‡
+35.1‡
+33.8‡
+17.7
+38.8‡
+40.8‡
+40.8‡
+47.1‡
+46.6‡
+TREC-COVID
+65.6
+40.6
+53.8
+71.3
+33.2
+65.4
+48.1
+61.9
+75.4
+72.3
+NFCorpus
+32.5
+28.3
+30.1
+32.8
+18.9
+23.7
+31.9
+31.9
+31.0
+30.7
+NQ
+32.9
+18.8
+39.8
+39.9
+47.4‡
+44.6
+46.3
+35.8
+51.5
+50.8
+HotpotQA
+60.3
+50.3
+49.2
+58.0
+39.1
+45.6
+58.4
+53.4
+58.0
+56.0
+FiQA-2018
+23.6
+19.1
+19.8
+29.1
+11.2
+29.5
+30.0
+30.8
+31.8
+29.5
+ArguAna
+31.5
+30.9
+27.9
+34.9
+17.5
+41.5
+42.9
+49.3
+38.5
+34.9
+Touch´e-2020
+36.7
+15.6
+17.5
+34.7
+13.1
+24.0
+16.2
+18.2
+22.9
+24.7
+CQADupStack
+29.9
+26.8
+25.7
+32.5
+15.3
+29.6
+31.4
+34.7
+33.5
+30.6
+Quora
+78.9
+69.1
+63.0
+80.2
+24.8
+85.2
+83.5
+83.0
+84.2
+81.4
+DBPedia
+31.3
+17.7
+31.4
+33.1
+26.3
+28.1
+38.4
+32.8
+37.7
+35.9
+SCIDOCS
+15.8
+12.4
+12.6
+16.2
+07.7
+12.2
+14.9
+14.3
+14.8
+14.4
+FEVER
+75.3
+35.3
+59.6
+71.4
+56.2
+66.9
+70.0
+66.9
+76.7
+76.9
+Climate-FEVER
+21.3
+06.6
+08.2
+20.1
+14.8
+19.8
+22.8
+17.5
+23.5
+22.5
+SciFact
+66.5
+63.0
+58.2
+67.5
+31.8
+50.7
+64.3
+64.4
+59.8
+55.5
+AVG (w/o MSMARCO)
+43.0
+31.0
+35.5
+44.4
+25.5
+40.5
+42.8
+42.5
+45.7
+44.0
+Table 10: In-domain and zero-shot retrieval performance on BEIR benchmark [Thakur et al., 2021], as
+measured by Recall@100. All the baseline number in the table are taken from Thakur et al. [2021]. ‡
+indicates in-domain retrieval performance. ∗ indicates capped recall following original benchmark setup. We
+exclude (in-domain) MSMARCO from average computation as common practice.
+Model (→)
+Lexical
+Sparse
+Dense
+Dataset (↓)
+BM25
+DeepCT
+SPARTA
+docT5query
+DPR
+ANCE
+TAS-B
+GenQ
+SentenceBERT
+(our teacher)
+EmbedDistill
+(ours)
+MS MARCO
+65.8
+75.2‡
+79.3‡
+81.9‡
+55.2
+85.2‡
+88.4‡
+88.4‡
+91.7‡
+90.6‡
+TREC-COVID
+49.8∗
+34.7∗
+40.9∗
+54.1∗
+21.2∗
+45.7∗
+38.7∗
+45.6∗
+54.1∗
+48.8∗
+NFCorpus
+25.0
+23.5
+24.3
+25.3
+20.8
+23.2
+28.0
+28.0
+27.7
+26.7
+NQ
+76.0
+63.6
+78.7
+83.2
+88.0‡
+83.6
+90.3
+86.2
+91.1
+89.9
+HotpotQA
+74.0
+73.1
+65.1
+70.9
+59.1
+57.8
+72.8
+67.3
+69.7
+68.3
+FiQA-2018
+53.9
+48.9
+44.6
+59.8
+34.2
+58.1
+59.3
+61.8
+62.0
+60.1
+ArguAna
+94.2
+93.2
+89.3
+97.2
+75.1
+93.7
+94.2
+97.8
+89.2
+87.8
+Touch´e-2020
+53.8
+40.6
+38.1
+55.7
+30.1
+45.8
+43.1
+45.1
+45.3
+45.5
+CQADupStack
+60.6
+54.5
+52.1
+63.8
+40.3
+57.9
+62.2
+65.4
+63.9
+61.3
+Quora
+97.3
+95.4
+89.6
+98.2
+47.0
+98.7
+98.6
+98.8
+98.5
+98.1
+DBPedia
+39.8
+37.2
+41.1
+36.5
+34.9
+31.9
+49.9
+43.1
+46.0
+42.6
+SCIDOCS
+35.6
+31.4
+29.7
+36.0
+21.9
+26.9
+33.5
+33.2
+32.5
+31.5
+FEVER
+93.1
+73.5
+84.3
+91.6
+84.0
+90.0
+93.7
+92.8
+93.9
+93.8
+Climate-FEVER
+43.6
+23.2
+22.7
+42.7
+39.0
+44.5
+53.4
+45.0
+49.3
+47.6
+SciFact
+90.8
+89.3
+86.3
+91.4
+72.7
+81.6
+89.1
+89.3
+88.9
+87.2
+AVG (w/o MSMARCO)
+63.4
+55.9
+56.2
+64.7
+47.7
+60.0
+64.8
+64.2
+65.1
+63.5
+8https://github.com/beir-cellar/beir
+30
+
+F.5
+Additional results with single-stage trained teachers
+Hereby we evaluate EmbedDistill with a simple single-stage trained teachers instead of teachers trained in
+complex multi-stage frameworks, in order to test the generalizability of the method.
+Similar to Table 1, we conducted an experiment on top of single-stage trained teacher based on RoBERTa-
+base instead of AR2 [Zhang et al., 2022] in the main text. We also changed the student to be based on
+DistilRoBERTa or RoBERTa-mini accordingly for simplicity to use same tokenizer.
+Table 11 demonstrates that EmbedDistill provides a significant boost of the performance on top of standard
+distillation techniques similar to what we observed in Table 1.
+Table 11: Full recall performance of various student DE models on NQ dev set, including symmetric DE
+student model, and asymmetric DE student models. All students used the same in-house teacher (124M
+parameter RoBERTa-base models as both encoders), with the full Recall@5 = 64.6, Recall@20 = 81.7, and
+Recall@100 = 91.5.
+Method
+6-Layer (82M)
+4-Layer (16M)
+R@5 R@20 R@100
+R@5 R@20 R@100
+Train student directly
+41.9
+64.5
+82.0
+39.5
+59.9
+76.3
++ Distill from teacher
+48.3
+67.2
+80.9
+44.9
+61.1
+74.8
++ Inherit doc embeddings
+56.9
+74.3
+85.4
+47.2
+64.0
+77.0
++ Query embedding matching
+61.8
+78.7
+89.0
+56.7
+74.6
+85.9
++ Query generation
+61.7
+79.4
+89.6
+57.1
+75.2
+86.7
+Train student using only
+embedding matching and
+inherit doc embeddings
+63.7
+80.3
+90.3
+57.9
+74.6
+85.7
++ Query generation
+64.1
+80.5
+90.4
+58.9
+76.0
+86.6
+Furthermore, we also consider a in-house trained teacher (RoBERTa-base) for MSMARCO reranking task.
+Table 12 demonstrates a similar pattern to Table 3, providing evidence of generalizability of EmbedDistill.
+Table 12: Reranking performance of various DE models on MSMARCO dev set. We utilize a RoBERTa-base
+in-house trained teacher achieving MRR@10 of 33.1 and nDCG@10 of 38.8 is used. The table shows
+performance of the symmetric DE student model and asymmetric DE student models.
+Method
+MRR@10
+nDCG@10
+82M
+16M
+82M
+16M
+Train student directly
+29.7
+26.3
+35.2
+31.4
++ Distill from teacher
+31.6
+28.4
+37.2
+33.5
++ Inherit doc embeddings
+32.4
+30.2
+38.0
+35.8
++ Query embedding matching
+32.8
+31.9
+38.6
+37.6
++ Query generation
+33.0
+32.0
+38.8
+37.7
+Train student only using embedding
+matching and inherit doc embeddings
+32.7
+31.8
+38.5
+37.5
++ Query generation
+33.0
+31.8
+38.9
+37.5
+These result showcase that our method brings performance boost orthogonal to how teacher was trained,
+whether single-staged or multi-staged.
+31
+
+G
+Embedding analysis
+G.1
+DE to DE distillation
+Traditional score matching-based distillation might not result in transfer of relative geometry from teacher to
+student. To assess this, we look at the discrepancy between the teacher and student query embeddings for all
+q, q′ pairs: ∥embt
+q − embt
+q′∥ − ∥embs
+q − embs
+q′∥. Note that the analysis is based on NQ, and we focus on the
+teacher and student DE models based on BERT-base and DistilBERT, respectively. As evident from Fig. 4,
+embedding matching loss significantly reduces this discrepancy.
+1.0
+0.5
+0.0
+0.5
+1.0
+Discrepancy in distance from teacher
+0
+5
+10
+15
+Density
+Standard
+distillation
+Embedding
+matching
+Figure 4: Histogram of teacher-student distance discrepancy in queries.
+G.2
+CE to DE distillation
+We qualitatively look at embeddings from CE model in Fig. 5. The embedding embt
+q,d from [CLS]-pooled
+CE model does not capture semantic similarity between query and document as it is solely trained to classify
+whether the query-document pair is relevant or not. In contrast, the (proxy) query embeddings embt
+q←(q,d)
+from our Dual-pooled CE model with reconstruction loss do not degenerate and its embeddings groups same
+query whether conditioned on positive or negative document together. Furthermore, other related queries are
+closer than unrelated queries. Such informative embedding space would aid distillation to a DE model via
+embedding matching.
+32
+
+q1: macy credit card 
+      phone number
+q2: phone number to 
+     experian credit bureau
+q4: is phosphorus diatomic
+q5: what is a cancer 
+     doctor called 
+q3: colloids chemistry 
+     definition
+q6: physiological disease 
+     examples 
+  All positive
+            pairs
+All negative
+            pairs
+[CLS]-pooled CE model
+Dual pooled 
+CE model
+Pairwise distance matrix
+Dual pooled
+[CLS]-pooled
+Figure 5: Illustration of geometry expressed by [CLS]-pooled CE and our Dual-pooled CE model on 6
+queries from MSMARCO and 12 passages based on pairwise distance matrix across these 72 pairs. [CLS]-
+pooled CE embeddings degenerates as all positive and negative query-document pairs almost collapse to two
+points and fail to capture semantic information. In contrast, our Dual-pooled CE model leads to much richer
+representation that can express semantic information.
+33
+

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@@ -0,0 +1,1636 @@
+arXiv:2301.08694v1  [math.PR]  20 Jan 2023
+Almost everywhere continuity of conditional
+expectations
+Alberto Alonso ∗,a and Fernando Brambila-Paz †,a
+aDepartment of Mathematics, Faculty of Science - UNAM, Mexico
+Abstract
+A necessary and sufficient condition on a sequence {An}n∈N of σ-subalgebras that assures convergence almost
+every where of conditional expectations is given.
+Keywords:
+1.
+Introduction
+Let A be a σ-algebra of a set X with a probability measure µ. Given a sequence {An}n∈N of σ-subalgebras of A, it it
+a natural problem to establish conditions under which the conditional expectations converge almost everywhere.
+In [] we analyzed the case for a sequence to converge in Lp and provided necessary an sufficient conditions for that
+to happen. We defined two σ-subalgebras Aµ and A⊥ and proved that we have convergence in Lp if and only if
+Aµ = A⊥. We also established a chain of σ-subalgebras.
+A ⊂ Aµ ⊂ A⊥ ⊂ A.
+where A and A are the inferior and the superior limit of the the σ-subalgebras {An}n∈N. That is A = �∞
+m=1
+�∞
+n=m An
+and A = �∞
+m=1
+�∞
+n=m. As a corollary we had convergence in Lp in the special case A = A which was a result previ-
+ously obtained by Fetter []. In that paper she pondered if, as in the case of monotone convergence of σ-algebras,
+we have also convergence a.e.. However, in [] a negative answer was provided by establishing a counterexample.
+In this paper we will analyze a chain of families of sets
+A ⊂ Aw.a.e. ⊂ Aµ ⊂ A⊥ ⊂ A⊥a.e. ⊂ A.
+We will show that we have a.e. convergence if and only if Aw.a.e. = A⊥a.e. provided that these two sets are σ-
+subalgebras.
+2.
+Previous-Works
+As usual, we will use the notation E (f |A) for the conditional expectation of f given the σ-algebra A. Some well-
+known results for a.e.-convergence and Lp-convergence (1 ≤ p < ∞) are :
+Theorem 2.1. (Martingales). If {An}n∈N is monotone increasing sequence of σ-subalgebras of A, that is, An ⊂ An+1 for
+any n ∈ N, then
+E (f |An)
+a.e.
+−→
+Lp E
+
+f
+�������
+∞
+�
+n=1
+An
+
+,
+∗Email address: alonsoalber@gmail.com
+†Email address: fernandobrambila@gmail.com
+1
+
+for every f ∈ Lp(A), where �∞
+n=1 An stands for the minimum σ-algebra that contains �∞
+n=1 An.
+�
+Or if {An}n∈N is
+monotone decreasing, that is, An ⊃ An+1, then E (f |An)
+a.e.
+−→
+Lp E
+�
+f
+����∞
+n=1 An
+� �
+.
+In [1], Boylan introduced a Hausdorff metric of the space of σ-algebras. It gives us a relationship between Cauchy
+sequences of σ-subalgebras and Lp-convergence of conditional expectations.
+Theorem 2.2. (Boylan, Equiconvergence). Let {An}n∈N be a Cauchy sequence on the space of σ-algebras with the
+Hausdorff metric, that is,
+d(An,Am) = sup
+A∈An
+�
+inf
+B∈Am
+µ(A △ B)
+�
++ sup
+B∈An
+�
+inf
+A∈Am
+µ(A △ B)
+�
+,
+here A △ B = (A \ B) ∪ (B \ A). There is a subalgebra D such that
+lim
+n→∞d(An,D) = 0,
+and
+E (f |An)
+Lp
+−→ E (f |D),
+for every f ∈ Lp(A), 1 ≤ p < ∞ (see [2,3]).
+Another approach was given by Fetter([]). She proved that if the lim sup of a sequence of σ-algebras coincides
+with the lim inf, then we have convergence in Lp. Indeed
+Theorem 2.3. (Fetter). If {An}n∈N is such that A = A where
+A =
+∞
+�
+m=1
+∞
+�
+n=m
+An
+and
+A =
+∞
+�
+m=1
+∞
+�
+n=m
+An,
+then
+E (f |An) −→
+Lp E
+�
+f
+���A
+�
+,
+for every f ∈ Lp(A), 1 ≤ p < ∞ (see [1,4]).
+Since the condition of the above theorem is fulfilled when we have monotone sequences of σ-algebras, Fetter�s
+result implies that of the martingale theorem in the case of Lp convergence. However, we point out that in [4] it
+was proved that the condition A = A does not imply convergence almost everywhere.
+Given a sequence {An}n∈N of σ-algebras, two relevant σ-algebras were defined in []:
+Aµ =
+�
+A ∈ A : ∃An ∈ An, lim
+n→∞µ(An △ A) = 0
+�
+,
+and if
+W =
+�
+g ∈ L2(A) : ∃Ank ∈ Ank,with χAnk
+weakly
+−→ g
+�
+A⊥ was defined as the minimum complete σ-algebra such that g is A⊥ measurable for all g ∈ W . The importance
+of these σ-algebras is clear due to the following theorem:
+2
+
+Theorem 2.4. (Alonso-Brambila). Let {An}n∈N be such that there is a σ-subalgebra A∞ with the property that An
+µ→ A∞
+and An
+⊥→ A∞. Then and only then
+E (f |An)
+Lp(A)
+−→ E (f |A∞ ),
+for every f ∈ Lp(A), 1 ≤ p < ∞.
+Since these σ-algebras satisfy the relationship
+A ⊆ Aµ ⊆ A⊥ ⊆ A.
+We have that if A = A Fetter�s theorem becomes an immediate corollary.
+Finally we point out that none of the theorems but theorem 2.1 deal with the problem of convergence a.e.
+3.
+A sigma-subalgebra
+In [5] it was introduced the concept for a sequence of σ-subalgebras {An} to µ-approach a σ-subalgebra D if for
+each D ∈ D there were An ∈ An such that µ(An △D) → 0. It was established that in such a case and only in that case
+we have for f ∈ Lp(D) (1 ≤ p < ∞) that
+E(f |An)
+Lp
+−→ E(f |D) = f .
+It is clear that in order to have convergence a.e. we need to introduce a more stronger concept for sets in {An} to
+approach those of D.
+As usual, in the following Ac will stand for X \ A, χA for the characteristic function of a set A, A △ B for the
+symmetric difference of the sets A and B, and A = B a.e for µ(A △ B) = 0.
+We begin by first establishing the following lemma.
+Lemma 3.1. Let (X,A,µ) a probability space with σ-algebra A and measure µ. If {An}n∈N is a sequence of elements in
+A and A ∈ A, the following statements are equivalent:
+i) χAn −→
+a.e χA.
+ii) A =
+∞
+�
+N=1
+�
+n≥N
+An =
+∞
+�
+N=1
+�
+n≥N
+An a.e.
+iii)
+lim
+N→∞µ
+
+
+�
+n>N
+(An △ A)
+
+ = 0.
+Proof. Notice that χAn → χA a.e implies that, for all x ∈ X\M, where M is a set of measure zero, there is an Nx ∈ N
+such that if n > Nx
+���χAn(x) − χA(x)
+��� < 1
+2.
+(1)
+To prove i) → ii), we first take a look at the elements of A \ M. Since
+���χAn(x) − χA(x)
+��� =
+���χAcn(x) − χAc(x)
+���,
+(2)
+(1) and (??) tell us that for n > Nx and x ∈ A \ M; χAcn(x) < 1/2, and so x ∈ An. Therefore
+A \ M ⊂
+∞
+�
+N=1
+�
+n>N
+An.
+3
+
+Using the same argument for Ac we get
+Ac \ M ⊂
+∞
+�
+N=1
+�
+n>N
+Ac
+n.
+Thus
+A ∪ M ⊃
+∞
+�
+N=1
+�
+n>N
+An.
+Since
+∞
+�
+N=1
+�
+n>N
+An ⊂
+∞
+�
+N=1
+�
+n>N
+An,
+we have
+
+
+∞
+�
+N=1
+�
+n>N
+An
+
+ ∩ Mc ⊂ A ∩ Mc ⊂
+∞
+�
+N=1
+�
+n>N
+An ⊂
+∞
+�
+N=1
+�
+n>N
+An.
+So
+
+
+∞
+�
+N=1
+�
+n>N
+An
+
+ ∩ Mc =
+
+
+∞
+�
+N=1
+�
+n>N
+An
+
+ ∩ Mc = A ∩ Mc.
+Therefore i → ii).
+We will prove now that iii) → i). Let
+M =
+∞
+�
+N=1
+�
+n>N
+(An △ A).
+Then, by hypothesis µ(M) = 0. Now, as
+Mc = X \ M =
+∞
+�
+N=1
+�
+n>N
+(An △ A)c.
+if x ∈ X \ M there is an N ∈ N such that
+x ∈
+�
+n>N
+(An △ A)c,
+and hence x ∈ (An △ A)c for all n > N. That is
+0 = χAn△A(x) =
+���χA(x) − χAn(x)
+��� < ǫ,
+for n > N.
+Finally, to prove that ii) implies iii), we notice that:
+µ
+
+
+�
+n>N
+(An △ A)
+
+ = µ
+
+
+
+
+�
+n>N
+An
+
+ ∩ Ac
+
+ + µ
+
+
+
+
+�
+n>N
+Ac
+n
+
+ ∩ A
+
+.
+4
+
+Since by hypothesis
+0 = µ
+
+A △
+∞
+�
+N=1
+�
+n>N
+An
+
+ ≥ µ
+
+A \
+∞
+�
+N=1
+�
+n>N
+An
+
+ = µ
+
+A ∩
+∞
+�
+N=1
+�
+n>N
+Ac
+n
+
+ = lim
+N→∞µ
+
+A ∩
+�
+n>N
+Ac
+n
+
+,
+and
+0 = µ
+
+A △
+∞
+�
+N=1
+�
+n>N
+An
+
+ ≥ µ
+
+
+∞
+�
+N=1
+�
+n>N
+An \ A
+
+ = lim
+N→∞µ
+
+
+�
+n>N
+An ∩ Ac
+
+,
+we get
+lim
+N→∞µ
+
+
+�
+n>N
+(An △ A)
+
+ = 0
+Given a sequence {An}n∈N of σ-subalgebras of A we will define the family of sets F as
+F =
+A ∈ A : A =
+∞
+�
+N=1
+�
+n>N
+An =
+∞
+�
+N=1
+�
+n>N
+An a.e. for a sequence {An} such that An ∈ An
+.
