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This paper extends geometric deep learning by introducing order-equivariant neural networks (OENN) that unify graph message passing and sheaf neural networks via equivariant bundles over face posets. It characterizes linear order-equivariant maps, builds OENN layers, and proves universal approximation theorems for continuous order-equivariant maps. It further generalizes the framework to category-equivariant neural networks (CENN) using categorical symmetry, and illustrates results on graph and sheaf models.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/foundations-of-equivariant-deep-learning-unifying-graph-and-sheaf-neural-networks-paper/327781/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/foundations-of-equivariant-deep-learning-unifying-graph-and-sheaf-neural-networks-paper/327781.png","ImageObject",300,407,{"name":92,"@type":93},"Asher","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-31","2026-09-21",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",15,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What problem does the paper address in geometric deep learning?","Question",{"text":113,"@type":114},"It addresses how to exploit richer symmetry structures beyond standard group actions or simple permutation settings, focusing on symmetry induced by poset-like hierarchical or incidence structures.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How are order-equivariant neural networks (OENN) constructed and what do they unify?",{"text":118,"@type":114},"OENN generalize graph message passing and sheaf neural networks using the theory of equivariant bundles over face posets, providing a unified framework for these structured domains.",{"name":120,"@type":111,"acceptedAnswer":121},"What theoretical results does the paper prove?",{"text":122,"@type":114},"The paper characterizes linear order-equivariant maps, builds OENN layers, and proves universal approximation theorems for continuous order-equivariant maps, including new results for sheaf neural networks where no prior UAT was known.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},327781,1790228792,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":145,"read_time":146,"preview_page_count":4},687197207639,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","Foundations of Equivariant Deep Learning: Unifying Graph and Sheaf Neural Networks  \nYoshihiro Maruyama 1  \narXiv :2607 .03798v 1 [ cs .LG] 4 Jul 2026  \nAbstract  \nSymmetry is everywhere in nature and society.  \nGeometric deep learning exploits symmetries in data to improve the performance and efficiency of deep learning systems. In this paper, we extend geometric deep learning to utilize richer symmetry structures. Specifically, we develop orderequivariant neural networks (OENN), which generalize standard graph message passing and sheaf neural networks via the theory of equivariant bundles over face posets (face categories) . We (i) characterize all linear order-equivariant maps,(ii) build OENN layers, and (iii) prove universal approximation theorems (UATs) for continuous order-equivariant maps, which are new resultseven when restricted to sheaf neural networks (for which no UAT was known before) . We illustrate the framework on graph and sheaf models. Our results can alsobe seen as extending the known UAT for graph neural networks to a more general setting that subsumes sheaf neural networks as well.  \nIn addition, we show that OENN can be extended further to CENN, Category-Equivariant Neural Network, which gives the general form of equivariant neural networks as well as of equivariant universal approximation theorems, allowing us to leverage categorical symmetry in data (e.g., noninvertible symmetries on multiple objects with compositional relations on those symmetries) .  \n1. Introduction  \nMotivation. Symmetry has long been recognized as a powerful inductive bias in deep learning. Group-equivariant architectures explicitly exploit the symmetries of an index set to share parameters, improve sample efficiency, and en-  \n1Department of Mathematical and Information Sciences, Kyoto University, Kyoto, Japan. Correspondence to: Yoshihiro Maruyama \u003C[maruyama@i.h.kyoto-u.ac.jp](maruyama@i.h.kyoto-u.ac.jp) >.  \nProceedings of the 43 rd International Conference on Machine Learning, Seoul, South Korea. PMLR 306, 2026 . Copyright 2026 by the author(s) .  \nsure consistent behavior under transformations of the input domain (Cohen and Welling, 2016 ; 2017 ; Weiler and Cesa, 2019 ; Fuchs et al., 2020 ; Satorras et al., 2021) . Classical convolutional networks are equivariant to translations, while group-equivariant CNNs generalize this idea to arbitrary compact groups. More recently, this principle has been extended to permutation-invariant and permutation-equivariant models such as DeepSets (Zaheer et al., 2017) and graph neural networks GNNs (Gilmer et al., 2017 ; Xu et al., 2019 ; Maron et al., 2019), which achieve equivariance with respect to the automorphism group of a graph. However, many data domains are not merely sets or graphs but possess richer hierarchical or incidence structures. Examples include faceposets of graphs, simplicial or CW complexes, and general partially ordered collections of cells or regions. Such domains are naturally indexed by posets, and their symmetries form automorphism groups G = Aut(P,≤) that may involve both permutation and structural (non-invertible or higher-order) relations. This observation motivates the development of order-equivariant neural networks (OENN) that respect the combinatorial structure of a poset rather than a pure group action.  \nBackground and context. The study of equivariance has unified diverse neural architectures under the umbrella of geometric deep learning (Bronstein et al., 2021 ; Shumanet al., 2013 ; Bruna et al., 2014 ; Defferrard et al., 2016) . Group-and permutation-equivariant networks (Cohen and Welling, 2016 ; Zaheer et al., 2017 ; Gilmer et al., 2017 ; Xu et al., 2019 ; Maron et al., 2019) rely on the algebraic structure of groups acting on index sets, while higher-order analogs exploit topological constructions such as simplicial, cellular, or combinatorial complexes (Ebli et al., 2020 ; Bodnar et al., 2022 ; Hansen and Gebhart, 2020 ; Barbero et al., 2022) . This i","cbCaihM7JeMiMWtm","https://ap.wps.com/l/cbCaihM7JeMiMWtm","pdf",534440,23,"English","# Abstract\n# 1. Introduction\n## Motivation\n## Background and context\n## Contributions","[{\"question\":\"What problem does the paper address in geometric deep learning?\",\"answer\":\"It addresses how to exploit richer symmetry structures beyond standard group actions or simple permutation settings, focusing on symmetry induced by poset-like hierarchical or incidence structures.\"},{\"question\":\"How are order-equivariant neural networks (OENN) constructed and what do they unify?\",\"answer\":\"OENN generalize graph message passing and sheaf neural networks using the theory of equivariant bundles over face posets, providing a unified framework for these structured domains.\"},{\"question\":\"What theoretical results does the paper prove?\",\"answer\":\"The paper characterizes linear order-equivariant maps, builds OENN layers, and proves universal approximation theorems for continuous order-equivariant maps, including new results for sheaf neural networks where no prior UAT was known.\"}]","Foundations of Equivariant Deep Learning - Unifying Graph and Sheaf Neural Networks - Paper | PDF",1789978027,58]