[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-119203-en":3,"doc-seo-119203-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},119203,1099514068035,"Ezra","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","A Unified Framework to Enforce, Discover, and Promote Symmetry in Machine Learning","Symmetry is a foundational principle spanning nature and playing an increasingly central role in physics and machine learning. The framework in this work targets symmetry’s key contribution to extrapolation, enabling models to leverage invariances such as translation invariance for improved performance with fewer parameters and smaller datasets. It unifies three goals—enforcing known symmetry, discovering unknown symmetry, and promoting symmetry during training—using a shared mathematical structure centered on Lie derivatives for fiber-linear Lie group actions on vector bundles. Enforcement and discovery are shown to be dual linear-algebraic tasks, and promotion is achieved via convex regularization and nuclear-norm relaxation, applicable to regression, dynamical systems, neural networks, and neural operators.","arXiv :2311 .00212v2 [ cs .LG] 19 Aug 2024  \nA Unified Framework to Enforce, Discover, and Promote Symmetry in Machine Learning  \nSamuel E. Otto∗ [s.otto@cornell.edu](s.otto@cornell.edu)  \nAI Institute in Dynamic Systems University of Washington Seattle, WA 98195-4322, USA  \nNicholas Zolman [nzolman@uw.edu](nzolman@uw.edu)  \nAI Institute in Dynamic Systems University of Washington Seattle, WA 98195-4322, USA  \nJ. Nathan Kutz [kutz@uw.edu](kutz@uw.edu)  \nAI Institute in Dynamic Systems University of Washington Seattle, WA 98195-4322, USA  \nSteven L. Brunton [sbrunton@uw.edu](sbrunton@uw.edu)  \nAI Institute in Dynamic Systems University of Washington Seattle, WA 98195-4322, USA  \nAbstract  \nSymmetry is present throughout nature and continues to play an increasingly central role in physics and machine learning. Fundamental symmetries, such as Poincaré invariance, allow physical laws discovered in laboratories on Earth to be extrapolated to the farthest reaches of the universe.  \nSymmetry is essential to achieving this extrapolatory power in machine learning applications. For example, translation invariance in image classification allows models with fewer parameters, such as convolutional neural networks, to be trained on smaller data sets and achieve state-of-the-art performance. In this paper, we provide a unifying theoretical and methodological framework for incorporating symmetry into machine learning models in three ways: 1. enforcing known symmetry when training a model; 2. discovering unknown symmetries of a given model or data set; and  \n3. promoting symmetry during training by learning a model that breaks symmetries within a user-specified group of candidates when there is sufficient evidence in the data. We show that these tasks can be cast within a common mathematical framework whose central object is the Lie derivative associated with fiber-linear Lie group actions on vector bundles. We extend and unify several existing results by showing that enforcing and discovering symmetry are linear-algebraic tasks that are dual with respect to the bilinear structure of the Lie derivative. We also propose a novel way to promote symmetry by introducing a class of convex regularization functions based on the Lie derivative and nuclear norm relaxation to penalize symmetry breaking during training of machine learning models. We explain how these ideas can be applied to a wide range of machine learning models including basis function regression, dynamical systems discovery, neural networks, and neural operators acting on fields.  \nKeywords: Symmetries, machine learning, Lie groups, manifolds, invariance, equivariance, neural networks, deep learning  \n∗ . Present affiliation: Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY, USA  \nContents  \n1 Introduction 3  \n2 Executive summary 5  \n2.1 Enforcing symmetry .................................... 5  \n2.2 Discovering symmetry ................................... 5  \n2.3 Promoting symmetry .................................... 5  \n3 Related work 6  \n3.1 Enforcing symmetry .................................... 6  \n3.2 Discovering symmetry ................................... 6  \n3.3 Promoting symmetry .................................... 7  \n3.4 Additional approaches and applications .......................... 8  \n4 Elementary theory of Lie group actions 8  \n4.1 Lie groups and subgroups ................................. 8  \n4.2 Group representations, actions, and infinitesimal generators .............. 10  \n5 Fundamental operators for studying symmetry 12  \n6 Enforcing symmetry with linear constraints 14  \n6.1 Multilayer perceptrons ................................... 14  \n6.2 Neural operators acting on fields ............................. 15  \n7 Discovering symmetry by computing nullspaces 17  \n7.1 Symmetries of submanifolds ................................ 17  \n7.2 Symmetries of functions as symmetries of submanifolds ................. 19  \n7.3 Symmetries an","cbCaim04lgOvBVUB","https://ap.wps.com/l/cbCaim04lgOvBVUB","pdf",3444533,1,64,"English","en",105,"# Introduction\n# Executive summary\n## Enforcing symmetry\n## Discovering symmetry\n## Promoting symmetry\n# Related work\n## Additional approaches and applications\n# Elementary theory of Lie group actions\n# Fundamental operators for studying symmetry\n# Enforcing symmetry with linear constraints\n## Multilayer perceptrons\n## Neural operators acting on fields\n# Discovering symmetry by computing nullspaces\n## Symmetries of submanifolds\n# Promoting symmetry with convex penalties\n## Discrete symmetries\n## Continuous symmetries\n# Numerical study of sample complexity to recover symmetric functions\n# Discretizing the operators\n# Generalization to sections of vector bundles\n# Conclusion","[{\"question\":\"What three symmetry-related tasks does the paper unify for machine learning models?\",\"answer\":\"It unifies enforcing known symmetry, discovering unknown symmetries, and promoting symmetry during training.\"},{\"question\":\"How does the framework model these tasks mathematically?\",\"answer\":\"The approach casts them into a common framework centered on the Lie derivative associated with fiber-linear Lie group actions on vector bundles.\"},{\"question\":\"How is symmetry promotion formulated during training?\",\"answer\":\"It introduces convex regularization functions based on the Lie derivative and nuclear-norm relaxation to penalize symmetry breaking, when training a model over candidate symmetry groups.\"}]","A Unified Framework to Enforce, Discover, and Promote Symmetry in Machine Learning | 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three symmetry-related tasks does the paper unify for machine learning models?","Question",{"text":76,"@type":77},"It unifies enforcing known symmetry, discovering unknown symmetries, and promoting symmetry during training.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the framework model these tasks mathematically?",{"text":81,"@type":77},"The approach casts them into a common framework centered on the Lie derivative associated with fiber-linear Lie group actions on vector bundles.",{"name":83,"@type":74,"acceptedAnswer":84},"How is symmetry promotion formulated during training?",{"text":85,"@type":77},"It introduces convex regularization functions based on the Lie derivative and nuclear-norm relaxation to penalize symmetry breaking, when training a model over candidate symmetry 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