+We will prove that F is a σ-algebra. To do this we will need the following lemma.
+Lemma 3.2. If for any r > 0 there is a sequence {Arn : Arn ∈ An} and Nr ∈ N such that
+µ
+
+
+�
+n>Nr
+(Ar
+n △ B)
+
+ < r,
+then B ∈ F .
+Proof. Let Sm = 1/2m for m ∈ N and {Qm}m∈N a strictly increasing sequence such that Qm ≥ NSm. As Qn ≥ NSm we
+have
+µ
+
+
+�
+n>Qm
+(ASm
+n △ B)
+
+ ≤ µ
+
+
+�
+n>NSm
+(ASm
+n △ B)
+
+ < Sm.
+For n > Q1 define the sequence {An} by:
+An = ASk
+n
+if
+Qk < n ≤ Qk+1,
+since ASk
+n ∈ An so is An. We have
+µ
+
+
+�
+n>Qm
+(An △ B)
+
+ =µ
+
+
+∞
+�
+k=m
+�
+Qk<n≤Qk+1
+(ASk
+n △ B)
+
+ ≤
+∞
+�
+k=m
+µ
+
+
+�
+Qk<n≤Qk+1
+(ASk
+n △ B)
+
+
+≤
+∞
+�
+k=m
+µ
+
+
+�
+n>Qk
+(ASk
+n △ B)
+
+ <
+∞
+�
+k=m
+Sk =
+1
+2m−1 ,
+and therefore
+5
+
+lim
+N→∞µ
+
+
+�
+n>N
+(An △ B)
+
+ = 0.
+The proof follows from lemma 3.1
+We can now prove that F is a σ-algebra.
+Proposition 3.3. F is a σ-subalgebra of A
+Proof. It is clear that ∅, X are elements of F since we can take An = ∅ for all n ∈ N or An = X for all n ∈ N.
+If A ∈ F , by Proposition 3.1 there are {An ∈ An} such that
+χAn −→ χA a.e.
+Since Ac
+n ∈ An and
+χAcn = 1 − χAn
+a.e.
+−→ 1 − χA = χAc,
+we have that Ac ∈ F .
+Let A,B ∈ F . By proposition 3.1 there are two sequences {An}, {Bn}, with An,Bn ∈ An for all n ∈ N, such that
+χAn → χA a.e. and χBn → χB a.e. Since An ∩Bn ∈ An and χAn∩Bn = χAnχBn −→
+a.e. χAχB = χA∩B we have that A∩B ∈ F .
+Thus F is an algebra.
+To show that is is a σ-algebra, let B = �∞
+k=1 Dk with Dk ∈ F . Since F is an algebra, we can consider the sets {Dk}
+disjoint.
+Define EM = �M
+k=1 Dk and let r > 0. Since the sets Dk are disjoint, there is an M1 ∈ N such that
+µ
+
+
+∞
+�
+k>M1
+Dk
+
+ < r
+2.
+On the other hand, since EM1 ∈ F . There is a sequence {An ∈ An} such that
+lim
+N→∞µ
+
+
+�
+n>N
+(An △ EM1)
+
+ = 0.
+Therefore, there is an N1 ∈ N such that µ
+��
+n>N1(An △ EM1)
+�
+< r/2.
+As EM1 ⊂ B, An \ B ⊂ An \ EM1. So
+µ
+
+
+�
+n>N1
+An \ B
+
+ ≤ µ
+
+
+�
+n>N1
+(An \ EM1)
+
+ ≤ µ
+
+
+�
+n>N1
+(An △ EM1)
+
+ < r
+2.
+Also, as B is the union of the disjoint sets EM1 and �∞
+k>M1 Dk we have
+µ
+
+
+�
+n>N1
+(B \ An)
+
+ =µ
+
+
+�
+n>N1
+(EM1 \ An)
+
+ + µ
+
+
+�
+n>N1
+
+
+∞
+�
+k>M1
+Dk
+
+ \ An
+
+
+≤µ
+
+
+�
+k>N1
+(EM1 △ An)
+
+ + µ
+
+
+∞
+�
+k>M1
+Dk
+
+ < r
+2 + r
+2.
+Therefore µ
+��
+n>N1(An △ B)
+�
+< r.
+So, by lemma 3.2 B is in F .
+6
+
+Given a sequence of σ-subalgebras of A in [5] we found for the σ-algebra Aµ defined by
+Aµ =
+�
+A ∈ A : ∃An ∈ An, lim
+n→∞µ(An △ A) = 0
+�
+,
+the relationships
+∞
+�
+m=1
+∞
+�
+n=m
+An ≡ A ⊂ Aµ ⊂ A ≡
+∞
+�
+m=1
+∞
+�
+n=m
+An.
+It is clear by its definition that F ⊂ Aµ. Now, let A ∈ �∞
+n=N An, take An = A for n > N, then trivially χAn −→
+a.e. χA and
+thus A ∈ F . Since F is a σ-algebra, we have
+F ⊃
+∞
+�
+N=1
+∞
+�
+n=N
+An.
+Therefore
+A ⊂ F ⊂ Aµ ⊂ A.
+What we have proven is that given a sequence of σ-algebras {An}, if A is a set F measurable, there are fn func-
+tions An measurable such that fn → χA a.e. However that does not imply that E(χA|An) → χA a.e. even when the
+conditional expectations do so in Lp (1 ≤ p < ∞).
+In the example shown in [4] we have the property that A = A and hence A = F , and a Borel set for which
+the conditional expectations do not converge a.e. to its characteristic function. Let us show an easier example
+for which the conditional expectations for a characteristic function in F do not converge a.e. but they do so in
+Lp (1 ≤ p < ∞).
+Let X = [0,1] and A the Lebesgue measurable sets. We are going to define a sequence {An} that lie in
+� 1
+2,1
+�
+and
+such that χAn → χ[ 1
+2 ,1) a.e. {Jk} will be a sequence that lies in
+� 1
+2,1
+�
+\ An but with measure half its size
+Specifically
+An =
+�1
+2,1 − 1
+2n
+�
+,
+Jn =
+�
+1 −
+1
+2n+1 ,1
+�
+,
+n ∈ N.
+On
+�
+0, 1
+2
+�
+define a sequence of intervals that go back and forth in the interval and in each turn diminishing its size
+In,k =
+�
+k
+4 · 2n , k + 1
+4 · 2n
+�
+with 0 ≤ k < 2 · 2n,
+n ∈ N.
+We have that for each n ∈ N
+2·2n−1
+�
+k=0
+In,k =
+�
+0, 1
+2
+�
+.
+Let Bn,k = In,k ∪ Jn and Cn,k = (An ∪ Bn,k)c = Ac
+n ∩ Bc
+n,k.
+Consider the sequence of σ-subalgebras
+An,k = �∅,X,An,Bn,k,Cn,k
+�,
+Since
+7
+
+E
+�
+χ[ 1
+2 ,1)
+���An,k
+�
+=
+⟨χ[ 1
+2 ,1),χAn⟩
+µ(An)
+χAn +
+⟨χ[ 1
+2 ,1),χBn,k ⟩
+µ(Bn,k)
+χBn,k +
+⟨χ[ 1
+2 ,1),χCn,k ⟩
+µ(Cn,k)
+χCn,k,
+we have
+E
+�
+χ[ 1
+2 ,1)
+���An,k
+�
+=χAn +
+1
+2n+1
+3
+2
+1
+2n+1
+χBn,k +
+1
+2n+1
+1
+2 +
+1
+2n+2
+χCn,k
+=χAn + 2
+3χBn,k +
+2
+1 + 2n+1 χCn,k
+=χAn + 2
+3χIn,k + 2
+3χJn +
+2
+1 + 2n+1 χCn,k .
+Since χAn → χ[ 1
+2 ,1) a.e., 2
+3χJn → 0 a.e. and
+2
+1 + 2n+1 χCn,k → 0 a.e. but 2
+3χIn,k does not convergence a.e., we have that
+� 1
+2,1
+�
+∈ F but E
+�
+χ[ 1
+2 ,1)
+���An,k
+�
+↛ χ[ 1
+2 ,1) a.e.
+4.
+Necessary and sufficient conditions for almost everywhere convergence.
+4.1.
+On almost everywhere convergence of characteristics functions
+Let B be a σ-subalgebra of A.
+Definition 4.1. Define the seminorm ∥ · ∥B for f ∈ L∞(dµ) as
+∥f ∥B = ∥E (f |B)∥∞ .
+In the following we are going to use the notation ∥A∥B = ∥χA∥B for any set A ∈ A. We have the property
+Lemma 4.2. Let A be a measurable set in A and χA its characteristic function. Then
+∥A∥B = ∥χA∥B = sup
+B ∈ B
+µ(B) > 0
+µ(A ∩ B)
+µ(B)
+.
+Proof. Let B ∈ B. We have then
+µ(A ∩ B) =
+�
+χA∩Bdµ =
+�
+χAχBdµ
+=
+�
+E(χA|B)χBdµ ≤ ∥E(χA|B)∥∞ µ(B).
+Therefore, for any B ∈ B with µ(B) > 0
+µ(A ∩ B)
+µ(B)
+≤ ∥E(χA|B)∥∞ .
+Since the case ∥E(χA|B)∥∞ = 0 is trivial, take any ε > 0 such that
+∥E(χA|B)∥∞ − ε > 0.
+By definition the set
+8
+
+C = �x : E(χA|B) > ∥E(χA|B)∥∞ − ε�
+is B-measurable and
+�
+C
+E(χA|B)dµ ≥ (∥E(χA|B)∥∞ − ε)µ(C).
+as
+�
+C
+E(χA|B)dµ =
+�
+χCE(χA|B)dµ =
+�
+χCχAdµ = µ(C ∩ A),
+we have then
+µ(C ∩ A)
+µ(C)
+≥ ∥E(χA|B)∥∞ − ε.
+Definition 4.3. We say that A ∈ A is uniformly covered by the sequence of σ-subalgebras {An}n∈N if there is a sequence
+{An ∈ An}n∈N such that
+i) A =
+∞
+�
+N=1
+�
+n≥N
+An =
+∞
+�
+N=1
+�
+n≥N
+An.
+ii)
+���χA\An
+���An → 0 as
+n → ∞.
+Next lemma shows that actually, we can relax a little bit condition ii).
+Lemma 4.4. A is uniformly covered by {An}n∈N if and only if for any r > 0 there is a sequence {Ar
+n}n∈N, Ar
+n ∈ An and
+M ∈ N such that
+i) A =
+∞
+�
+N=1
+�
+n>N
+Ar
+n =
+∞
+�
+N=1
+�
+n>N
+Ar
+n .
+ii) ∥A \ Ar∥An < r
+if
+n > Mr.
+Proof. By lemma 3.1 the condition i) means that
+lim
+N→∞µ
+
+
+�
+n>N
+(A △ Ar
+n)
+
+ = 0.
+Thus, for r = 1 let N1 > M1 and such that
+µ
+
+
+�
+n>N1
+(A △ Ar
+n
+
+ < 1
+.
+In general let rk = 2−k and ˜Nk such that
+µ
+
+
+�
+n> ˜Nk
+(A △ Ak
+n)
+
+ < rk,
+9
+
+and N′
+k such that
+���A \ Arkn
+���An < rk
+for n > N′
+k.
+It is clear that we can define a strictly increasing sequence Nk such that Nk > max( ˜Nk,N′
+k)
+Define An ∈ An as Arkn if Nk ≤ n ≤ Nk+1. Then
+∥A \ An∥An =
+���A \ Arkn
+���An < 1
+2k ,
+we also have
+µ
+
+
+�
+n≥�
+Nk′
+(A △ An)
+
+ = µ
+
+
+∞
+�
+k=k′
+�
+�
+Nk≤n<�
+Nk+1
+(A △ An)
+
+ ≤
+∞
+�
+k=k′
+µ
+
+
+�
+�
+Nk≤n<�
+Nk+1
+A △ Arkn
+
+ <
+∞
+�
+k=k′
+1
+2k =
+1
+2k′−1 .
+Therefore since µ(�
+n>N A △ An) is monotone
+lim
+N→∞µ
+
+
+�
+n≥N
+A △ An
+
+ = 0.
+(3)
+and thus finally
+lim
+n→∞∥A \ An∥An = 0.
+We are going to say that a sequence of sets {An ∈ An} uniformly covers a set A ∈ A if conditions i) and ii) of
+definition XXX are satisfied. A couple of results regarding uniform covering are:
+Lemma 4.5. If {An ∈ An} uniformly covers a set A ∈ A and {A′n ∈ An} is such that An ⊂ A′n for all n ∈ N and
+χA′n −→ χA a.e. then it uniformly covers A.
+Proof. The proof is trivial since for 0 ≤ f ≤ g, and B σ-subalgebra E(f |B) ≤ E(g|B) and so, as A \ A′n ⊂ A \ An,
+E(χA\A′n|An) ≤ E(χA\An|An)
+Lemma 4.6. If A and B are sets in A uniformly covered by {An}n∈N. Then so are A�B and A�B.
+Proof. Let An ∈ An and Bn ∈ An sequences of sets that uniformly cover A and B respectively. Consider first the
+sequence Cn = An
+�Bn ∈ An. Since by property i) of the definition of uniform covering we have that χAn
+a.e.
+−→ χA
+and χBn
+a.e.
+−→ χB:
+χAn
+�Bn = χAnχBn
+a.e.
+−→ χAχB = χA�B
+We have also that
+A ∩ B \ (An ∩ Bn) = A ∩ B ∩ (Ac
+n ∪ Bc
+n) ⊂ (A ∩ Ac
+n) ∪ (B ∩ Bc
+n)
+Therefore
+∥A ∩ B \ (An ∩ Bn)∥An ≤ ∥(A ∩ Ac
+n) ∪ (B ∩ Bc
+n)∥An ≤ ∥A ∩ Ac
+n∥An + ∥B ∩ Bc
+n∥An −→ 0a.e.
+For the case of the union of two sets we have
+χAn
+�Bn = χAn + χBn − χAnχBn
+a.e.
+−→ χA + χB − χAχB = χA�B
+and
+(A ∪ B) \ (An ∪ Bn) = (A ∪ B) ∩ (Ac
+n ∩ Bc
+n) ⊂ (A ∩ Ac
+n) ∪ (B ∩ Bc
+n)
+10
+
+Lemma 4.7. If A is uniformly covered by {An}n∈N. Then
+lim
+n→∞(E(χA|An))(x) ≤ χA(x) a.e.
+Proof. Let {An} be a sequence with the properties of Definition 4.8. Then
+E(χA|An) = E(χAχAn + χAχAcn|An)
+= χAnE(χA|An) + E(χA\An|An)
+≤ χAn +
+���χA\An
+���An .
+as A is uniformly covered by the sequence {An} then i) of Definition 4.8 implies that
+χAn
+a.e.
+−→ χA.
+So
+lim
+n→∞E(χA|An) ≤ χA a.e.
+xxx222
+In view of the proof we can actually relax somewhat the condition of uniformly covering to get a similar result.
+Lemma 4.8. Let {An}n∈N be a sequence of σ-subalgebras . If A ∈ A is such that there is a sequence {An ∈ An}n∈N
+satisfying
+i) A =
+∞
+�
+N=1
+�
+n≥N
+An =
+∞
+�
+N=1
+�
+n≥N
+An.
+ii) E(χA\An|An) → 0 a.e as n → ∞.
+Then
+lim
+n→∞(E(χA|An))(x) ≤ χA(x) a.e.
+Proof. The proof is exactly the same as the one of the above lemma.
+E(χA|An) = E(χAχAn + χAχAcn|An)
+= χAnE(χA|An) + E(χA\An|An)
+≤ χAn + E(χA\An|An).
+Notice that if A is uniformly covered by {An} the conditions of lemma 4.8xx are satisfied.
+The interesting case occurs when both A and Ac are uniformly covered by {An} or satisfy the conditions of the
+above lemma.
+Lemma 4.9. If A ∈ A is such that A and Ac are uniformly covered by {An} (or satisy the conditions of lemma 4.9x) then
+E(χA|An)
+a.e.
+−→ χA.
+11
+
+Proof.
+lim
+n→∞E(χAc|An) = lim
+n→∞E(1 − χA|An)
+= lim
+n→∞(1 − E(χA|An))
+= 1 − lim
+n→∞
+E(χA|An).
+Since Ac is uniformly covered
+χAc = 1 − χA ≥ 1 − lim
+n→∞
+E(χA|An),
+and so
+lim
+n→∞
+E(χA|An) ≥ χA a.e.
+finally, as A is uniformly covered
+lim
+n→∞E(χA|An) ≤ χA ≤ lim
+n→∞
+E(χA|An) a.e.
+And then
+E(χA|An)
+a.e.
+−→ χA.
+Now the necessary lemma for a.e. convergence of characteristic functions.
+Lemma 4.10. Let A ∈ A and {An}n∈N σ-subalgebras such that
+E(χA|An)
+a.e.
+−→ χA.
+Then A and Ac are uniformly covered by {An}n∈N.
+Proof. Let 0 < r < 1. Define An ∈ An as
+An = {x ∈ X : E(χA|An)(x) ≥ r},
+since E(χA|An)
+a.e.
+→ χA. We have that
+χAn
+a.e.
+−→ χA,
+and therefore
+A =
+∞
+�
+N=1
+�
+n>N
+An =
+∞
+�
+N=1
+�
+n>N
+An.
+It is also clear that
+0 ≤E(χA\An|An) = E(χAχAcn|An)
+=χAcnE(χA|An) < χAcnr ≤ r.
+Thus
+���E(χA\An|An)
+���∞ ≤ r and by the above lemma A is uniformly covered.
+Since E(χA|An)
+a.e.
+−→ χA implies that E(χAc|An)
+a.e.
+−→ χAc, we have that Ac is also uniformly covered.
+12
+
+From Lemma 4.9 and Lemma 4.10 we have the following theorem.
+Theorem 4.11. Let {An}n∈N be a sequence of σ-algebras and let A ∈ A. Then
+E(χA|An)
+a.e.
+−→ χA,
+if and only if A and Ac are uniformly covered by {An}n∈N.
+In view of the above theorem it is clear that it is convenient to establish the following definition.
+Definition 4.12. Define Aµ.a.e. as
+Aµ.a.e. = {A ∈ A : A and Ac are uniformly covered by {An}n∈N}
+Before we proceed we observe that if {An ∈ An} is a sequence of sets that uniformly covers A and {Cn ∈ An} is a
+sequence of sets that uniformly covers Ac, the sequence An
+�Cn uniformly covers A and Ac.
+We have then that Aµ.a.e. is an algebra of sets. XXXX No de hecho necesitamos que acomplemento este. Si ese es el
+caso si pero ¿por que?
+Notice that if Aµ.a.e. is σ-subalgebra then
+A ⊂ Aµ.a.e. ⊂ F ⊂ Aµ ⊂ A⊥ ⊂ A.
+Lemma 4.13. If Aµ.a.e. is σ-subalgebra then
+i) f ∈ L∞ �
+Aµ.a.e.
+�
+we have that
+E(f |An)
+a.e.
+−→ E(f |Aµ.a.e.) = f .
+ii) If there is a σ-subalgebra B such that for all f ∈ L∞ �
+Aµ.a.e.
+�
+E(f |An)
+a.e.
+−→ E(f |B) thenB ⊂ Aµ.a.e..
+Proof. Let ǫ > 0. As f ∈ L∞ �
+Aµ.a.e.
+�
+, there is a simple function g, Aµ.a.e.-measurable such that ∥f − g∥∞ < ǫ/2. It is
+clear that Lemma 4.10 implies that E(g|An)
+a.e.
+−→ g. Thus
+|E(f |An) − f |(x) ≤|E(f |An) − E(g|An)|(x) + |E(g|An) − g|(x) + |g − f |(x)
+≤|E(f − g|An)|(x) + |E(g|An) − g|(x) + ∥g − f ∥∞
+≤ǫ + |E(g|An) − g|(x).
+So lim
+n→∞|E(f |An) − f |(x) ≤ ǫ a.e. for every ǫ and therefore lim
+n→∞E(f |An) = f a.e.
+To prove ii), let A ∈ B. By hypothesis E(χA|An) −→ χA a.e..
+For 0 < ǫ < 1 define An = {x : E(χA|An) > ǫ} ∈ An.
+Since for almost all x ∈ A, E(χA|An)(x) −→
+n→∞ 1, x ∈ An for n big enough. So χA(x)χAn(x) −→ χA(x) a.e.. The case
+for Ac is similar. In this case, for almost all x � A, E(χA|An)(x) −→
+n→∞ 0. Hence x � An for n big enough. Thus
+χAc(x)χAn(x) −→ 0 a.e.. And thus χAn(x) −→ χA(x) a.e..
+A =
+∞
+�
+N=1
+�
+n>N
+An =
+∞
+�
+N=1
+�
+n>N
+An.
+we also have
+∥A \ An∥An =
+���E(χAχAcn|An)
+���∞ =
+���χAcnE(χA|An)
+���∞ < ǫ
+���χAcn
+���∞ < ǫ.
+And so by Lemma 3.2 A is uniformly covered by by {An}.
+13
+
+We can improve the above lemma by considering functions in Lp.
+Lemma 4.14. If Aµ.a.e. is σ-subalgebra and 1 ≤ p < ∞ then for all f ∈ L2 �
+Aµ.a.e.
+�
+we have that
+E(f |An)
+a.e.
+−→ E(f |Aµ.a.e.) = f .
+How do the above results look in the case of An are monotone?. In the case we have a monotone decreasing
+sequence of σ-subalgebras it is clear that AN = �∞
+n=N An and that for any N, �∞
+n=N An = �∞
+n=1 An. Therefore
+∞
+�
+n=1
+An = A ⊂ F ⊂ A ⊂
+∞
+�
+n=1
+An
+So if A ∈ (F) the sequence An = A ∈ An and trivially uniformly covers A.
+XXXXXX falta la demostracion
+XXXXXXXXXXXXXXXXXXXXX
+4.2.
+Necessary and sufficient conditions to have
+In [4.x] we defined a σ-subalgebra A⊥ by first considering the set W
+W =
+�
+g ∈ L2(A) : ∃Ank ∈ Ank with χAnk
+L2−weakly
+−→
+g
+�
+,
+A⊥ was defined as the minimum σ-algebra generated by W. We are going to do something similar. First we notice
+the obvious fact χAn = E(χAn|An). Define then, given a sequence {An}n∈N σ-subalgebras, the set
+CN =
+h ∈ L1(µ) : h =
+�
+k≥N
+E(χBk|Ak), Bk ∈ A, disjoint, and such that there are only finite many of such B′
+ks
+.
+Notice that if h ∈ CN
+∥h∥1 =
+� �
+n≥N
+E(χBn|An)dµ =
+�
+n≥N
+µ(Bn) = µ
+
+
+�
+n≥N
+Bn
+
+ ≤ 1.
+XXX”””XXX”””
+We will define
+Definition 4.15.
+W ⊥a.e. =
+�
+f ∈ L∞(µ) : for every subsequence
+�
+hNk
+�
+, hNk ∈ CNk,
+�
+f ,hNk
+�
+−→
+k→∞ 0
+�
+.
+Theorem 4.16. If f ∈ L∞(µ) then
+E(f |An)
+a.e.
+−→ 0,
+(4)
+if and only if f ∈ W ⊥a.e..
+Proof. ⇒) Let f ∈ L∞(µ). Without loss of generality we can assume that ∥f ∥∞ ≤ 1. First, notice that if h ∈ CN we
+have
+�h,f � =
+�
+n≥N
+�
+E(χBn,An),f
+�
+=
+�
+n≥N
+�
+χBn,E(f |An)
+�
+.
+14
+
+Let ǫ > 0. Since E(f ,An)
+a.e.
+−→ 0, Egoroff’s theorem implies that there is a set Mǫ and N1 ∈ N such that m ≥ N1
+µ(Mc
+ǫ) < ǫ
+2
+and
+���χMǫE(f |An)
+��� < ǫ
+2.
+Thus if N > N1
+|�hN,f �| =
+�������
+�
+n≥N
+�
+χBn,E(f ,An)
+�
+�������
+≤
+�
+n≥N
+�����
+�
+χBn,χMǫE(f ,An)
+����� +
+����
+�
+χBnχMcǫ,E(f |An)
+�����
+�
+≤ ǫ
+2
+�
+n≥N
+�
+χBn,χMǫ
+�
++
+�
+n≥N
+∥E(f |An)∥∞
+���χBnχMcǫ
+���1
+≤ ǫ
+2µ(Mǫ) + ∥f ∥∞ µ(Mc
+ǫ) ≤ ǫ,
+thus �hN,f � −→
+N→∞ 0.
+⇐) To prove the theorem in the other direction first we notice that since we can take f or −f , without loss of
+generality we can assume that there is an ǫ such that
+µ
+
+
+∞
+�
+N=1
+�
+n≥N
+{x ∈ X : E(f |An)(x) ≥ ǫ}
+
+ > 0.
+That is, there is an r > 0 such that for any N there is an M with
+µ
+
+
+�
+N≤n≥M
+{x ∈ X : E(f |An)(x) ≥ ǫ}
+
+ > r.
+Let An = {x ∈ X : |E(f |An)(x)| > ǫ} and as usual
+BN = AN,
+Bk = Ak \ �k−1
+j=N Bj for N < k < M,
+(5)
+{Bk} is a disjoint family and
+�
+N≤n<M
+Bn =
+�
+N≤n<M
+An.
+Let
+hN =
+�
+N≤n<M
+E(χBn|An),
+we have that
+�hN,f � =
+�
+N≤n<M
+�
+χBn|E(f |An)
+�
+> ǫ
+�
+N≤n<M
+µ(χBn) = ǫµ
+
+
+�
+N≤n<M
+Bn
+
+ > ǫr.
+We have then constructed a sequence {hN}N∈N such that for all N, �hN,f � > ǫr. Therefore f � W ⊥a.e..
+15
+
+In XXX we defined the orthogonal conditional expectation induced by D as the operator E⊥
+B = I − EB, which in
+L2(A) is the orthogonal projection E⊥
+B : L2(A) → L2(B)⊥. Let D be the following family of σ-subalgebras
+D =
+�
+B ∈ A : E⊥
+Bf ∈ W ⊥a.e. for all f ∈ L∞(A)
+�
+It is clear that D is not empty since it trivially contains A.
+An immediate property is the following.
+Proposition 4.17. Let C,B be two σ-subalgebras. If C ∈ D and B ⊃ C, then B ∈ D
+Proof. Notice that in this case E⊥
+B = E⊥
+CE⊥
+B. So, as f in L∞ implies that E⊥
+Bf is also in L∞,
+E⊥
+Bf = E⊥
+C(E⊥
+Bf ) ∈ W ⊥a.e.
+Definition 4.18. Amin will be the minimum complete σ-subalgebra such that contains the set
+˜W =
+�
+g ∈ L2(A) : there is
+�
+hNk
+�
+,hNk ∈ CNk such that hNk −→ g weakly in L2�
+Lemma 4.19. If B ∈ D then Amin ⊂ B.
+Proof. Let g ∈ ˜W and hN ∈ CN such that hN
+w
+−→ g. Since B is in D we have that for all f in Lp ∩ L∞
+�f ,E(g|B)� = �E(f |B),g� = lim
+N→∞
+�E(f |B),hN
+� = lim
+N→∞
+�
+(I − E⊥
+B)f ,hN
+�
+= lim
+N→∞
+�
+f ,(I − E⊥
+B)hN
+�
+= lim
+N→∞
+�f ,hN
+� = �f ,g�
+Therefore g is equal almost everywhere to E(g|B) and so is B measurable. Since by definition Amin is the minumum
+σ-subalgebra such that makes all such g measurable so Amin ⊂ B.
+Definition 4.20. We will call the sequence {An}n∈N 2-bounded if for all hN in CN sup∥hN∥2 < ∞.
+Lemma 4.21. If {An} is 2-bounded and B is a σ-subalgebra, then B ∈ D if and only if Amin ⊂ B.
+Proof. In view of lemma xxx we only need to prove the assertion in the sense (⇐) Assume Amin ⊂ B and suppose
+that Amin is not in D. That means that there is an f ∈ L∞ such that its orthogonal projection with respect to Amin
+is not in W ⊥a.e..That is, there is a sequence {hN ∈ CN} such that �f ,hN
+� does not converge to zero. Hence, there is
+an ǫ > 0 and a subsequence
+�
+hNk
+�
+such that
+����
+�
+E⊥
+Aminf ,hNk
+����� > ǫ. By hypothesis the sequence of σ-subalgebras is 2-
+bounded, so there is a subsequence, which we still denote by hNk, that weakly converges to an h in L2. By definition
+h is in ˜W and therefore is Amin measurable. But that leads us to a contradiction since 0 < ǫ <
+����
+�
+E⊥
+Aminf ,h
+����� =
+����
+�
+f ,E⊥
+Aminh
+����� = 0.
+Of course the 2-boundeness condition is a very strong restriction. In the following we will show that the main
+property is the fact that D has a minimum σ-subalgebra. In those cases we will denote that minimum by A⊥a.e..
+Of course in the case the sequence is 2-bounded A⊥a.e. = Amin
+HOOOOOOOOOOOOOOOLA XXXXXXXXXXXXXXXXXXXXX
+4.3.
+Necessary and sufficient conditions to have comvergence a.e.
+Theorem 4.22. If {An} is a sequence of σ-subalgebras such that
+i)
+ii)
+for all f ∈ L∞, E(f |An) converges a.e. then Aµa.e. is a σ-subalgebra, D has a minimum σ-subalgebra and Aµa.e. = A⊥a.e.
+16
+
+Proof. By definition Aµa.e. = {A ∈ A|A and Ac are univormly covered} Let us recall that for xxx Aµa.e. is an algebra.
+Thus, in order to prove that is a σ−subalgebra we need only to check that if we have a sequence of pairwise disjoint
+sets {Bk}, with Bk ∈ Aµa.e. , then ∪∞
+k=1Bk ∈ Aµa.e..
+In view of theorem 4.11, we need to prove that E(χ∪∞
+k=1Bk|An)
+a.e.
+−→ χ∪∞
+k=1Bk. Now, since the Bk are in Aµa.e. for any
+natural M we have
+E(χ∪∞
+k=1Bk|An) ≥ E(χ∪M
+k=1Bk|An)
+a.e.
+−→ χ∪M
+k=1Bk
+By hypothesis, the conditional expectations converge almost everywhere for any f ∈ L∞. Therefore there is a
+g ∈ L∞ such that
+E(χ∪∞
+k=1Bk|An)
+a.e.
+−→ g
+and thus g ≥ χ∪M
+k=1Bk. Since this is true for any M, we conclude that g ≥ χ∪∞
+k=1Bk.
+Using the dominated convergence theorem we have that
+0 ≤
+�
+1,g − χ∪∞
+k=1Bk
+�
+= lim
+�
+1,E(χ∪∞
+k=1Bk|An) − χ∪∞
+k=1Bk
+�
+= µ(∪∞
+k=1Bk) − µ(∪∞
+k=1Bk) = 0
+0 ≤
+�
+g − χ∪∞
+k=1Bk dµ = lim
+�
+E(χ∪∞
+k=1Bk|An) − χ∪∞
+k=1Bk dµ = µ(∪∞
+k=1Bk) − µ(∪∞
+k=1Bk) = 0
+And hence g = χ∪∞
+k=1Bk a.e..
+17
+

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+CiT: Curation in Training for Effective Vision-Language Data
+Hu Xu
+Saining Xie
+Po-Yao Huang
+Licheng Yu
+Russell Howes
+Gargi Ghosh
+Luke Zettlemoyer
+Christoph Feichtenhofer
+FAIR, Meta AI
+https://github.com/facebookresearch/CiT
+Abstract
+Large vision-language models are generally applicable
+to many downstream tasks, but come at an exorbitant train-
+ing cost that only large institutions can afford. This pa-
+per trades generality for efficiency and presents Curation
+in Training (CiT), a simple and efficient vision-text learning
+algorithm that couples a data objective into training. CiT
+automatically yields quality data to speed-up contrastive
+image-text training and alleviates the need for an offline
+data filtering pipeline, allowing broad data sources (includ-
+ing raw image-text pairs from the web). CiT contains two
+loops: an outer loop curating the training data and an inner
+loop consuming the curated training data. The text encoder
+connects the two loops. Given metadata for tasks of interest,
+e.g., class names, and a large pool of image-text pairs, CiT
+alternatively selects relevant training data from the pool by
+measuring the similarity of their text embeddings and em-
+beddings of the metadata. In our experiments, we observe
+that CiT can speed up training by over an order of magni-
+tude, especially if the raw data size is large.
+1. Introduction
+Vision-language models have demonstrated success for
+fine-tuning and zero-shot transfer to downstream tasks [12,
+21,26] by training on a general-purpose large-scale dataset
+instead of a small task-level dataset. While general, large-
+scale pre-training is computationally expensive (e.g. CoCa
+[36] trains on 2048 TPUs for 5 days) and typically per-
+formed on a pre-filtered dataset (e.g. WIT400M [21] used
+by CLIP [21] is created by searching for image-text pairs
+with text containing a set of 500,000 queries and [24] uses
+this model to create another dataset).
+Such filtering pipelines usually involve manual labor-
+intensive efforts to remove data that is unlikely useful for
+downstream tasks [12, 21]. Recent effort has been made to
+curate data for high-quality image-text pairs (such as CC3M
+[25], CC12M [3], YFCC15M [21,29], WIT400M [21] and
+LAION [23, 24]). Nevertheless, research is typically tied
+to the static datasets or model weights (if the data is not re-
+Text 
+Model
+Vision
+Model
+CLIP Pre-training
+Filtering Pipeline
+RawData
+StaticTrainingData
+DynamicTrainingData
+Curation in Training
+Text 
+Model
+Vision 
+Model
+Training Loop
+Training Loop
+Data Curation Loop
+Text 
+Model
+Metadata
+RawData
+Figure 1.
+A conceptual illustration of CLIP training vs. CiT.
+Vanilla CLIP training uses static data from offline human filter-
+ing (e.g. cleaned YFCC15M or WIT400M [21]) and optimizes the
+model. Instead, our CiT incorporates dynamic data curation into
+training in two loops: (i) an outer curation loop improving data (for
+downstream tasks) given the current model; (ii) an inner loop op-
+timizing the model given the curated data. The trained text model
+connects the loops by providing embeddings for curation.
+leased) and is not able to access or change the data pipelines
+or model architectures. Further, work is limited by the pro-
+hibitive cost of training on these large image-text datasets
+(e.g. the CLIP model is trained on WIT400M for 12 days
+using 256 GPUs).
+In this work, our goal is to empower training with the ca-
+pability of adjusting the data distribution. Our intention is to
+dynamically curate the data during training and our key idea
+is to use the learned text representation of vision-language
+models to measure relevance of the data w.r.t. the task of in-
+terest. Given metadata (from downstream tasks e.g. a class
+name such as “chicken”), we measure its embedding simi-
+larity to the training data. This similarity can guide us for
+the decision of including this data into our training process.
+For example a caption containing the word “giraffe” will
+have higher embedding similarity to “chicken” than a cap-
+tion such as “throwback Thursday”.
+arXiv:2301.02241v1  [cs.CV]  5 Jan 2023
+
+Driven by this idea, we presents a simple algorithm that
+incorporates data Curation in Training (CiT), aiming at im-
+proving both data efficiency and model performance. CiT
+works as follows. Given a large source of image-text pairs
+and metadata (e.g. a list of class names used in this paper),
+CiT alternatively performs curation of the data and training
+on that curated data. As shown in Figure 1, CiT contains
+two loops: an outer loop to curate data given the current
+model and an inner loop trains the model given the curated
+data. Similar as Locked image Tuning (LiT [38]), CiT uses
+pre-trained image and text encoders and freezes the image
+one. The text model connects the two loops by serving cu-
+rated data to inner loop for training which in turn learns
+good representations for the outer loop for curation.
+CiT can speed up training by multiple orders of mag-
+nitude, especially if the raw data size is large; e.g. when
+trained on LAION-400M data, CiT reaches similar Ima-
+geNet zero-shot1 accuracy as OpenCLIP [31], while being
+37.7× faster in training. Since CiT changes the training
+data distribution that focuses on one or more tasks of in-
+terest, it can even handle image-text pairs from any (noisy)
+source with unknown distribution. Our experiments reveal
+that vanilla CLIP/LiT training fails on raw random image-
+text pairs crawled from the web, while CiT trains easily.
+2. Related Work
+Vision-Language Learning. Contrastive learning was ini-
+tially popular in vision self-supervision [4,11,32] and later
+adopted for cross-modal learning [16, 18, 19,21, 33]. CLIP
+[21] populates the idea of contrastive learning from image-
+text pairs (used before e.g. in ConVIRT [40]) at scale and
+shows a strong performance of zero-shot transfer to image
+classification and retrieval tasks. SLIP [20] combines image
+self-supervision and language supervision. LiT [38] shows
+that when a good pre-trained vision encoder is adopted, it
+is better to lock (freeze) the well pre-trained vision encoder
+to protect vision representations from being corrupted by
+noisy language supervision. Flamingo also use pre-trained
+models for various tasks [1].
+Vision-Language Data.
+Large-scale vision-language
+learning is typically coupled to a data pipeline to yield high-
+quality data for efficient training [12, 26, 37]. For exam-
+ple, CC3M [25] heavily filters web crawled pairs and only
+keeps 0.1% of the raw data.
+Both CC3M and CC12M
+[3] leverage Google Cloud APIs with models predicting a
+large number of classes (on the order of 105) [25] to filter
+out mismatched image-text pairs. YFCC100M is curated
+from Yahoo Flicker using text fields (such as title, descrip-
+1Zero-shot refers to not seeing any training examples of the target
+dataset. We note that our approach uses extra information of the down-
+stream task, such as class names; however, this metadata is easy to acquire
+and can be of various forms as shown in experiments.
+tion, etc.). This ensures certain data quality but limits the
+scale. Later YFCC100M is further cleaned as YFCC15M
+to contain English-only image-text pairs by [21]. Due to
+the limited scale, CLIP further curates a WebImageText
+dataset (WIT400M) by formulating queries from Wikipedia
+and performing searching them online to obtain image-text
+pairs. Florence [37] curates a dataset with the extra multi-
+label signals to improve supervision. ALIGN [12] relaxes
+CC12M filtering to show that training on 1.8B noisy pairs
+can achieve CLIP-level performance. FLAVA [26] com-
+bines existing human annotated datasets of smaller scale for
+high-quality image-text pairs. Different to related research,
+CiT improves data within the training algorithm, and not as
+a pre-filtering. We demonstrate that such approach allows
+us to effectively learn from raw image-text pairs.
+Related Areas. Our work is related to research in other do-
+mains. In NLP, there are existing works on domain-adaptive
+finetuning and retrieval [9, 14, 15, 34, 35, 39]. In machine
+learning research, subset selection [13, 30] cast data selec-
+tion as a discrete bi-level optimization problem.
+3. Method
+In CLIP pre-training, the training objective (e.g. con-
+trastive image-text correspondence) operates as a training
+proxy that approximates downstream tasks (e.g. classifica-
+tion accuracy). Our CiT introduces a data proxy to fit the
+data distribution to downstream tasks. In this section, we
+first go through the details of the CiT algorithm in §3.1,
+training loop in §3.2 and the data proxy for the curation
+loop in §3.3.
+3.1. CiT Algorithm
+CiT contains two loops: the curation loop curates data
+given the current weights of the model and the training loop
+optimizing the weights given the curated data.
+Let D = {(xi
+img, xi
+txt)}N
+i=1, be the set of source of image-
+text pairs. Then DC ⊆ D is the actual training data we
+aim to curate from the source. We define two functions:
+(i) Curation(D; Θ), and (ii) Training(Θ; DT ), for curation
+and training loops, respectively. Importantly, the weights
+of the learned model Θ connects the two loops and serves
+the curation loop with the updated representations from the
+training loop. There is no notion of a fixed dataset or train-
+ing epochs over D; instead, we view the data source as an
+online data stream. CiT uses a sequential setup that alter-
+natively performs curation for every s pairs of training.
+CiT is shown in Algorithm 1. It takes 3 inputs: a data
+source D, the pre-trained weights Θ and a training budget
+b, which can be training time, resources consumed, etc. We
+simply use steps of weight updates as the training cost in
+this paper. Line 1 initializes the training budget. Line 2
+determines if current training exceeds that training budget.
+
+Algorithm 1: CiT (see §A.1.1 for pseudo code)
+Input: D: data source
+Θ: model’s pre-trained weights
+b: training budget
+1 c ← 0
+2 while c < b do
+3
+DT ← Curation
+�
+D; Θ
+�
+4
+Θ, n ← Training(Θ; DT )
+5
+c ← c + n
+6 end
+Algorithm 2: Curation
+Input
+: Θ: model’s current weights
+D: data source
+Constant: m(·; ·): model architecture
+Tmeta: metadata for tasks of interests
+s: number of expected pairs
+1 xmeta ← m(Tmeta; Θ)
+2 DC ← ∅
+3 while |DC| < s and Draw ⊂ D do
+4
+Draw,txt ← {xi
+txt|(xi
+img, xi
+txt) ∈ Draw}
+5
+xtxt ← m(Draw,txt; Θ)
+6
+f ← DataProxy(Draw, xtxt, xmeta)
+7
+DC ← f(Draw; Θ, Draw, Tmeta) ∪ DC
+8 end
+9 return DC
+The main framework of CiT is to alternatively perform cu-
+ration and training in line 2-4. To recap CLIP pre-training,
+we first detail the training function next.
+3.2. Training
+The core of CLIP [21] training is the contrastive cross-
+modal objective serving as the proxy to approximates down-
+stream tasks (e.g. higher classification accuracy). This ob-
+jective pulls embeddings of positive image-text pairs closer
+and pushes negative pairs from other examples in a training
+batch apart; thus it creates a proxy for classification, which
+has one example per class and the rest of the batch are other
+classes described by natural language.
+The training loop is shown in Algorithm 3, with the
+training data DC, delivered from curation. We let m(·; ·)
+denote the image-text model.
+We use sim(ximg, xtxt) =
+ximgx⊤
+txt/(∥ximg∥∥xtxt∥) in line 3 to compute the image-to-
+text cosine similarity, divided by a trainable temperature τ.
+Our CiT training objective has almost the same structure as
+in CLIP, except that we only use an image-to-text (and no
+text-to-image) contrastive loss (Limg2txt) in line 4. We ab-
+late this loss versus the averaged bidirectional contrastive
+loss (used by CLIP) in our experiments. Line 5 updates the
+model parameters and line 6 counts training cost.
+3.3. Curation
+CiT also has a data objective that curates data using the
+(previously updated) model. Encoding the data with an up-
+Algorithm 3: Training
+Input
+: DC: curated training data
+Θ: model’s weights
+Constant: m(·; ·): model architecture
+1 foreach Dbatch ⊂ DC do
+2
+ximg, xtxt ← m(Dbatch; Θ)
+3
+l ← sim(ximg, xtxt)/τ
+4
+Limg2txt ← CrossEntropy(l, arange(|Dbatch|))
+5
+Θ ← Limg2txt(Θ; Dbatch)
+6
+n ← n + 1
+7 end
+8 return Θ, n
+dated model allows for better representation of the data.
+Akin to the contrastive objective for training, the core func-
+tion in curation is a data proxy (or objective) that selects
+data based on the metadata (e.g. a list of class names).
+We detail the curation loop in Algorithm 2. It takes the
+following inputs: model weights Θ, a data source D, the
+model architecture, the metadata for downstream tasks Tmeta
+and an expected size of curated data s. Tmeta is a list con-
+taining a pre-defined taxonomy; (e.g. ImageNet WordNet
+lemmas or a combination from a group of tasks in our ex-
+periments), but could be generalized to other forms of text.
+Algorithm 2 first obtains the embeddings for the meta-
+data in line 1. Then it sets up the curated set DC for the next
+round of training and keeps curating data in line 3-7. Line 3
+gets the next batch of raw image-text pairs. Line 4 obtains
+its text part and line 5 computes the text embedding from
+the current model. Line 6 is the data proxy, which approx-
+imates the data distribution for the downstream tasks (de-
+tailed in the next subsection). Lastly, we merge the newly
+curated subset into the curated set DC.
+Data Proxy.
+We use language-based metadata and the
+text encoder to measure the relevance of training data. This
+favors efficiency because the text encoders are typically sig-
+nificantly cheaper to evaluate (e.g. the text encoder only
+uses ∼4.6% of the ViT-L image-encoders’ compute).
+In DataProxy(Draw, xtxt, xmeta) of Algorithm 2, we first
+compute the similarities of text embeddings (xtxt) over em-
+beddings of the metadata (xmeta):
+vi
+max = max
+j (sim(xi
+txt, xj
+meta)),
+(1)
+where sim(xi
+txt, xj
+meta) = xi
+txtxj,⊤
+meta/(∥xi
+txt∥∥xj
+meta∥) is the
+cosine similarity between embeddings of sample i and
+metadata j. Here the highest similarity over all metadata
+vi
+max is used to measure the sample quality.
+Let Dt = {(xi
+img, xi
+txt)|(xi
+img, xi
+txt) ∈ Draw and vi
+max > t}
+denote a subset, where all samples have a maximum simi-
+larity above a curation threshold t. Given the best possible
+match to metadata, we use a mixed strategy to determine if
+
+a sample shall be used:
+�
+Dt
+if
+|Dt|
+|Draw| > γ,
+arg topki(vi
+max, k = γ|Draw|),
+otherwise,
+(2)
+where
+|Dt|
+|Draw| is the ratio of curation with γ being a pre-
+defined minimal ratio of curation. If enough samples meet
+the threshold t, Dt is used. Otherwise, we use a minimal
+ratio γ of samples, that represent the top-k matching ones
+(with k = γ|Draw|) in terms of similarity across metadata.
+The threshold t is crucial for CiT to balance the tradeoff
+between data quality and quantity. A higher t leads to high
+data quality, but can lead a lower ratio of curation. We adopt
+this mixed strategy because line 3 in Algorithm 2 could be-
+come a near infinite loop if the ratio of curation is low and
+not enough data that meets t can be found. This could hap-
+pen because the threshold is set too high, or the data source
+has low metadata correspondence. The otherwise part in
+equation 2 resolves this by selecting the γ (typically set to
+around 1% - 5%) best possible matches for training. See
+§A.1.1 for PyTorch pseudo code of CiT.
+4. Experiments
+We use training data from two categories shown below;
+clean data that involves human-based offline filter pipelines
+and raw data that has not undergone cleaning.
+4.1. Cleaned Training Data
+YFCC15M. We use the 15M subset of YFCC100M [29]
+(filtered by [21]) as the main evaluation dataset as it is
+widely adopted in existing literatures [20–22, 38]. It con-
+sists of English-only titles, descriptions, and tags. We sim-
+ply refer to this as YFCC15M in this paper. Except for ap-
+plying the script from [20] to remove HTML formatting,
+we do not perform any extra filtering or preprocessing. In
+contrast, LiT [38] performs extra filtering such as remov-
+ing titles that start with “DSC”, “IMG” and “Picture”, or
+removing them if more than half of them contain digits.
+CC12M. Since YFCC15M may lack enough training data,
+LiT [38] also combines YFCC15M with Conceptual Cap-
+tions 12M (CC12M) [3], which is filtered and transformed
+from image & alt-text pairs from web pages. CC12M in-
+volves cleaning by supervised models from Google Cloud
+APIs to match the image’s prediction over classes with text.
+LAION400M [24] contains 400M English only image-text
+pairs. It is crawled from 2 and later filtered by a CLIP [21]
+model. Thus, LAION400M implicitly carries the data filter
+pipeline of WIT400M on which CLIP has been trained.
+2https://commoncrawl.org
+4.2. Raw Training Data
+YFCC100M. We use the raw YFCC100M (the source
+of YFCC15M) to compare with YFCC15M. Note that
+YFCC100M is multilingual, whereas YFCC15M is English.
+Raw Image-Text Crawl. To challenge CiT with real-world
+data, we further collect raw (unfiltered) image-text pairs
+from Common Crawl.
+We only perform de-duplication
+and NSFW filtering, but no filtering on image-text associ-
+ation. This ended with 1.2B multilingual image-text pairs
+and 28.56% pairs are English (identified by our language
+identification system but this information is not used for CiT
+training). As such, about 343M image-text pairs are in En-
+glish, which are slightly smaller than the scale of WIT400M
+or LAION400M, but much more noisy.
+4.3. Implementation and Training
+Our training recipe uses a global batch size of 16,384,
+which is trained in 16 Nvidia V100 32GB GPUs. Our vi-
+sion encoder corresponds to ViT [7] of various sizes and
+the text encoder defaults to BERTbase-SimCSE [6,8] with a
+maximum token length of 32, similar to LiT [38]. Unless
+specified, we set a budget of training to be within b = 5000
+steps (81M image-text pairs). We report hyper-parameters
+and an extra low-cost single-GPU setting in §A.1.
+We use pre-trained vision and text encoders and join
+them via two randomly initialized projection layers. Fol-
+lowing LiT, we freeze the vision encoder and make the text
+encoder and two projection layers trainable. One can either
+use the text representation before, or after the projection
+layer for computing cosine similarity during curation. We
+ablate these two choices in §4.6.
+4.4. Evaluation
+We evaluate zero-shot (0-shot) transfer accuracy of CiT
+on 26 benchmarks, following [20,21]. In our ablation stud-
+ies, we use YFCC15M as the main data source for training
+and ImageNet-1K (IN-1K) as the downstream task. We use
+prompts from CLIP for all 26 tasks and additionally use the
+extra 2 prompts from LiT [38] for ImageNet for a fair com-
+parison with LiT. Following CLIP, we perform prompt en-
+sembling by averaging the class embeddings for each class
+across the prompt templates. For classification, cosine sim-
+ilarity is computed between an image embedding and the
+averaged class embeddings and the class with the highest
+cosine similarity is CiT’s prediction. We perform validation
+every 500 training steps and stop training if the accuracy
+does not increase over the previous validation. The corre-
+sponding total training time (including curation and train-
+ing) is reported along with the validation accuracy. We esti-
+mate the training time of baselines by re-running them un-
+der the same setup as CiT (i.e. 16 GPUs) and maximize the
+GPU usage for best throughput. More results are in §A.2.
+
+Curation
+Acc
+online
+61.4
+offline
+57.5
+no
+53.8
+(a) Curation effect
+# of steps
+Acc
+50
+61.0
+100
+61.4
+200
+61.5
+300
+61.1
+(b) Curation freq.
+Feature of Curation
+Acc
+pooled encoder
+61.4
+projection output
+60.7
+w/ prompts
+61.4
+(c) Curation feature
+Threshold t
+Acc
+0.5
+60.9
+t = 0.55
+61.4
+0.6
+61.1
+0.7
+59.7
+(d) Threshold t
+Text Variants
+Acc
+BERT max len. 32
+61.4
+BERT max len. 77
+61.2
+w/o YFCC tag
+59.1
+w/o YFCC tag aug.
+60.8
+BERT first 6 layers
+60.2
+(e) Text variants
+Table 1. Ablation experiments. We use MoCo-v3 / BERTbase-SimCSE, YFCC15M as data source and report IN-1K Accuracy.
+4.5. Choice of Pre-trained Models
+We first study the effects of pre-trained encoders.
+As vision encoder, we consider (1) ViT-B/16 [7] (patch
+size of 16×16 pixels) with pre-trained weights from
+self-supervised MoCo-v3 [5], DINO [2] and MAE [10],
+all trained on IN-1K but without any labels.
+To be
+consistent with LiT [38], we also consider (2) super-
+vised ViT(AugReg) [28] B/32, B/16, and L/16 trained
+on ImageNet-21K3. Finally, we also explore weakly-
+supervised ViT-B/16 and ViT-H/14 SWAG [27].
+Vision Model
+Pre-train Obj.
+Pre-train Data
+IN-1K Acc.
+MoCo-v3 [5]
+Contrastive
+IN-1K
+61.4
+DINO [2]
+Contrastive
+IN-1K
+60.3
+MAE [10]
+Masking
+IN-1K
+42.4
+AugReg [28]
+Supervised
+IN-21K
+69.4
+SWAG [27]
+Weakly-Supervised
+IG 3.6B
+67.5
+Table 2.
+Ablation study of different vision encoders on ViT-
+B/16 with text encoder as BERTbase-SimCSE on YFCC15M. Pre-
+training objective matters for CiT training.
+Results for different vision encoder weights under the
+same ViT-B/16 architecture are in Table 2. We notice that
+the accuracy of MoCo-v3 (61.4%) and DINO (60.3%) pre-
+training are close, while MAE is worse (42.4%), presum-
+ably because the representations learned by instance dis-
+crimination (MoCo-v3 and DINO), which learns different
+embeddings for different images, is closer to zero-shot clas-
+sification than MAE’s training objective. AugReg performs
+best with 69.4% accuracy, presumably because the super-
+vised pre-training on IN-21K is superior to unsupervised
+IN-1K pre-training. Finally, SWAG is worse than AugReg,
+but better than MoCo-v3. In the following experiments of
+this section, we will show larger variants.
+For text encoder, we consider self-supervised base mod-
+els from (1) language models BERT [6]; and contrastive
+tuned (2) BERT-SimCSE and RoBERTa-SimCSE [8], as
+shown in Table 3.
+Text Model
+Pre-training obj.
+IN-1K Acc.
+BERTbase(uncased) [6]
+from scratch
+57.7
+BERTbase(uncased) [6]
+SimCSE [8]
+61.4
+BERTbase(uncased) [6]
+BERT NSP [6]
+59.9
+RoBERTabase [17]
+SimCSE [8]
+59.7
+Table 3. Ablation study of different text encoders with MoCo-v3
+on YFCC15M: contrastive pre-training yields better accuracy.
+3We follow LiT here, but note that using IN-21K is not strictly a zero-
+shot setting, because 999 of the 1000 classes in IN-1K are in IN-21K.
+We observe similar trends as for vision: SimCSE trained
+BERT is better than vanilla BERT or RoBERTa, proba-
+bly because contrastively trained [CLS] token by Sim-
+CSE can perform better text similarity than BERT’s pair-
+wise (a.k.a, next sentence prediction) trained [CLS] token
+or RoBERTa’s no training on [CLS] token.
+4.6. Ablations
+We adopt the combination of MoCo-v3 ViT B/16 and
+BERT-SimCSE as our default setting. We summarize abla-
+tion experiments of CiT in Table 1.
+Stage of Curation. We first ablate the effects of curation
+in Table 1a. We see that CiT has a 7.6% boost compared
+to no curation. We further ablate a offline curation before
+training. This is sub-optimal as the SimCSE purely pre-
+trained from the text may not learn good representations for
+semantic-level similarity (discussion in §3.1).
+Frequency of Curation. Next, we are interested in how
+frequently curation needs to be performed. Table 1b varies
+the number of steps (and therefore pairs s when multiplied
+with the batch-size) for curation (in Alg. 2). We found that
+curating too frequent or infrequent yields sub-optimal re-
+sults, but the change is marginal so we chose 100 steps as
+default.
+Feature for Curation. In Table 1c, we find that using the
+feature before the projection layer (e.g. the direct output
+of SimCSE) is better than the features from the projection
+layer. This is probably because the projection layer tends
+to be more unstable during training (e.g. randomly initial-
+ized and needs longer training to align with the visual repre-
+sentation), whereas the SimCSE embedding is already pre-
+trained for text similarity.
+Threshold. In Table 1d we ablate the threshold t, which
+controls the trade-off for data quality and quantity. A lower
+threshold adds more low-quality data and a higher threshold
+reduces data quantity, so t = 0.55 is a good balance.
+Text Variants. We ablate the length of text encoders in
+Table 1e to understand the memory/text sequence length
+tradeoff. We find that longer text sequences (77) (we re-
+duce batch size per GPU to half and double the number of
+GPUs) are slightly worse. We also ablate the effectiveness
+of YFCC15M tag augmentation, adopted from LiT. Lastly,
+we are wondering if a shallow (6 layers) BERT-SimCSE is
+also a good text encoder. We obtain 1.2% worse results.
+
+Pre-train Data
+Method
+Vision Encoder
+Vision Initialization
+w/ Labels
+Total Time
+IN-1K Acc
+YFCC15M
+CLIP [21]
+ResNet-50
+scratch
+
+25 hrs
+31.3
+OpenCLIP [31]
+ResNet-50
+scratch
+
+25 hrs
+32.7
+LiT [38]
+ViT-B/16
+DINO [2]
+
+n/a
+55.4
+LiT [38]
+ViT-B/16
+MoCo-v3 [5]
+
+n/a
+55.5
+LiT [38]
+ViT-B/16
+AugReg [28]
+IN-21K
+n/a
+55.9†
+LiT [38]
+ViT-B/32
+AugReg [28]
+IN-21K
+64 hrs
+59.9*
+CiT
+ViT-B/16
+DINO [2]
+
+n/a
+60.3
+CiT
+ViT-B/16
+MoCo-v3 [5]
+
+5 hrs
+61.4
+CiT
+ViT-B/32
+AugReg [28]
+IN-21K
+11 hrs
+63.3
+CiT
+ViT-B/16
+AugReg [28]
+IN-21K
+8 hrs
+69.4
+CiT
+ViT-L/16
+AugReg [28]
+IN-21K
+8 hrs
+72.0
+CiT
+ViT-H/14
+SWAG [27]
+IG hashtags
+11 hrs
+73.7
+YFCC15M+CC12M
+LiT [38]
+ViT-L/16
+AugReg [28]
+IN-21K
+112 hrs
+72.2*
+CiT
+ViT-L/16
+AugReg [28]
+IN-21K
+32 hrs
+75.6
+YFCC100M
+LiT [38]
+ViT-B/32
+AugReg [28]
+IN-21K
+153 hrs
+58.9*
+CiT
+ViT-B/16
+MoCo-v3 [5]
+
+48 hrs
+64.6
+CiT
+ViT-B/32
+AugReg [28]
+IN-21K
+64 hrs
+65.6
+CiT
+ViT-B/16
+AugReg [28]
+IN-21K
+66 hrs
+72.2
+CiT
+ViT-L/16
+AugReg [28]
+IN-21K
+66 hrs
+74.8
+CiT
+ViT-H/14
+SWAG [27]
+IG hashtags
+62 hrs
+75.5
+Table 4. Comparison to existing methods on YFCC and CC12M. Under identical vision encoders, CiT achieves +3.2% higher accuracy
+with YFCC100M than using the human-cleaned YFCC15M subset and +5.9% accuracy over LiT on YFCC15M. * indicates reproduced
+results with BERTbase (uncased) for fair comparison; see appendix for the implementation differences to original LiT [38]. Total time for
+training and curation is reported for 16 V100 GPUs and varies depending on quality of embeddings from the vision encoder.
+Method
+Vision Encoder
+Vision Initialization
+w/ Labeled Data
+Total Time
+IN-1K Acc
+OpenCLIP
+ViT-B/32
+scratch
+
+458 hrs
+62.9
+OpenCLIP
+ViT-B/16
+scratch
+
+981 hrs
+67.1
+OpenCLIP
+ViT-L/14
+scratch
+
+6803 hrs
+72.8
+LiT [38]
+ViT-B/32
+AugReg [28]
+IN-21K
+31 hrs
+62.8
+CiT
+ViT-B/16
+MoCo-v3 [5]
+
+26 hrs
+67.1
+CiT
+ViT-B/32
+AugReg [28]
+IN-21K
+62 hrs
+67.5
+CiT
+ViT-B/16
+AugReg [28]
+IN-21K
+63 hrs
+73.1
+CiT
+ViT-L/16
+AugReg [28]
+IN-21K
+27 hrs
+75.8
+CiT
+ViT-H/14
+SWAG [27]
+IG hashtags
+26 hrs
+76.4
+Table 5. CiT on LAION400M: CiT reaches OpenCLIP-level accuracy with 37× total training time improvement.
+4.7. Comparison to prior work on ImageNet
+We compare CiT with existing contrastive cross-modal
+models in Tables 4 (YFCC and CC12M), 5 (LAION400M)
+and 6 (raw image-text crawl). We report the pre-training
+method (CLIP/LiT/CiT), vision encoder and initialization,
+usage of human-annotated labels, total training time in our
+setup (16 GPUs), as well as the ImageNet 0-shot accuracy.
+YFCC. In Table 4 we report several data points for LiT
+and CiT training with various vision encoders and initial-
+ization.
+On YFCC15M, CiT outperforms LiT on self-
+supervised MoCo-v3 vision encoders by +5.9% accuracy.
+With ViT-B/32 trained with supervised AugReg on IN-21K,
+CiT yields a +3.4% gain over LiT. On YFCC15M+CC12M
+data with ViT-L/16 models, CiT outperforms LiT by +3.4%.
+On YFCC100M we observe that LiT underperforms
+compared to YFCC15M (58.9 vs 59.9), due to cleaning [21]
+of the 15M subset. CiT however can reverse the trend. CiT
+outperforms its counterpart from YFCC15M by 3%+ when
+using the less curated YFCC100M. This indicates human
+cleaning of YFCC100M by CLIP [21] is sub-optimal. The
+performance of CiT on YFCC100M is even +2.6% bet-
+ter than LiT on YFCC15M+CC12M. This trend holds for
+larger image model sizes (ViT-L/H) and stronger initializa-
+tion (AugReg/SWAG), which lead to better accuracy.
+LAION400M. In Table 5 we see that CiT performs better
+than OpenCLIP on LAION400M, while being substantially
+faster. For example, CiT with ViT-B/16 MoCo-v3 vision
+encoder performs as good as OpenCLIP but is 37.7×faster
+in training. With more advanced initialization and larger
+ViT-L models, CiT is 283× faster and 3% more accurate,
+producing 75.8% in 1.1 days with a 16 GPU setup, while
+OpenCLIP would take ∼283 days for an accuracy of 72.8%.
+We note that this extreme speedup comes with the caveat
+that our approach curates data with respect to downstream
+tasks; therefore, CiT only uses 26 hours for training, com-
+pared to 981 hours for OpenCLIP pre-training.
+Raw Image-Text Crawl.
+We further test CiT on our
+raw image-text crawl containing 1.2B unfiltered image-text
+pairs from the web (about 343M pairs have English text).
+
+Method
+Vision Encoder
+Vision Initialization
+w/ Labeled Data
+Total Time
+IN-1K Acc.
+OpenCLIP
+ViT-B/16
+from scratch
+
+n/a
+NaN loss
+LiT
+ViT-B/16
+MoCo-v3 [5]
+
+n/a
+NaN loss
+LiT (English filter)
+ViT-B/16
+MoCo-v3 [5]
+
+65 hrs
+56.7
+CiT
+ViT-B/16
+MoCo-v3 [5]
+
+39 hrs
+68.7
+CiT
+ViT-B/32
+AugReg [28]
+IN-21K
+69 hrs
+68.4
+CiT
+ViT-B/16
+AugReg [28]
+IN-21K
+72 hrs
+75.2
+CiT
+ViT-L/16
+AugReg [28]
+IN-21K
+105 hrs
+77.9
+CiT
+ViT-H/14
+SWAG [27]
+IG hashtags
+43 hrs
+77.4
+Table 6. CiT on Raw Image-Text Crawl: CiT is able to produce strong results when learning from raw image-text data. The raw data
+contains 1.2B image-text pairs. An English language filter, which reduces the data to 343M pairs, is required to stabilize LiT training.
+Time
+Food-101
+CIFAR10
+CIFAR100
+CUB
+SUN397
+Cars
+Aircraft
+DTD
+Pets
+Caltech-101
+Flowers
+MNIST
+FER-2013
+STL-10
+EuroSAT
+RESISC45
+GTSRB
+KITTI
+Country211
+PCAM
+UCF101
+Kinetics700
+CLEVR
+HatefulMemes
+SST2
+ImageNet
+Avg
+CLIP [20,21]
+27
+50.6 66.0 34.5 38.8 51.1 4.0
+5.4 21.2 28.5 60.9 53.3 8.4 17.3 90.5 30.2 21.5 6.1 35.1 10.5 53.5 28.5 22.1 10.8 52.4 50.7 37.6 34.2
+SLIP [20]
+41
+59.5 78.6 45.2 38.7 53.4 5.4
+5.7 26.1 31.1 71.0 56.6 9.8 19.6 94.4 20.3 28.9 14.5 34.0 11.6 55.4 37.7 26.9 17.5 52.8 51.1 42.8 38.0
+CiT-1K-meta
+5
+45.6 81.0 49.9 30.4 44.9 6.3
+8.3 26.8 80.0 71.2 25.1 7.3 26.0 95.2 19.1 14.3 6.9 22.2 6.2 54.1 34.7 24.7 13.4 50.7 50.1 61.2 38.5
+CiT-21K-meta
+15
+51.2 84.4 53.5 45.7 52.3 7.6
+9.0 31.6 69.2 73.8 56.1 10.6 24.5 95.7 30.1 23.4 7.9 28.5 9.2 51.0 39.5 28.7 15.0 49.3 49.1 57.4 40.6
+CiT-multi-meta
+11
+51.3 81.8 50.5 50.7 51.6 9.5 14.6 30.8 75.6 73.3 58.7 10.3 26.2 95.6 23.2 19.1 7.8 14.6 9.4 50.8 39.7 28.0 14.7 52.8 50.0 58.8 40.4
+CiT-sep.-meta
+7
+59.1 82.2 55.2 56.6 50.7 13.0 13.1 32.8 74.8 77.6 65.9 16.9 13.8 96.3 17.1 21.6 7.6 40.6 9.4 53.5 42.7 27.8 14.2 52.2 50.9 50.7 42.2
+Table 7. CiT on 26 zero-shot benchmarks when trained on YFCC15M. We vary metadata from IN-1K, IN-21K, combined (multi) as well
+as separate (sep.) across 26 tasks. All methods use ViT-B/16 and we use MoCo-v3 vision initialization. Larger encoders are in Table 12.
+The data contains a large degree of noise. Results are shown
+in Table 6. To understand the challenge of training on raw
+image-text pairs, we run CLIP and LiT training on the raw
+image-text pairs. This yields unstable training that quickly
+reaches NaN loss for both a CLIP and LiT training. We be-
+lieve some noisy pairs are unhealthy for training. By using
+our English filter to clean the text, we can train LiT and it
+reaches 56.7% IN-1K zero-shot accuracy. Training our CiT
+(without even using an English filter) achieves 68.7% which
+is +12.0% higher. This indicates raw and very noisy image-
+text pairs lead to poor accuracy, but CiT can overcome this
+and curate high-quality data for vision-language learning.
+Surprisingly,
+as shown in Table 5,
+CiT achieves
+much better performance than OpenCLIP trained on
+LAION400M. CiT on raw image-text reaches 77.9%, which
+is +5.1% better than OpenCLIP ViT-L/14 (cf. Table 5).
+Note that our source is raw, with multilingual texts, whereas
+LAION400M is a curated English-only dataset filtered by
+the CLIP model. The training data used by CiT (e.g. 131M
+for 77.9%) is just around 1/5 of the scale of LAION400M
+dataset (one epoch), showing the effectiveness of curating
+training data.
+4.8. Comparison across 26 benchmarks
+We extend CiT to 26 common 0-shot evaluation tasks for
+CLIP/SLIP models [20] on the public dataset YFCC15M.
+We provide more comparisons with further encoders as well
+as pre-training on LAION400M in the appendix. We evalu-
+ate with prompts from CLIP/SLIP. For ImageNet, we drop
+the extra prompts used by LiT for a fair comparison with the
+baselines. We use three setups of metadata: (i) IN-1K, (ii)
+IN-21K, and (iii) multi-task CiT that combines class names
+from all 26 tasks (iv) we run every task separately on a sin-
+gle GPU as a low-compute setup (this trains a model for
+each task with separate metadata). Results are in Table 7
+and discussed next.
+We first evaluate CiT trained with IN-1K metadata on all
+26 tasks. As expected accuracy on ImageNet and Pets is
+highest among the metadata variants (i-iv). Overall, we ob-
+serve that CiT 1K meta already exhibits certain generality
+to all tasks and can outperform CLIP (34.2 vs. 38.5%) and
+is similar to SLIP, but 8.2× faster (5 vs. 41 hours), demon-
+strating its efficiency.
+Next, we explore the WordNet lemma from ImageNet-
+21K as a relatively general metadata for training CiT. In
+Table 7, CiT-21K-meta improves broadly over IN-1K lead-
+ing to 40.6% average accuracy, showing that a more general
+taxonomy works well across tasks.
+We combine the taxonomies from all 26 tasks in CiT-
+multi-meta. This allows us to curate training data for all 26
+tasks at again almost no extra training cost. We notice that
+multi-task CiT is on average similarly accurate as IN-21K
+metadata (40.4% vs. 40.6%) and converges faster because
+CiT is more targeted towards tasks of interest.
+Finally, we compare a setup that trains a model for
+each task with separate metadata.
+CiT-sep.-meta in Ta-
+ble 7 achieves overall the best average accuracy of 42.2%
+across tasks. This setup uses a restricted 1-GPU setting
+to save compute and could be boosted further with longer
+training. We think that this scenario might be quite practi-
+cal, where some domain data exists (e.g. on bird images in
+CUB) and one wants to build a classification system given
+a large amount of noisy image-text data from the web.
+
+Step (c)
+Text
+ImageNet Class
+Cosine Sim.
+0
+title: “Wollaston Beach”
+beach
+0.739
+100
+title: “tn_kif_3128”
+Vizsla
+0.779
+1000
+tag: “beach plumisland parker river national wildlife refuge newburyport massachusetts ocean”
+beach
+0.716
+2000
+desc: “These guys were nice, told me all about this and other planes of the show, but unfortunately...”
+military aircraft
+0.725
+3000
+title: “Turtle”
+terrapin
+0.725
+4000
+desc: “One of the fountains close by the south west entrance to the park”
+fountain
+0.734
+5000
+title: “butterfly”
+Papillon
+0.735
+5000
+tag: “ash;explosion;sakurajima;kagoshima;桜島;鹿児島県;volcano;tarumizu;垂水市;japan;eruption;日本”
+volcano
+0.645
+Table 8. Samples of curated text over training steps (c) from YFCC100M. CiT uses MoCo-v3 initialized vision encoder.
+4.9. Further Analysis
+Samples of Curated Data. We further investigate samples
+curated by CiT on YFCC100M dataset in Table 8. We show
+training steps, a sample text, the related ImageNet meta-
+data, as well as the cosine similarity in CiT’s data proxy.
+At step c = 0 CiT’s data proxy tends to select text with
+similar length as class names and string-matching behavior;
+the short-term run of CiT (e.g. c = 100) has some match-
+ing issues with many false positives. Later on, CiT starts to
+select texts of various lengths with similar semantics as the
+metadata. We do not observe any clearly less useful sam-
+ples such as file names after c = 2000. Interestingly, CiT
+can even use the English part of mixed language texts from
+YFCC100M (as in the last example).
+Speed/accuracy trade-off.
+In Figure 2, we show the
+speed/accuracy tradeoff of CiT vs. LiT [38], corresponding
+to results in Table 4). We see that CiT achieves a win-win
+scenario compared to LiT on identical AugReg ViT-B/32 vi-
+sion encoders: a +3.4% higher accuracy on ImageNet, and
+a 5× faster total training time (including the curation time).
+on data YFCC15M [21].
+0
+10
+20
+30
+40
+50
+60
+Total Training Time (hours)
+25
+30
+35
+40
+45
+50
+55
+60
+65
+ImageNet Acc.
+CiT
+LiT
+Figure 2. CiT on provides >5× speedup and +3.4% accuracy gain
+over LiT [38] on AugReg ViT-B/32 vision encoders. Training data
+is YFCC15M. Models are evaluated at 6 evenly sampled iterations.
+0
+1000
+2000
+3000
+4000
+Step
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Ratio of Curation for Training
+t=0.5
+t=0.55
+t=0.6
+Figure 3.
+Ratio of curation under different thresholds t.
+CiT
+broadly uses data first and curates more towards end of training.
+Ratio of Curation. We are interested in the training dy-
+namics of CiT. We use different curation thresholds t and
+inspect the amount of curated training data. In Figure 3, we
+see that the ratio of curation which corresponds to the frac-
+tion of used training samples from the raw data source, see
+§3.3, keeps changing over steps for curation/training. Ini-
+tially, CiT uses more data, e.g. for a threshold of t = 0.5,
+it peaks at about 75%. In this phase, the latent space of the
+text encoder is less aligned with the vision latents. Later on
+during training, CiT starts to produce embeddings that bet-
+ter represent the downstream task, producing a lower ratio.
+5. Conclusion
+This paper contributes CiT, a novel learning algorithm
+for efficient pre-training from noisy image-text data. CiT
+incorporates a curation process into learning to pull the
+training data distribution closer to downstream tasks. Our
+experiments demonstrate both significant accuracy and
+training time improvements when learning from either pub-
+lic or our own uncurated data from the web. We observe
+that training on the raw image-text pairs in YFCC can
+achieve better accuracy over the cleaned version from a
+hand-crafted filter pipeline. Further, we show that CiT can
+train with raw image-text pairs crawled from the web, which
+would lead to instability for vanilla pre-training objectives.
+Acknowledgement. We thank Norman Mu, Shang-Wen Li,
+Vasu Sharma, Wojciech Galuba and Max Bain for help.
+
+A. Appendix
+In this appendix, §A.1 contains implementation details
+and §A.2 contains further results as well as ablations.
+A.1. Implementation Details
+A.1.1
+PyTorch Pseudo Code
+To facilitate implementation of CiT, we provide the PyTorch
+pseudo-code in Algorithm 4 below.
+Algorithm 4: CiT: PyTorch Pseudo Code
+1
+# b: maximum training steps as budget.
+2
+# d: raw data loader.
+3
+# t_meta: textual metadata.
+4
+# bsz: batch_size.
+5
+# t: threshold.
+6
+# gamma: target radio for curation.
+7
+# s: number of expected pairs.
+8
+9
+c = 0
+10
+while c < b:
+11
+if c % int(s//bsz) == 0:
+12
+x_meta = model(t_meta)
+13
+x_meta = normalize(x_meta)
+14
+d_c = []
+15
+while len(d_c) < s:
+16
+x_imgs, x_txts = next(d)
+17
+x_txts = model(x_txts)
+18
+x_txts = normalize(x_txts)
+19
+v = x_txts @ x_meta.t()
+20
+sel = max(v) > t
+21
+b_ratio = sum(sel) / len(sel)
+22
+if b_ratio < gamma:
+23
+sel = max(v).topk(
+24
+k=int(bsz*gamma), dim=0)
+25
+d_c.extend((x_imgs[sel], x_txts[sel]))
+26
+27
+for (x_imgs, x_txts) in batchify(d_c):
+28
+x_imgs, x_txts = model(x_imgs, x_txts)
+29
+x_imgs, x_txts = normalize(x_imgs,x_txts)
+30
+# scale: learnable log logit scale
+31
+l = exp(scale) * x_imgs @ x_txts.t()
+32
+labels = arange(bsz)
+33
+loss = cross_entropy(l, labels)
+34
+loss.backward()
+35
+c += 1
+A.1.2
+Dataloader Implementation
+For efficiency, we only load text during the curation loop
+and the training loop uses the curated indices to reload the
+full image-text pairs. Our implementation also supports in-
+memory storage of curated image-text pairs in case the data
+source is not randomly accessible for (re-)loading curated
+data, where all s pairs of training data can be stored in
+the CPU memory with image tensors represented as uint8
+data. We use a larger batch size for curation (compared to
+training) to speed up CiT.
+Hyperparameter
+Value
+Optimizer
+AdamW
+Optimizer momentum
+β1 = 0.9, β2 = 0.999
+Optimizer ϵ
+1e-8
+Weight Decay (proj.)
+1.0
+Weight Decay (other)
+0.2
+Base Learning Rate
+5e-4
+Learning Rate Schedule
+cosine decay
+Minimum Learning Rate
+1e-5
+Gradient Clipping
+None
+Warm-up % of Train Steps
+4%
+Batch size
+16,384
+GPUs
+16 Nvidia V100 32GB GPUs
+precision
+float16
+Max BERT len.
+32
+Train Aug.
+RandomResizedCrop(224, scale=(0.5, 1.0))
+YFCC15M/YFCC100M Aug.
+shuffle/join tags [38]
+Eval Aug.
+Resize(256), CenterCrop(224)
+AugReg rgb Mean
+(0.5, 0.5, 0.5)
+AugReg rgb Std.
+(0.5, 0.5, 0.5)
+Other encoder rgb Mean
+(0.485, 0.456, 0.406)
+Other encoder rgb Std.
+(0.229, 0.224, 0.225)
+Table 9. Hyperparameters of CiT Training.
+Data Source
+Metadata
+b
+t
+γ
+YFCC15M
+IN-1K
+5K
+0.55 0.003
+YFCC15M
+IN-21K
+8K
+0.55 0.003
+YFCC15M
+multi.
+8K
+0.55 0.003
+YFCC100M
+IN-1K
+5K
+0.7
+0.01
+LAION400M
+IN-1K
+5K
+0.6
+0.01
+LAION400M
+IN-21K
+30K 0.65
+0.01
+LAION400M
+multi.
+16K
+0.6
+0.01
+RAW IMG-TXT IN-1K
+8K
+0.7
+0.003
+RAW IMG-TXT IN-21K
+60K 0.75 0.003
+RAW IMG-TXT multi.
+30K
+0.7
+0.003
+Table 10. Hyperparameters of CiT Curation.
+A.1.3
+Detailed Implementation Settings
+The hyper-parameters of CiT training are shown in Table 9.
+We mostly follow [20, 21, 38]. CiT is trained on 16 GPUs
+with a global batch size of 16,384 (1024 per GPU).
+Hyperparameters for CiT curation outlined in §3 of the
+main paper are shown in Table 10. We use different thresh-
+olds t and minimal ratios γ for each dataset/metadata com-
+bination to fit the training into a budget b shown in the table
+as well. We use the same values for all variants of vision en-
+coders. Due to smaller size, we use a lower t for YFCC15M
+and CC12M, whereas for YFCC100M and Raw Img-Text
+Crawl we use a higher t to focus on high-quality data from
+the raw data source, in order to roughly meet the budget b.
+Single GPU Setting.
+We provide more details on the im-
+plementation of the extremely efficient single GPU setup
+used for zero-shot evaluation on multiple tasks in Table 12.
+We can fit a batch size of 1,536 into a single 32GB V100
+GPU and train for b = 5000 steps. To ensure the training
+can be finished quickly, we set γ = 0.05. Further to re-
+duce the chance of using the minimal ratio during curation,
+
+we perform a pre-curation on YFCC15M for each task us-
+ing BERT-SimCSE with a threshold of 0.45 to remove pairs
+with low relevance.
+A.1.4
+Implementation Differences from LiT
+While we aim for a close reproduction of LiT [38], there are
+a few tricks that our implementation does not incorporate
+and we suspect the differences on our LiT reproduction on
+YFCC stem from those. Below we list some tricks known
+to us, but there could be more differences we are not aware
+of since we have no access to LiT’s full preprocessing and
+training code.
+Preprocessing. For the captions, LiT performs extra filter-
+ing and removes titles that start with “DSC”, “IMG”, “Pic-
+ture”. Also, LiT removes text consisting of only the word
+“image” or text that contains a large fraction of digits.
+Joint Contrastive Loss. LiT adopts a joint contrastive loss
+over 3 text fields in YFCC15M and shows the gain in Figure
+8 of the LiT paper [38]. Since this technique is specific to
+the type of captions in the specific YFCC data, we remove
+it from our implementation and randomly sample one of the
+three text fields to pair with a training image.
+Text encoder.
+LiT adopts various text encoders such
+as BERTbase and BERTlarge. This work consistently uses
+BERTbase for all main results to have a fair comparison.
+A.2. Additional Results
+This section extends the results of CiT in the main pa-
+per to full results across 26 CLIP/SLIP benchmarks on
+YFCC15M and LAION400M and an extra ablation study.
+A.2.1
+Full Results on YFCC15M
+We show the full results of Table 7 above in Table 12
+below.
+On average, CiT-multi-meta (52.6) is slightly
+better than CiT-21K-meta (51.7), which is better than
+CiT-sep-meta and CiT-1K-meta (47.2).
+It appears that
+the broader ImageNet-21K wordnet taxonomy works well
+across datasets, and combining metadata from all down-
+stream tasks is only slightly better than that. We note that
+training on the larger metadata does not introduce much
+extra curation compute since forwarding the raw exam-
+ples takes the majority of computation. Nevertheless, we
+observe that larger metadata takes longer to converge and
+therefore increase the training budget to b = 8000 for CiT-
+21K-meta and CiT-multi-meta. We expect larger budgets
+will lead to even better results.
+Besides what was already discussed in the main paper,
+we observe that CiT performs even better on larger mod-
+els or models trained with supervised (AugReg IN-21K) or
+weakly supervised (SWAG) data than the unsupervisedly
+Eval. Prompts
+Acc
+CLIP+LiT prompts
+61.4
+CLIP prompts only
+61.2
+(a) Evaluation Prompts
+Objective
+Acc
+img2txt obj.
+61.4
+CLIP obj.
+61.2
+(b) Training Objective
+Table 11.
+Additional ablation experiments.
+We use the de-
+fault setup (MoCo-v3 / BERTbase-SimCSE) and YFCC15M as data
+source and report IN-1K Accuracy.
+pre-trained MoCo-v3 on IN-1K. Out-of-domain issues (e.g.
+MNIST) are present even for larger vision encoders.
+A.2.2
+Full Results on LAION400M
+In Table 13, we show the result of CiT trained on
+LAION400M and evaluated on 26 CLIP/SLIP benchmarks.
+With a larger data source, we realize CiT takes more time
+to converge especially with more metadata, which can be
+attributed to more data meeting the curation criteria. We set
+b = 16000 for CiT-multi-meta and b = 30000 for CiT-21K-
+meta. The trend is similar to YFCC15M but with better
+performance aross the benchmarks. Similar as in Table 12,
+CiT-multi-meta is better than CiT-21K-meta, but this time
+the gap is larger. In addition to the longer training, we be-
+lieve that the combined metadata from 26 benchmarks are
+more effective on larger pre-training data.
+A.2.3
+Full Results on Raw Image-Text Crawl
+In Table 14, we show the result of CiT trained on our raw
+image-text crawl and evaluated on 26 benchmarks. With a
+larger raw data source, we realize CiT takes more time to
+converge. We set b = 30000 for CiT-multi-meta and b =
+60000 for CiT-21K-meta. The trend is similar to LAION-
+400M but raw Image-Text Crawl is not cleaned for vision-
+language association. Similar as in Table 13, CiT-multi-
+meta is better than CiT-21K-meta, but the gap is larger. We
+expect better accuracy for longer training.
+A.2.4
+Additional Ablations
+This section extends ablations in Table 1 of the main paper
+to (i) evaluation prompts and (ii) training objectives.
+Evaluation Prompts. We first verify the effects of LiT’s
+extra prompts on CiT in Table 11a. We obtain a +0.2% gain
+by adding them to the CLIP prompts.
+Training Objective. We ablate the Limg2txt training objec-
+tive which our approach uses (see §3.2 of the main paper).
+In Table 11a we see that this variant provides a +0.2% gain
+over CLIP’s objective that also incorporates a text2img loss.
+
+Vis. Encoder Init.
+Hrs
+Food-101
+CIFAR10
+CIFAR100
+CUB
+SUN397
+Cars
+Aircraft
+DTD
+Pets
+Caltech-101
+Flowers
+MNIST
+FER-2013
+STL-10
+EuroSAT
+RESISC45
+GTSRB
+KITTI
+Country211
+PCAM
+UCF101
+Kinetics700
+CLEVR
+HatefulMemes
+SST2
+ImageNet
+Avg
+CLIP [20,21]
+ViT-B/16
+scratch
+27
+50.6 66.0 34.5 38.8 51.1 4.0
+5.4 21.2 28.5 60.9 53.3 8.4 17.3 90.5 30.2 21.5 6.1 35.1 10.5 53.5 28.5 22.1 10.8 52.4 50.7 37.6 34.2
+ViT-L/16
+scratch
+189 59.5 72.9 41.5 40.3 53.6 6.9
+6.4 20.6 27.9 65.4 55.0 10.3 34.5 94.2 22.7 28.8 5.8 41.4 12.6 54.9 34.3 24.0 12.9 54.3 50.1 40.4 37.4
+SLIP [20]
+ViT-B/16
+scratch
+41
+59.5 78.6 45.2 38.7 53.4 5.4
+5.7 26.1 31.1 71.0 56.6 9.8 19.6 94.4 20.3 28.9 14.5 34.0 11.6 55.4 37.7 26.9 17.5 52.8 51.1 42.8 38.0
+ViT-L/16
+scratch
+284 64.4 87.8 56.4 39.8 58.9 8.6
+7.8 26.8 32.0 76.6 59.4 13.2 36.0 96.6 27.7 36.5 7.2 28.8 15.6 54.4 42.6 30.0 14.1 53.4 50.1 46.2 41.2
+CiT-1K-meta
+ViT-B/16
+MoCo-v3
+5
+45.6 81.0 49.9 30.4 44.9 6.3
+8.3 26.8 80.0 71.2 25.1 7.3 26.0 95.2 19.1 14.3 6.9 22.2 6.2 54.1 34.7 24.7 13.4 50.7 50.1 61.2 38.5
+ViT-B/16
+AugReg
+8
+57.9 92.3 74.2 36.9 52.5 7.7
+5.6 25.2 77.9 84.5 38.8 8.3 31.2 94.4 16.6 24.3 6.5 17.2 6.4 59.1 47.8 32.2 13.3 52.0 50.1 68.9 41.6
+ViT-L/16
+AugReg
+8
+60.0 93.6 77.8 36.3 54.0 9.0
+5.7 25.6 79.8 87.3 45.2 9.7 29.2 96.1 20.9 32.8 7.0 36.0 7.6 52.8 51.5 35.2 12.6 53.0 49.7 71.6 43.8
+ViT-H/14
+SWAG
+11
+79.0 91.6 68.1 35.3 56.9 26.2 12.5 30.0 88.8 86.4 47.6 8.1 31.3 97.8 27.6 46.4 7.3 34.2 14.5 50.3 54.7 43.8 12.3 51.8 51.0 73.3 47.2
+CiT-21K-meta
+ViT-B/16
+MoCo-v3 15
+51.2 84.4 53.5 45.7 52.3 7.6
+9.0 31.6 69.2 73.8 56.1 10.6 24.5 95.7 30.1 23.4 7.9 28.5 9.2 51.0 39.5 28.7 15.0 49.3 49.1 57.4 40.6
+ViT-B/16
+AugReg
+23
+75.3 93.8 75.7 57.8 59.8 9.7 10.1 35.4 68.3 87.9 74.3 12.1 27.4 97.1 30.8 30.6 7.3 24.3 9.9 50.5 54.7 37.4 13.6 53.8 50.1 63.7 46.6
+ViT-L/16
+AugReg
+29
+78.9 95.1 78.6 60.5 61.9 11.6 10.9 35.1 74.2 90.5 75.4 14.8 34.8 98.0 24.7 35.5 7.5 25.7 10.9 50.8 57.4 40.7 14.8 49.9 48.7 67.7 48.3
+ViT-H/14
+SWAG
+39
+92.2 92.9 70.9 59.0 64.7 36.9 14.9 40.3 87.7 90.9 77.4 10.1 32.7 99.1 38.8 53.2 9.3 15.9 20.5 50.7 62.2 49.4 12.9 46.8 44.2 71.4 51.7
+CiT-multi-meta
+ViT-B/16
+MoCo-v3 11
+51.3 81.8 50.5 50.7 51.6 9.5 14.6 30.8 75.6 73.3 58.7 10.3 26.2 95.6 23.2 19.1 7.8 14.6 9.4 50.8 39.7 28.0 14.7 52.8 50.0 58.8 40.4
+ViT-B/16
+AugReg
+11
+77.8 94.0 76.5 63.9 60.1 10.3 13.1 35.2 79.0 88.9 79.4 12.2 33.0 96.2 31.6 29.3 10.2 17.4 9.6 50.8 56.0 38.0 12.5 55.8 47.8 67.0 47.9
+ViT-L/16
+AugReg
+16
+80.4 95.3 79.4 65.6 61.9 13.3 11.3 35.1 79.9 90.6 80.1 10.7 37.8 97.4 29.3 35.0 7.8 13.8 10.7 49.7 59.5 41.3 13.0 54.5 47.9 70.5 48.9
+ViT-H/14
+SWAG
+31
+91.8 90.7 71.3 65.6 62.4 47.9 19.7 40.8 91.7 91.3 81.2 10.7 37.5 98.0 23.9 46.4 11.0 12.4 20.2 51.3 64.3 50.2 13.5 54.6 47.1 73.4 52.6
+CiT-sep.-meta (single GPU)
+ViT-B/16
+MoCo-v3
+4
+59.1 82.2 55.2 56.6 50.7 13.0 13.1 32.8 74.8 77.6 65.9 16.9 13.8 96.3 17.1 21.6 7.6 40.6 9.4 53.5 42.7 27.8 14.2 52.2 50.9 50.7 42.2
+ViT-B/16
+AugReg
+5
+79.1 94.4 75.2 73.8 60.6 19.4 17.4 36.6 78.1 88.0 79.8 12.4 39.2 97.0 31.1 29.1 11.1 30.1 9.9 51.9 54.9 37.1 19.2 52.5 50.0 56.8 49.4
+ViT-L/16
+AugReg
+7
+83.8 94.8 79.6 76.9 60.4 19.6 17.2 36.0 77.8 89.6 82.2 12.1 39.0 96.7 24.8 31.2 9.7 26.9 10.7 57.6 59.1 39.9 14.9 46.8 51.2 60.1 49.9
+ViT-H/14
+SWAG
+11
+92.1 89.9 71.8 71.3 65.4 52.0 20.9 38.7 90.6 90.4 84.8 15.1 30.6 92.8 26.8 47.1 13.4 34.8 20.8 59.4 65.8 50.1 14.0 48.5 51.7 67.0 54.1
+Table 12. CiT trained on YFCC15M and evaluated on 26 CLIP/SLIP benchmarks: we vary metadata on IN-1K, IN-21K and combined class
+names on 26 tasks (CiT-multi-meta) with a single training and run 26 separate training on each task with a single GPU (CiT-sep.-meta).
+Vis. Encoder Init.
+Hrs
+Food-101
+CIFAR10
+CIFAR100
+CUB
+SUN397
+Cars
+Aircraft
+DTD
+Pets
+Caltech-101
+Flowers
+MNIST
+FER-2013
+STL-10
+EuroSAT
+RESISC45
+GTSRB
+KITTI
+Country211
+PCAM
+UCF101
+Kinetics700
+CLEVR
+HatefulMemes
+SST2
+ImageNet
+Avg
+CLIP (WIT400M) [21]
+ViT-B/32
+scratch
+458
+84.4 91.3 65.1 37.8 63.2 59.4 21.2 44.5 87.0 87.9 66.7 51.9 47.3 97.2 49.4 60.3 32.2 39.4 17.8 58.4 64.5 47.8 24.8 57.6 59.6 63.2 56.9
+ViT-B/16
+scratch
+981
+89.2 91.6 68.7 39.1 65.2 65.6 27.1 46.0 88.9 89.3 70.4 56.0 52.7 98.2 54.1 65.5 43.3 44.0 23.3 48.1 69.8 52.4 23.4 61.7 59.8 68.6 60.1
+ViT-L/14
+scratch
+6803 92.9 96.2 77.9 48.3 67.7 77.3 36.1 55.3 93.5 92.6 78.7 87.2 57.5 99.3 59.9 71.6 50.3 23.1 32.7 58.8 76.2 60.3 24.3 63.3 64.0 75.3 66.2
+OpenCLIP *
+ViT-B-32
+scratch
+458
+n/a
+90.8 70.2
+n/a
+67.0 79.2 16.8 54.3 86.8 83.3 68.3 37.4 42.7 95.5 51.6
+n/a
+42.0 28.8 14.7 54.6
+n/a
+n/a
+16.3
+n/a
+52.6 62.9
+n/a
+ViT-B-16
+scratch
+981
+n/a
+91.7 71.0
+n/a
+69.6 83.7 17.5 51.3 89.2 83.5 69.3 66.6 42.9 97.0 50.3
+n/a
+43.5 19.0 18.1 60.5
+n/a
+n/a
+28.8
+n/a
+54.7 67.0
+n/a
+ViT-L-14
+scratch
+6803
+n/a
+94.7 77.4
+n/a
+72.6 89.6 25.1 60.3 91.9 84.2 75.4 76.4 50.1 98.0 61.8
+n/a
+50.0 20.8 23.1 48.6
+n/a
+n/a
+24.2
+n/a
+56.3 72.7
+n/a
+CiT-1K-meta
+ViT-B/16
+MoCo-v3
+26
+31.2 80.7 56.7 29.5 41.7 12.6 3.9 35.2 85.9 82.3 19.1 16.3 25.0 89.7 20.0 19.7 14.5 42.2 3.7 55.3 34.8 23.0 14.4 49.5 49.3 67.0 38.6
+ViT-B/32
+AugReg
+62
+45.0 86.6 68.8 34.5 48.1 12.1 3.8 35.3 87.0 87.6 34.5 10.2 29.2 89.8 19.7 23.0 10.5 33.1 4.4 50.6 45.5 27.7 15.2 48.5 50.4 67.5 41.1
+ViT-B/16
+AugReg
+63
+45.4 87.8 70.9 33.7 50.8 12.4 3.3 38.0 86.2 89.0 31.5 9.7 26.4 90.0 25.3 25.3 13.2 34.9 5.2 54.7 50.0 31.5 14.7 50.4 49.3 73.0 42.4
+ViT-L/16
+AugReg
+27
+45.3 90.6 76.3 36.3 54.7 13.6 5.0 35.9 87.2 92.1 32.0 10.2 20.0 91.3 28.2 31.2 10.6 21.4 5.5 51.7 50.9 33.6 16.1 48.9 50.1 75.7 42.9
+ViT-H/14
+SWAG
+26
+65.4 89.8 68.7 36.4 56.5 38.0 7.9 41.7 89.4 88.5 41.4 10.2 30.5 94.3 34.6 41.5 12.0 19.1 12.3 49.5 57.0 42.6 13.2 51.5 46.5 76.2 46.7
+CiT-21K-meta
+ViT-B/16
+MoCo-v3
+70
+64.8 85.0 63.1 59.5 56.3 26.2 8.1 40.2 87.6 87.1 60.6 17.8 34.5 95.9 29.4 30.3 10.9 33.0 6.4 54.5 48.8 31.2 15.1 47.9 50.1 64.1 46.5
+ViT-B/32
+AugReg
+57
+71.7 91.1 72.8 62.4 59.0 18.8 5.9 42.6 81.8 89.8 67.5 16.3 38.8 96.3 27.1 32.8 12.4 33.9 6.4 52.8 56.8 35.9 16.4 51.0 50.1 65.0 48.3
+ViT-B/16
+AugReg
+72
+77.1 92.8 74.7 68.9 61.9 20.6 8.3 41.5 85.7 91.2 73.8 21.7 38.3 97.0 26.2 36.4 15.1 41.8 7.1 52.4 56.8 38.3 12.1 51.0 50.5 71.2 50.5
+ViT-L/16
+AugReg
+97
+77.5 93.5 79.1 67.6 62.9 19.5 8.3 44.8 84.4 93.1 71.5 18.9 34.2 98.0 29.6 38.9 11.7 22.9 7.7 50.9 60.3 41.6 14.8 51.5 48.2 73.9 50.2
+ViT-H/14
+SWAG
+135
+89.2 91.5 72.1 68.2 64.0 36.9 10.4 43.9 88.2 92.1 75.8 7.1 41.7 97.4 29.2 49.6 10.7 34.6 15.0 50.9 62.6 46.4 13.2 52.3 49.7 76.1 52.6
+CiT-multi-meta
+ViT-B/16
+MoCo-v3
+31
+68.1 84.3 62.0 63.7 56.9 65.7 16.0 40.3 90.0 87.8 61.1 6.8 26.6 92.1 27.6 35.9 18.0 38.6 7.2 50.9 56.0 35.2 17.2 46.0 49.7 65.8 48.8
+ViT-B/32
+AugReg
+32
+75.2 90.0 72.2 70.9 60.2 43.9 11.8 42.8 86.6 90.2 74.6 29.2 21.6 93.0 31.7 33.3 13.5 44.7 6.9 51.1 61.7 38.7 14.9 49.9 50.1 66.2 51.0
+ViT-B/16
+AugReg
+51
+80.2 91.5 74.4 75.1 62.3 53.7 15.5 40.1 87.2 90.8 76.3 12.3 31.2 92.4 28.1 38.3 13.2 18.6 7.8 60.5 66.0 42.5 14.0 50.3 50.0 71.7 51.7
+ViT-L/16
+AugReg
+61
+81.6 92.7 79.2 72.3 63.8 56.9 15.7 42.6 88.5 92.9 73.9 22.6 33.3 94.1 30.9 38.4 16.9 27.7 8.7 56.7 68.4 45.5 16.4 50.0 48.3 74.8 53.6
+ViT-H/14
+SWAG
+54
+92.1 91.0 71.8 71.7 66.3 77.4 18.7 51.3 93.8 92.2 81.5 14.9 39.6 97.5 39.4 50.0 15.0 19.1 17.8 50.9 71.8 52.4 14.7 51.7 51.1 76.5 56.5
+Table 13. CiT trained on LAION400M and evaluated on 26 CLIP benchmarks: We vary metadata from IN-1K (CiT-1K-meta), IN-21K
+(CiT-21K-meta) and combined class names from 26 benchmarks (CiT-multi.-meta). We also list results from CLIP on WIT400M and
+OpenCLIP trained on LAION400M. *: from https://github.com/LAION-AI/CLIP_benchmark, with some results using
+VTAB benchmark evaluation/prompts.
+
+Vis. Encoder Init.
+Hrs
+Food-101
+CIFAR10
+CIFAR100
+CUB
+SUN397
+Cars
+Aircraft
+DTD
+Pets
+Caltech-101
+Flowers
+MNIST
+FER-2013
+STL-10
+EuroSAT
+RESISC45
+GTSRB
+KITTI
+Country211
+PCAM
+UCF101
+Kinetics700
+CLEVR
+HatefulMemes
+SST2
+ImageNet
+Avg
+CLIP (WIT400M) [21]
+ViT-B/32
+scratch
+458
+84.4 91.3 65.1 37.8 63.2 59.4 21.2 44.5 87.0 87.9 66.7 51.9 47.3 97.2 49.4 60.3 32.2 39.4 17.8 58.4 64.5 47.8 24.8 57.6 59.6 63.2 56.9
+ViT-B/16
+scratch
+981
+89.2 91.6 68.7 39.1 65.2 65.6 27.1 46.0 88.9 89.3 70.4 56.0 52.7 98.2 54.1 65.5 43.3 44.0 23.3 48.1 69.8 52.4 23.4 61.7 59.8 68.6 60.1
+ViT-L/14
+scratch
+6803 92.9 96.2 77.9 48.3 67.7 77.3 36.1 55.3 93.5 92.6 78.7 87.2 57.5 99.3 59.9 71.6 50.3 23.1 32.7 58.8 76.2 60.3 24.3 63.3 64.0 75.3 66.2
+CiT-1K-meta
+ViT-B/16
+MoCo-v3
+39
+29.0 86.0 56.5 17.6 41.3 12.4 5.8 25.7 83.8 77.0 10.6 10.8 24.9 95.1 22.3 20.8 6.8 35.6 4.2 50.8 27.7 20.5 17.2 48.9 50.1 68.4 36.5
+ViT-B/32
+AugReg
+69
+42.8 92.2 70.5 22.1 49.0 11.4 5.5 27.0 83.8 81.1 16.5 8.2 32.5 94.3 29.4 22.2 8.5 39.1 4.9 51.3 37.6 26.7 16.4 48.0 50.1 67.8 40.0
+ViT-B/16
+AugReg
+72
+43.9 92.1 73.4 20.4 50.0 10.9 4.5 31.3 84.6 83.0 18.8 7.1 21.5 96.2 23.3 22.4 11.2 29.4 5.2 52.3 41.9 29.4 17.0 50.6 50.1 74.9 40.2
+ViT-L/16
+AugReg
+105
+47.8 95.4 76.0 18.5 49.4 11.4 5.6 30.9 84.7 83.7 22.4 6.4 25.6 96.8 24.7 29.7 8.9 36.3 5.3 50.9 45.9 31.0 16.3 46.5 50.1 77.5 41.4
+ViT-H/14
+SWAG
+43
+57.2 93.2 68.5 19.8 47.2 25.6 5.9 32.4 81.3 82.5 25.3 8.2 28.8 97.4 17.6 42.2 8.1 29.2 10.3 50.9 53.7 38.8 14.5 48.0 53.2 77.1 43.0
+CiT-21K-meta
+ViT-B/16
+MoCo-v3 134
+57.1 87.1 60.3 57.1 54.0 10.5 6.0 37.0 84.6 82.8 59.9 9.8 26.8 96.8 31.8 30.8 8.3 41.2 7.4 59.9 37.9 25.9 20.8 48.2 50.1 62.8 44.4
+ViT-B/32
+AugReg
+148
+64.4 93.2 71.7 49.5 56.8 10.8 5.7 35.4 76.2 85.8 60.9 9.5 29.1 95.4 27.1 25.2 9.3 39.8 7.7 51.3 45.8 32.1 14.1 51.3 50.1 62.2 44.6
+ViT-B/16
+AugReg
+161
+70.0 93.6 75.9 58.2 59.9 11.7 5.2 37.7 74.9 89.3 61.7 9.8 32.6 97.9 29.5 29.4 11.2 40.9 9.0 51.1 49.6 36.1 13.6 48.9 50.1 69.4 46.8
+ViT-L/16
+AugReg
+228
+71.7 96.0 78.7 56.7 62.4 12.2 5.9 37.4 77.0 90.6 65.3 14.6 37.6 98.3 27.8 34.0 8.5 34.0 9.4 44.2 54.7 39.0 15.5 47.9 50.1 72.6 47.8
+ViT-H/14
+SWAG
+310
+80.4 93.2 72.0 58.4 60.8 25.6 5.5 36.0 78.4 89.1 70.6 7.8 34.7 98.9 28.4 41.7 10.8 29.9 14.0 50.8 57.5 41.9 12.4 45.8 52.6 75.5 49.0
+CiT-multi-meta
+ViT-B/16
+MoCo-v3
+91
+70.4 88.8 61.1 60.1 59.0 63.2 24.5 38.4 90.2 85.5 66.5 9.8 32.0 96.6 35.4 39.0 9.5 35.8 10.2 50.3 48.7 33.4 17.1 43.8 50.1 66.1 49.4
+ViT-B/32
+AugReg
+62
+72.7 92.9 71.0 51.0 58.9 30.9 10.9 36.3 86.6 87.4 67.5 9.8 36.3 94.5 29.1 29.4 8.5 33.4 8.6 54.9 51.6 36.3 14.8 49.2 50.0 64.4 47.6
+ViT-B/16
+AugReg
+62
+81.3 94.0 76.6 65.2 62.2 44.1 17.9 41.3 90.0 90.6 74.9 9.8 35.3 97.5 34.6 36.5 13.1 34.4 10.4 56.8 57.6 41.3 13.4 50.6 50.1 71.9 52.0
+ViT-L/16
+AugReg
+62
+82.4 96.1 79.2 62.4 64.1 44.5 15.8 41.2 89.3 91.3 74.9 9.8 34.7 98.2 27.9 38.7 8.9 33.4 11.1 55.9 61.3 44.0 11.9 48.9 50.1 74.4 51.9
+ViT-H/14
+SWAG
+203
+93.7 93.5 73.2 75.7 65.1 79.5 25.2 40.3 95.8 92.1 85.0 11.6 38.9 98.3 30.5 51.9 10.1 28.7 21.8 52.5 68.9 52.9 15.9 45.7 50.1 77.6 56.7
+Table 14. CiT trained on Raw Image-Text Crawl and evaluated on 26 CLIP benchmarks: We vary metadata from IN-1K (CiT-1K-meta),
+IN-21K (CiT-21K-meta) and combined class names from 26 benchmarks (CiT-multi.-meta). The budget b = 60000 for IN-21K and
+b = 30000 for combined class names. We also list results from CLIP on WIT400M.
+YFCC15M
+Food-101
+CIFAR10
+CIFAR100
+CUB
+SUN397
+Cars
+Aircraft
+DTD
+Pets
+Caltech-101
+Flowers
+MNIST
+FER-2013
+STL-10
+EuroSAT
+RESISC45
+GTSRB
+KITTI
+Country211
+PCAM
+UCF101
+Kinetics700
+CLEVR
+HatefulMemes
+SST2
+ImageNet
+# of classes
+101
+10
+100 200 397 196 100
+47
+37
+102 102
+10
+7
+10
+10
+45
+43
+4
+211
+2
+101 700
+8
+2
+2
+1000
+t > 0.55
+# pairs per class (k) 5.32 16.64 9.27 3.28 5.63 0.81 4.35 3.12
+3.66 6.71 6.51 11.9 5.43 18.21 8.59 10.57 2.22 23.63 2.33 0.53 4.28 3.9 20.96 5.48
+0.66
+3.69
+total keep rates (%) 3.66 1.13 6.31 4.46 15.2 1.08 2.96 0.999 0.922 4.66 4.52 0.81 0.259 1.24 0.585 0.324 0.65 0.644 3.34 0.007 2.95 18.6 1.14 0.075 0.009 25.1
+Table 15. Statistics of YFCC15M (title and description) coverage on 26 tasks of CLIP evaluation: Low coverage could explain the root
+cause of the poor performance of zero-shot transfer (e.g. Cars, PCAM, etc.).
+A.2.5
+Early Detection of Task Coverage
+One extra benefit of curation is being able to detect zero-
+shot transferability. Although existing scaled pre-trainings
+have huge success, the coverage of pre-training data distri-
+bution for downstream tasks is largely unknown. We dis-
+cuss this coverage issue below.
+Task Coverage. We obtain the statistics of curated data (of-
+fline in Table 1a) for the 26 tasks and show it in Table 15.
+We consider a sample with a maximum cosine similarity
+for one class as one sample belonging to that class/task. We
+note that this is a hard-matching which does not necessar-
+ily cover the full class to sample correlation. Breaking down
+YFCC15M for different tasks partially explains the low per-
+formance on some. For example, SST2 (a binary classifica-
+tion task) has low image-text pair matches, explaining the
+low performance (close to random) for all models.
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+

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+de Haas van Alphen oscillations in hybridization-gap insulators
+as a sudden change in the diamagnetic moment of Landau levels
+S. R. Julian
+Department of Physics, University of Toronto, 60 St.
+George Street, Toronto, Ontario, Canada M5S 1A7
+(Dated: January 16, 2023)
+Abstract
+This note revisits the semi-classical theory of quantum oscillations in hybridization-gap insu-
+lators, and shows that the physical origin of the oscillations, at T = 0 K, is a sudden change in
+the diamagnetic moment of each Landau level as it crosses the hybridized region of the valence
+band.
+1
+arXiv:2301.05366v1  [cond-mat.str-el]  13 Jan 2023
+
+I.
+INTRODUCTION
+For more than 80 years after the first observation of a quantum oscillatory magneti-
+zation, the de Haas-van Alphen (dHvA) effect was regarded as a signature of a metal,
+because the theory was formulated in terms of the emptying of quantized Landau levels
+as they sequentially cross the Fermi surface. In this picture, there should be no oscilla-
+tory magnetization for a filled or empty band, because there is no Fermi surface for the
+Landau levels to cross. Thus, the observation of dHvA oscillations in the Kondo insulator
+SmB6 [1, 2] was greeted with suprise. Almost equally surprising, a possible, very simple,
+theory of these quantum-oscillations-without-a-Fermi-surface was soon found: Knolle and
+Cooper [3] showed that quantum oscillations emerge in a straightforward calculation of the
+thermodynamic potential of a narrow-gap insulator with Landau-quantized states.
+The physical origin of these oscillations has, however, been unclear, and a source of
+some confusion. As Ref. [4] states, in the absence of a Fermi surface “it is not clear what
+surface is being measured.” The original Knolle and Cooper paper [3] does not suggest a
+physical origin for the oscillations: they emerge from the mathematics. Subsequent papers
+by these and other authors [5–8] simply state that the thermodynamic potential oscillates
+as Landau levels sequentially cross the region where the bands cross, although Pal [7] makes
+the enigmatic remark that, as B changes, the Landau levels “feel” the abrupt change in
+the slope of E(k) and “manifest as quantum oscillations”. Moreoever, some papers that
+offer competing theories, e.g. [9–11], state that the Knolle and Cooper dHvA oscillations
+are due to magnetic breakdown, in which the condition ℏωc ≳ E2
+g/EF, where ωc is the
+quasiparticle cyclotron frequency, Eg is the energy gap and EF is the Fermi energy, allows
+the quasiparticles to tunnel across the hybridization gap, which would mean that these
+are, essentially, conventional quantum oscillations.
+The purpose of this brief, somewhat pedagogical, note is to present a simple semi-
+classical picture that shows that the anomalous Knolle-Cooper dHvA oscillations in hy-
+bridzation gap insulators are produced by the sudden change in the diamagnetic moment
+of the Landau levels as they pass between regions of E(k) having different quasiparticle
+velocities. This is unrelated to magnetic breakdown, and indeed it shows that this is a
+novel mechanism for the dHvA effect. This paper does not address the issue of whether
+2
+
+this, as opposed to one of several competing theories (see Ref. 8 for a recent list), is actually
+correct for the case of SmB6.
+II.
+RESULTS AND DISCUSSION: SEMICLASSICAL CALCULATION OF THE
+DHVA EFFECT
+The situation considered by Knolle and Cooper, somewhat generalized in subsequent
+treatments [3, 5–8], is as pictured in Fig. 1a.
+The essential feature is that two bands
+with very different dispersions (in this case one electron-like and one hole-like) weakly
+hybridize where they cross, and the Fermi energy EF lies in the gap. An applied magnetic
+field B results in the formation of Landau levels. As B is increased, the Landau levels,
+illustrated in Fig. 1b, sequentially pass from the electron-like to the hole-like part of the
+valence band, and vice-versa for the conduction band. Working, for simplicity, at T = 0
+in two-dimensions with spinless fermions and no impurity scattering, the grand canonical
+potential can be expressed as a sum over all of the Landau levels of the valence band,
+Ω(µ, T = 0, B) = D
+�
+ℓ
+(Eℓ,v − EF) ,
+(1)
+where D = (eB/2πℏ) is the degeneracy of each Landau level per unit area of sample, and
+Eℓ,v is the energy of the ℓth Landau level in the valence band.
+In the conventional treatment of the dHvA effect in a metal (see e.g. Ref. 12), the sum
+in Eq. 1 is approximated in such a way that the oscillatory part, ˜Ω, which involves only
+states in the immediate vicinity of EF, can be extracted. Then it is straightforward to
+calculate the oscillations in any thermodynamic quantity. For example, the dHvA effect
+measures the oscillatory magnetization
+˜
+M = −∂ ˜Ω
+∂B .
+(2)
+One could approach the problem differently, however, obtaining the total magnetization
+by taking the derivative of Ω, rather than ˜Ω. Consider first a simple parabolic band of
+non-interacting electrons, so that the dispersion relation is E(k) = ℏ2k2/2m. Then, as in
+the case of electrons in free space, the Landau levels are quantized in energy as
+Eℓ = ℏωc(ℓ + 1/2),
+(3)
+3
+
+2.0 × 109
+0
+2.0 × 109
+k  (m
+1)
+50
+0
+50
+E  (meV)
+(a)
+Ec(k)
+Ev(k)
+-5
+0
+5
+E  (meV)
+(b)
+1.50
+1.52
+1.54
+1.56
+1.58
+k  (m
+1)
+1e9
+50
+0
+   (2
+B/(elect.)
+(c)
+FIG. 1. (a) Band structure for a hybridization-gap insulator, with the conduction band in red,
+and valence band in blue (see §IV, Methods, for details.) (b) A zoomed-in view of the circled
+region from (a), where the bands cross and hybridize, with Landau levels shown as blue and red
+points for the valence and conduction band respectively. The green and pink lines respectively
+are the unhybridized electron and hole bands at zero field. (c) Shows the diamagnetic moment
+per electron in the Landau levels near the band crossing in the valence (blue) and conduction
+(red) bands, in units of double Bohr magnetons ℏe/me. The hybridization strength is moderate,
+so that the band dispersion changes rather gradually compared to the Landau level spacing. At
+T = 0 K the conduction-band Landau levels are unoccupied.
+where the cyclotron frequency is ωc = eB/m.
+Substituting this in Eq. 1, and taking
+the derivative with respect to B using the non-obvious step of applying the chain rule to
+4
+
+separate the derivative inside the sum from the derivative of the prefactor D, gives
+M = − D
+ℓmax
+�
+ℓ=0
+eℏ
+m (ℓ + 1/2)
+(4a)
+− Dℏωc
+B
+ℓmax
+�
+ℓ=0
+[(ℓ + 1/2) − X].
+(4b)
+The new variables are X ≡ EF/ℏωc, and ℓmax, which is the quantum number of the
+highest occupied Landau level at a given field B, which satisfies (ℓmax + 1/2) ≤ EF/ℏωc ≤
+(ℓmax + 3/2). ℓmax changes by one every time a Landau level crosses EF.
+In Fig. 2a the two terms of Eq. 4 are plotted separately (top), and summed (bottom). It
+can be seen that, to a good approximation, the first term is comprised of an oscillatory part
+superposed on a large background, while the second term cancels (or very nearly cancels)
+the large background of the first term. (The second term also has an oscillatory part, but
+it is normally ignored, being typically less than 1% of the oscillatory part of the first term.)
+That is, the oscillations are almost entirely contained in the first term, which has a simple
+physical interpretation: the Landau diamagnetic dipole moment of an electron in the ℓth
+Landau level is µ = −(eℏ/m)(ℓ + 1/2) (see below). Thus the oscillatory magnetization
+comes from the sum over the diamagnetic dipole moments of the electrons in their Landau
+levels, and in particular the sharp step in the ‘sawtooth’ pattern of M is due to the loss of
+the Landau diamagnetic moment of the electrons in the ℓmax Landau level, when the level
+suddenly empties as it passes EF with increasing B.
+Shoenberg, in his book, [12] credits Brian Pippard with this insight, but it must have
+been known to Landau and others. Pippard’s treatment, contained in the proceedings of
+a summer school that was held in Vancouver, Canada, in 1967 [13], is nevertheless useful.
+In a departure from the usual Landau gauge approach, he uses the cylindrical gauge to
+construct wave-functions, for the Landau quantized electrons, that in real space are sharply
+peaked (provided ℓ is not too small) at the classical cyclotron radius, which encloses a real-
+space area Ar. Since the electrons circulate with period T = 2π/ωc, their diamagnetic
+dipole moment is
+µ = −eωc
+2π Ar = − e2B
+2πmAr.
+(5)
+5
+
+B    T
+0.14
+0.13
+M  (mJ/T m2)
+(a)  conventional
+Eq. 4a
+(Eq. 4b)
+0.11
+0.10
+(b)   anomalous
+Eq. 10a
+(Eq. 10b)
+10
+10.5
+11
+B  (T)
+0.5
+0.0
+0.5
+M
+(2
+B/e. )
+(Eq. 4a)+(Eq. 4b)
+10
+10.5
+11
+B  (T)
+0.5
+0.0
+0.5
+(Eq. 10a)+(Eq. 10b)
+FIG. 2.
+(a), top panel, plots Eq. 4a and the negative of 4b for the conventional dHvA effect
+in an isolated metallic band of mass m = 1me, showing that Eq. 4a contains the quantum
+oscillations, while 4b, to a very good approximation, cancels the quasi-linear background, leaving
+a total magnetization (lower panel) that is dominated by the oscillations. (See §IV, Methods, for
+details.) (b) shows the corresponding plot for Eq. 10, the hybridization-gap insulator of Fig. 1,
+with a very weak hybridization so that the band dispersion changes very suddenly, compared to
+the Landau level spacing.
+Pippard merely remarks that one can reformulate the dHvA effect directly in terms of the
+diamagnetic magnetization.
+For an arbitrary band structure, applying the semi-classical relation between the real-
+space, Ar,ℓ, and the k-space, Ak,ℓ, areas of a Landau orbit [12, 14],
+Ar,ℓ(B) =
+ℏ2
+e2B2Ak,ℓ(B) = 2πℏ
+eB (ℓ + γ),
+(6)
+where γ = 1/2 for circularly-symmetric two-dimensional bands such as we use here [15],
+gives
+µ = −
+ℏe
+m∗
+ℓ(B)(ℓ + γ),
+(7)
+where
+m∗
+ℓ(B) ≡ ℏ2
+2π
+∂Ak
+∂E
+����
+ℓ,B
+.
+(8)
+It is the factor 1/m∗
+ℓ in Eq. 7 that is of interest. The sign of m∗
+ℓ determines the direction
+in which an electron circulates in its Landau orbit, and thus determines the sign of the
+6
+
+dipole moment; meanwhile the magnitude of m∗, which in this case is really a proxy for
+the speed of the electrons in the Landau orbit, determines the magnitude of the moment:
+fast electrons (low m∗
+ℓ) produce a strong diamagnetic moment, slow electrons (high m∗
+ℓ)
+produce a weak diamagnetic moment. The diamagnetic moments of the Landau levels near
+the hybridization gap are illustrated in Fig. 1c. On the light electron-like part of the bands
+the circulating electrons have a large negative diamagnetic moment, but on the heavy
+hole-like part (where m∗
+ℓ is negative) they have a weak positive moment. The diamagnetic
+moment of the electrons in a Landau level in the valence band thus undergoes a sudden
+change when they cross the hybridized region, which is located at the Fermi wave-vector
+of the unhybridized bands.
+Returning to Eq. 1 and using
+∂Eℓ
+∂B = ∂E
+∂Ak
+∂Ak
+∂B =
+eℏ
+m∗
+ℓ(B)(ℓ + γ),
+(9)
+gives the generalized version of Eq. 4:
+M = − D
+occ.
+�
+ℓ
+ℏe
+m∗
+ℓ(B)(ℓ + γ)
+(10a)
+− D/B
+occ.
+�
+ℓ
+(Eℓ − EF).
+(10b)
+In Fig. 2b the sum in Eq. 10 is evaluated for the band structure of Fig. 1a. Despite
+the fact that the Landau levels do not cross the Fermi energy, but rather remain in the
+valence band for all values of ℓ, it can be seen that the results for the ‘anomalous’ and
+‘conventional’ dHvA effects are remarkably similar: the pattern is the same, the period
+of the oscillations is the same, and again the oscillations are contained in the sum over
+the diamagnetic moments of the electrons in the occupied Landau levels. There are slight
+differences: the magnitude of the quantum oscillations is slightly larger in the anomalous
+case, and the cancellation of background is not quite as good.
+Eq. 10a, and Fig. 1c, demonstrate the physical mechanism of the Knolle-Cooper dHvA
+oscillations: as each subsequent Landau level in the valence band crosses the hybridized
+region at kF, its diamagnetic moment undergoes a sudden change in sign and magnitude
+as the electrons suddenly start circulating in the opposite sense at a different speed.
+The actual size of the step in M depends on the details of the band structure. In the
+case chosen here, the velocity of the electrons in a Landau level reverses, and slows by a
+7
+
+0
+1
+2
+3
+F  (kT)
+0
+A
+= 1 eV
+0.1 meV
+0.3 meV
+(a)
+10
+11
+B  (T)
+0.5
+0.0
+0.5
+M
+0.0
+0.1
+0.2
+0.3
+0.4
+   (meV)
+Amax
+(b)
+T = 0 K
+0
+2
+4
+6
+T   (K)
+Amax
+(c)
+= 1 eV
+0.1 meV
+0.2 meV
+0.4 meV
+FIG. 3.
+(a) shows the effect of changing hybridization strength. The inset compares oscillations
+for λ = 1 µeV (blue line) with λ = 0.1 meV (red line).
+The main figure shows the Fourier
+transform of M vs 1/B. The frequency of the fundamental peak, F ∼ 780 T, corrsponds via the
+Onsager relation, F = ℏAkF /2πe, to the Fermi wave-vector of the unhybridized bands. (See §IV,
+Methods, for details.) (b) shows the amplitude of the fundamental peak vs. λ. The black line is a
+decaying exponential fit to the points with λ > 0.15 meV. (c) shows the temperature dependence
+of the oscillations for selected values of λ.
+factor of 10, as the Landau level passes from the electron-like to the hole-like part of the
+valence band, so the Landau diamagnetism goes from a negative value to a 10× smaller,
+positive value, so the step-change in M is slightly larger than in the conventional case,
+where the diamagnetic moment merely disappears when the Landau level empties as it
+crosses EF. In the original Knolle and Cooper paper [3] the hole band had infinite mass,
+so the diamagnetic moment suddenly drops to zero at kF – exactly as if the Landau level
+had emptied.
+In Figs. 3a and 3b the effect of increasing the hybridization strength λ is shown. There
+is an exponential fall in dHvA amplitude with increasing λ as long as λ is not too small.
+This was noted by Knolle and Cooper, but some authors [9–11] mistakenly attributed this
+to magnetic breakdown, because it has a similar dependence on the energy gap Eg. But in
+8
+
+fact, as noted in Ref. 7, the physical effect (recall that the conduction band is unoccupied in
+this calculation) has to do with the width of the hybridization-crossover region, compared
+with the Landau level spacing. If one Landau level crossing the hybridized region changes
+its diamagnetic moment before the next level enters, there is a large oscillatory effect. If,
+on the other hand, several Landau levels are crossing this region at the same time (the
+large λ case), as pictured in Fig. 1c, then the oscillations are small. It is of course a feature
+of degenerate perturbation theory that the larger the gap, the wider the region in k-space
+over which the bands change their slope.
+Finally, in Fig. 2c the temperature dependence of the peak in the amplitude spectrum
+for several values of λ is shown. Similar results were found by Knolle and Cooper, and
+discussed by them. The key point is that when kBT ≫ Eg the hybridization gap no longer
+matters, and conventional Lifshitz-Kosevich temperature dependence is found.
+III.
+CONCLUSIONS
+By showing that the quantum oscillatory magnetization is contained in the sum over the
+diamagnetic moments of the electrons in occupied Landau levels, it follows that at T = 0 K
+the oscillations in the anomalous dHvA effect arise, not because the highest Landau level
+empties as in the conventional dHvA effect, but rather because its diamagnetic moment
+changes suddenly when the slope of the valence band changes in the hybridization region.
+This gives a simple interpretation of the quantum oscillatory magnetism found by Knolle
+and Cooper, and reinforces that this is a novel mechanism for quantum oscillations.
+IV.
+METHODS
+In Fig. 1a two bands, with E1(k) = ℏ2k2/2m1 and E2(k) = W + ℏ2k2/2m2, where
+m1 = 1me, m2 = −10me and W = 0.1 eV, are hybridized with a coupling of strength λ.
+(The negative mass for m2 is needed for consistency of notation, particularly with Eq. 8.)
+The resulting bands are
+Ei(k) = Eav(k) ±
+�
+∆E(k)2 + λ2,
+(11)
+9
+
+where Eav(k) = 0.5(E1(k) + E2(k)), ∆E = 0.5(E1(k) − E2(k)), and Ei(k) is Ec(k)(Ev(k))
+for the +(−) solution. Since the band structure is circularly symmetric, the wave-vector
+corresponding to a given Landau level ℓ at a field B can be obtained from πkℓ(B)2 =
+Ak,ℓ(B) = 2πeB(ℓ + γ)/ℏ, and then m∗
+ℓ(B) for i = v, c is obtained from Eq. 8. In Figs. 1b
+and 1c the Landau levels are evaluated at B = 10 T.
+For Figs. 2 and 3, Eqs. 8 and 10 were numerically evaluated. A spherical Brillouin zone
+boundary was used, with a gradual logic-function cutoff to avoid spurious zone-boundary
+dHvA oscillations. Care must be taken with this spherical zone boundary when taking
+derivatives.
+For Fig. 2a, the hole-like band is absent, leaving only a metallic band, with m = 1me.
+EF is the same as for the anomalous dHvA case, so kF occurs at the crossing-point of the
+unhybridized bands in Fig. 1b. In Fig. 2b, lower panel, the same normalization was used
+as in (a). That is, only electrons in the electron-like part of the valence band are counted,
+leaving out the electrons in the Landau levels of the hole-like part, which are of course also
+occupied. This makes comparison with Fig. 2a meaningful.
+To get the temperature dependence for Fig. 3c the usual expression for Ω was used:
+Ω = −DkBT
+�
+ℓ
+ln
+�
+1 + e(EF −Eℓ)/kBT�
+,
+(12)
+which leads to the generalization of Eq. 10:
+M = − D
+�
+ℓ
+f(Eℓ, T)
+ℏe
+m∗
+ℓ(B)(ℓ + γ)
+(13a)
++ kBTD
+B
+�
+ℓ
+ln
+�
+1 + e(EF −Eℓ)/kBT�
+,
+(13b)
+where f(Eℓ, T) is the Fermi-Dirac distribution function, and the sum now includes both
+the conduction and the valence bands.
+10
+
+V.
+ACKNOWLEDGEMENTS
+This research was funded by NSERC (RGPIN-2019-06446).
+[1] G. Li, Z. Xiang, F. Yu, T. Asaba, B. Lawson, P. Cai, C. Tinsman, A. Berkley, S. Wolgast,
+Y. S. Eo, D.-J. Kim, C. Kurdak, J. W. Allen, K. Sun, X. H. Chen, Y. Y. Wang, Z. Fisk, and
+L. Li, Science 346, 1208 (2014).
+[2] B. S. Tan, Y.-T. Hsu, B. Zeng, M. Ciomaga Hatnean, N. Harrison, Z. Zhu, M. Hartstein,
+M. Kiourlappou, A. Srivastava, M. D. Johannes, T. P. Murphy, J.-H. Park, L. Balicas, G. G.
+Lonzarich, G. Balakrishnan, and S. E. Sebastian, Science 349, 287 (2015).
+[3] J. Knolle and N. R. Cooper, Phys. Rev. Lett. 115, 146401 (2015).
+[4] P. Ram and B. Kumar, Phys. Rev. B 96, 075115 (2017).
+[5] J. Knolle and N. R. Cooper, Phys. Rev. Lett. 118, 176801 (2017).
+[6] H. K. Pal, F. Pi´echon, J.-N. Fuchs, M. Goerbig, and G. Montambaux, Phys. Rev. B 94,
+125140 (2016).
+[7] H. K. Pal, Phys. Rev. B 95, 085111 (2017).
+[8] A. Panda, S. Banerjee, and M. Randeria, Proc. Natl. Acad. Sci. USA 119, e2208373119
+(2022).
+[9] O. Erten, P. Ghaemi, and P. Coleman, Phys. Rev. Lett. 116, 046403 (2016).
+[10] I. Sodemann, D. Chowdhury, and T. Senthil, Phys. Rev. B 97, 045152 (2018).
+[11] A. Ghazaryan, M. N. Emilian, O. Erten, and P. Ghaemi, New J. Phys. 23, 123042 (2021).
+[12] D. Shoenberg, Magnetic oscillations in metals (Cambridge University Press, 1984).
+[13] A. Pippard, Solid State Physics, vol. 1, Electrons in Metals, edited by J. F. Cochran and
+R. R. Haering (Gordon and Breach, 1968).
+[14] N. W. Ashcroft and N. D. Mermin, Solid state physics (Hold, Rinehart and Winston, 1976)
+Chap. 12 and 14.
+[15] L. Roth, Phys. Rev. B 145, 434 (1966).
+11
+

QdE4T4oBgHgl3EQf-w6Q/content/tmp_files/load_file.txt

@@ -0,0 +1,335 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf,len=334
+page_content='de Haas van Alphen oscillations in hybridization-gap insulators  as a sudden change in the diamagnetic moment of Landau levels  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Julian  Department of Physics, University of Toronto, 60 St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  George Street, Toronto, Ontario, Canada M5S 1A7  (Dated: January 16, 2023)  Abstract  This note revisits the semi-classical theory of quantum oscillations in hybridization-gap insu-  lators, and shows that the physical origin of the oscillations, at T = 0 K, is a sudden change in  the diamagnetic moment of each Landau level as it crosses the hybridized region of the valence  band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='05366v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='str-el]  13 Jan 2023  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  INTRODUCTION  For more than 80 years after the first observation of a quantum oscillatory magneti-  zation, the de Haas-van Alphen (dHvA) effect was regarded as a signature of a metal,  because the theory was formulated in terms of the emptying of quantized Landau levels  as they sequentially cross the Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' In this picture, there should be no oscilla-  tory magnetization for a filled or empty band, because there is no Fermi surface for the  Landau levels to cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Thus, the observation of dHvA oscillations in the Kondo insulator  SmB6 [1, 2] was greeted with suprise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Almost equally surprising, a possible, very simple,  theory of these quantum-oscillations-without-a-Fermi-surface was soon found: Knolle and  Cooper [3] showed that quantum oscillations emerge in a straightforward calculation of the  thermodynamic potential of a narrow-gap insulator with Landau-quantized states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  The physical origin of these oscillations has, however, been unclear, and a source of  some confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' As Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' [4] states, in the absence of a Fermi surface “it is not clear what  surface is being measured.” The original Knolle and Cooper paper [3] does not suggest a  physical origin for the oscillations: they emerge from the mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Subsequent papers  by these and other authors [5–8] simply state that the thermodynamic potential oscillates  as Landau levels sequentially cross the region where the bands cross, although Pal [7] makes  the enigmatic remark that, as B changes, the Landau levels “feel” the abrupt change in  the slope of E(k) and “manifest as quantum oscillations”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Moreoever, some papers that  offer competing theories, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' [9–11], state that the Knolle and Cooper dHvA oscillations  are due to magnetic breakdown, in which the condition ℏωc ≳ E2  g/EF, where ωc is the  quasiparticle cyclotron frequency, Eg is the energy gap and EF is the Fermi energy, allows  the quasiparticles to tunnel across the hybridization gap, which would mean that these  are, essentially, conventional quantum oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  The purpose of this brief, somewhat pedagogical, note is to present a simple semi-  classical picture that shows that the anomalous Knolle-Cooper dHvA oscillations in hy-  bridzation gap insulators are produced by the sudden change in the diamagnetic moment  of the Landau levels as they pass between regions of E(k) having different quasiparticle  velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' This is unrelated to magnetic breakdown, and indeed it shows that this is a  novel mechanism for the dHvA effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' This paper does not address the issue of whether  2  this, as opposed to one of several competing theories (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 8 for a recent list), is actually  correct for the case of SmB6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  RESULTS AND DISCUSSION: SEMICLASSICAL CALCULATION OF THE  DHVA EFFECT  The situation considered by Knolle and Cooper, somewhat generalized in subsequent  treatments [3, 5–8], is as pictured in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  The essential feature is that two bands  with very different dispersions (in this case one electron-like and one hole-like) weakly  hybridize where they cross, and the Fermi energy EF lies in the gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' An applied magnetic  field B results in the formation of Landau levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' As B is increased, the Landau levels,  illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 1b, sequentially pass from the electron-like to the hole-like part of the  valence band, and vice-versa for the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Working, for simplicity, at T = 0  in two-dimensions with spinless fermions and no impurity scattering, the grand canonical  potential can be expressed as a sum over all of the Landau levels of the valence band,  Ω(µ, T = 0, B) = D  �  ℓ  (Eℓ,v − EF) ,  (1)  where D = (eB/2πℏ) is the degeneracy of each Landau level per unit area of sample, and  Eℓ,v is the energy of the ℓth Landau level in the valence band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  In the conventional treatment of the dHvA effect in a metal (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 12), the sum  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 1 is approximated in such a way that the oscillatory part, ˜Ω, which involves only  states in the immediate vicinity of EF, can be extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Then it is straightforward to  calculate the oscillations in any thermodynamic quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' For example, the dHvA effect  measures the oscillatory magnetization  ˜  M = −∂ ˜Ω  ∂B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  (2)  One could approach the problem differently, however, obtaining the total magnetization  by taking the derivative of Ω, rather than ˜Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Consider first a simple parabolic band of  non-interacting electrons, so that the dispersion relation is E(k) = ℏ2k2/2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Then, as in  the case of electrons in free space, the Landau levels are quantized in energy as  Eℓ = ℏωc(ℓ + 1/2),  (3)  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='0 × 109  0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='0 × 109  k  (m  1)  50  0  50  E  (meV)  (a)  Ec(k)  Ev(k)  5  0  5  E  (meV)  (b)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='52  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='54  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='56  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='58  k  (m  1)  1e9  50  0  (2  B/(elect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=')  (c)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' (a) Band structure for a hybridization-gap insulator, with the conduction band in red,  and valence band in blue (see §IV, Methods, for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=') (b) A zoomed-in view of the circled  region from (a), where the bands cross and hybridize, with Landau levels shown as blue and red  points for the valence and conduction band respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' The green and pink lines respectively  are the unhybridized electron and hole bands at zero field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' (c) Shows the diamagnetic moment  per electron in the Landau levels near the band crossing in the valence (blue) and conduction  (red) bands, in units of double Bohr magnetons ℏe/me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' The hybridization strength is moderate,  so that the band dispersion changes rather gradually compared to the Landau level spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' At  T = 0 K the conduction-band Landau levels are unoccupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  where the cyclotron frequency is ωc = eB/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  Substituting this in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 1, and taking  the derivative with respect to B using the non-obvious step of applying the chain rule to  4  separate the derivative inside the sum from the derivative of the prefactor D, gives  M = − D  ℓmax  �  ℓ=0  eℏ  m (ℓ + 1/2)  (4a)  − Dℏωc  B  ℓmax  �  ℓ=0  [(ℓ + 1/2) − X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  (4b)  The new variables are X ≡ EF/ℏωc, and ℓmax, which is the quantum number of the  highest occupied Landau level at a given field B, which satisfies (ℓmax + 1/2) ≤ EF/ℏωc ≤  (ℓmax + 3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' ℓmax changes by one every time a Landau level crosses EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 2a the two terms of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 4 are plotted separately (top), and summed (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' It  can be seen that, to a good approximation, the first term is comprised of an oscillatory part  superposed on a large background, while the second term cancels (or very nearly cancels)  the large background of the first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' (The second term also has an oscillatory part, but  it is normally ignored, being typically less than 1% of the oscillatory part of the first term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=')  That is, the oscillations are almost entirely contained in the first term, which has a simple  physical interpretation: the Landau diamagnetic dipole moment of an electron in the ℓth  Landau level is µ = −(eℏ/m)(ℓ + 1/2) (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Thus the oscillatory magnetization  comes from the sum over the diamagnetic dipole moments of the electrons in their Landau  levels, and in particular the sharp step in the ‘sawtooth’ pattern of M is due to the loss of  the Landau diamagnetic moment of the electrons in the ℓmax Landau level, when the level  suddenly empties as it passes EF with increasing B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  Shoenberg, in his book, [12] credits Brian Pippard with this insight, but it must have  been known to Landau and others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Pippard’s treatment, contained in the proceedings of  a summer school that was held in Vancouver, Canada, in 1967 [13], is nevertheless useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  In a departure from the usual Landau gauge approach, he uses the cylindrical gauge to  construct wave-functions, for the Landau quantized electrons, that in real space are sharply  peaked (provided ℓ is not too small) at the classical cyclotron radius, which encloses a real-  space area Ar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Since the electrons circulate with period T = 2π/ωc, their diamagnetic  dipole moment is  µ = −eωc  2π Ar = − e2B  2πmAr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  (5)  5  B    T  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='14  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='13  M  (mJ/T m2)  (a)  conventional  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 4a  (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 4b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='11  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='10  (b)   anomalous  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 10a  (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 10b)  10  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='5  11  B  (T)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='5  M  (2  B/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' )  (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 4a)+(Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 4b)  10  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='5  11  B  (T)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='5  (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 10a)+(Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 10b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  (a), top panel, plots Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 4a and the negative of 4b for the conventional dHvA effect  in an isolated metallic band of mass m = 1me, showing that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 4a contains the quantum  oscillations, while 4b, to a very good approximation, cancels the quasi-linear background, leaving  a total magnetization (lower panel) that is dominated by the oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' (See §IV, Methods, for  details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=') (b) shows the corresponding plot for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 10, the hybridization-gap insulator of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 1,  with a very weak hybridization so that the band dispersion changes very suddenly, compared to  the Landau level spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  Pippard merely remarks that one can reformulate the dHvA effect directly in terms of the  diamagnetic magnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  For an arbitrary band structure, applying the semi-classical relation between the real-  space, Ar,ℓ, and the k-space, Ak,ℓ, areas of a Landau orbit [12, 14],  Ar,ℓ(B) =  ℏ2  e2B2Ak,ℓ(B) = 2πℏ  eB (ℓ + γ),  (6)  where γ = 1/2 for circularly-symmetric two-dimensional bands such as we use here [15],  gives  µ = −  ℏe  m∗  ℓ(B)(ℓ + γ),  (7)  where  m∗  ℓ(B) ≡ ℏ2  2π  ∂Ak  ∂E  ����  ℓ,B  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  (8)  It is the factor 1/m∗  ℓ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 7 that is of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' The sign of m∗  ℓ determines the direction  in which an electron circulates in its Landau orbit, and thus determines the sign of the  6  dipole moment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' meanwhile the magnitude of m∗, which in this case is really a proxy for  the speed of the electrons in the Landau orbit, determines the magnitude of the moment:  fast electrons (low m∗  ℓ) produce a strong diamagnetic moment, slow electrons (high m∗  ℓ)  produce a weak diamagnetic moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' The diamagnetic moments of the Landau levels near  the hybridization gap are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' On the light electron-like part of the bands  the circulating electrons have a large negative diamagnetic moment, but on the heavy  hole-like part (where m∗  ℓ is negative) they have a weak positive moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' The diamagnetic  moment of the electrons in a Landau level in the valence band thus undergoes a sudden  change when they cross the hybridized region, which is located at the Fermi wave-vector  of the unhybridized bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  Returning to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 1 and using  ∂Eℓ  ∂B = ∂E  ∂Ak  ∂Ak  ∂B =  eℏ  m∗  ℓ(B)(ℓ + γ),  (9)  gives the generalized version of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 4:  M = − D  occ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  �  ℓ  ℏe  m∗  ℓ(B)(ℓ + γ)  (10a)  − D/B  occ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  �  ℓ  (Eℓ − EF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  (10b)  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 2b the sum in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 10 is evaluated for the band structure of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Despite  the fact that the Landau levels do not cross the Fermi energy, but rather remain in the  valence band for all values of ℓ, it can be seen that the results for the ‘anomalous’ and  ‘conventional’ dHvA effects are remarkably similar: the pattern is the same, the period  of the oscillations is the same, and again the oscillations are contained in the sum over  the diamagnetic moments of the electrons in the occupied Landau levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' There are slight  differences: the magnitude of the quantum oscillations is slightly larger in the anomalous  case, and the cancellation of background is not quite as good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 10a, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 1c, demonstrate the physical mechanism of the Knolle-Cooper dHvA  oscillations: as each subsequent Landau level in the valence band crosses the hybridized  region at kF, its diamagnetic moment undergoes a sudden change in sign and magnitude  as the electrons suddenly start circulating in the opposite sense at a different speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  The actual size of the step in M depends on the details of the band structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' In the  case chosen here, the velocity of the electrons in a Landau level reverses, and slows by a  7  0  1  2  3  F  (kT)  0  A  = 1 eV  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='1 meV  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='3 meV  (a)  10  11  B  (T)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='5  M  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='4  (meV)  Amax  (b)  T = 0 K  0  2  4  6  T   (K)  Amax  (c)  = 1 eV  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='1 meV  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='2 meV  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='4 meV  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  (a) shows the effect of changing hybridization strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' The inset compares oscillations  for λ = 1 µeV (blue line) with λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='1 meV (red line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  The main figure shows the Fourier  transform of M vs 1/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' The frequency of the fundamental peak, F ∼ 780 T, corrsponds via the  Onsager relation, F = ℏAkF /2πe, to the Fermi wave-vector of the unhybridized bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' (See §IV,  Methods, for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=') (b) shows the amplitude of the fundamental peak vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' The black line is a  decaying exponential fit to the points with λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='15 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' (c) shows the temperature dependence  of the oscillations for selected values of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  factor of 10, as the Landau level passes from the electron-like to the hole-like part of the  valence band, so the Landau diamagnetism goes from a negative value to a 10× smaller,  positive value, so the step-change in M is slightly larger than in the conventional case,  where the diamagnetic moment merely disappears when the Landau level empties as it  crosses EF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' In the original Knolle and Cooper paper [3] the hole band had infinite mass,  so the diamagnetic moment suddenly drops to zero at kF – exactly as if the Landau level  had emptied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 3a and 3b the effect of increasing the hybridization strength λ is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' There  is an exponential fall in dHvA amplitude with increasing λ as long as λ is not too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  This was noted by Knolle and Cooper, but some authors [9–11] mistakenly attributed this  to magnetic breakdown, because it has a similar dependence on the energy gap Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' But in  8  fact, as noted in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 7, the physical effect (recall that the conduction band is unoccupied in  this calculation) has to do with the width of the hybridization-crossover region, compared  with the Landau level spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' If one Landau level crossing the hybridized region changes  its diamagnetic moment before the next level enters, there is a large oscillatory effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' If,  on the other hand, several Landau levels are crossing this region at the same time (the  large λ case), as pictured in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 1c, then the oscillations are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' It is of course a feature  of degenerate perturbation theory that the larger the gap, the wider the region in k-space  over which the bands change their slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  Finally, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 2c the temperature dependence of the peak in the amplitude spectrum  for several values of λ is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Similar results were found by Knolle and Cooper, and  discussed by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' The key point is that when kBT ≫ Eg the hybridization gap no longer  matters, and conventional Lifshitz-Kosevich temperature dependence is found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  CONCLUSIONS  By showing that the quantum oscillatory magnetization is contained in the sum over the  diamagnetic moments of the electrons in occupied Landau levels, it follows that at T = 0 K  the oscillations in the anomalous dHvA effect arise, not because the highest Landau level  empties as in the conventional dHvA effect, but rather because its diamagnetic moment  changes suddenly when the slope of the valence band changes in the hybridization region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  This gives a simple interpretation of the quantum oscillatory magnetism found by Knolle  and Cooper, and reinforces that this is a novel mechanism for quantum oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  METHODS  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 1a two bands, with E1(k) = ℏ2k2/2m1 and E2(k) = W + ℏ2k2/2m2, where  m1 = 1me, m2 = −10me and W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='1 eV, are hybridized with a coupling of strength λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  (The negative mass for m2 is needed for consistency of notation, particularly with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=')  The resulting bands are  Ei(k) = Eav(k) ±  �  ∆E(k)2 + λ2,  (11)  9  where Eav(k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='5(E1(k) + E2(k)), ∆E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='5(E1(k) − E2(k)), and Ei(k) is Ec(k)(Ev(k))  for the +(−) solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Since the band structure is circularly symmetric, the wave-vector  corresponding to a given Landau level ℓ at a field B can be obtained from πkℓ(B)2 =  Ak,ℓ(B) = 2πeB(ℓ + γ)/ℏ, and then m∗  ℓ(B) for i = v, c is obtained from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 1b  and 1c the Landau levels are evaluated at B = 10 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  For Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 2 and 3, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 8 and 10 were numerically evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' A spherical Brillouin zone  boundary was used, with a gradual logic-function cutoff to avoid spurious zone-boundary  dHvA oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Care must be taken with this spherical zone boundary when taking  derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  For Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 2a, the hole-like band is absent, leaving only a metallic band, with m = 1me.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  EF is the same as for the anomalous dHvA case, so kF occurs at the crossing-point of the  unhybridized bands in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 2b, lower panel, the same normalization was used  as in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' That is, only electrons in the electron-like part of the valence band are counted,  leaving out the electrons in the Landau levels of the hole-like part, which are of course also  occupied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' This makes comparison with Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 2a meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  To get the temperature dependence for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 3c the usual expression for Ω was used:  Ω = −DkBT  �  ℓ  ln  �  1 + e(EF −Eℓ)/kBT�  ,  (12)  which leads to the generalization of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 10:  M = − D  �  ℓ  f(Eℓ, T)  ℏe  m∗  ℓ(B)(ℓ + γ)  (13a)  + kBTD  B  �  ℓ  ln  �  1 + e(EF −Eℓ)/kBT�  ,  (13b)  where f(Eℓ, T) is the Fermi-Dirac distribution function, and the sum now includes both  the conduction and the valence bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  10  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  ACKNOWLEDGEMENTS  This research was funded by NSERC (RGPIN-2019-06446).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Li, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Xiang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
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+page_content='  [14] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Ashcroft and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Mermin, Solid state physics (Hold, Rinehart and Winston, 1976)  Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' 12 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  [15] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Roth, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content=' B 145, 434 (1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}
+page_content='  11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE4T4oBgHgl3EQf-w6Q/content/2301.05366v1.pdf'}