[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82528-en":3,"doc-seo-82528-105":29,"detail-sidebar-cat-0-en-105":91},{"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":13,"seo_description":14,"update_tm":27,"read_time":28},82528,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","Group-Equivariant Poincaré Convolutional Networks","Advances such as Poincaré ResNet learn visual representations in hyperbolic space, yet optimization is slowed by computationally intensive Riemannian gradients and hard constraints imposed by manifold boundaries. Existing hyperbolic networks also treat spatial transformations as unrelated hierarchies, wasting parameters and weakening signals. Equivariant Poincaré ResNets combine Poincaré geometry with discrete symmetry groups (C4, D4). The work develops geometrically safe tensor reshaping, left-regular hyperbolic group convolutions, and joint-orientation Poincaré midpoint batch normalization to preserve boundary constraints and maintain spatial-group equivariance, drastically shrinking the optimization space and accelerating convergence.","arXiv :2607 .00556v 1 [ cs .LG] 1 Jul 2026  \nGroup-Equivariant Poincaré Convolutional  \nNetworks  \nAiden Durrant 1 , Rahul Baburajan2 , and Georgios Leontidis2  \n1 School of Computing Sciences, University of East Anglia, NR4 7TJ, Norwich, UK [aiden.durrant@uea.ac.uk](aiden.durrant@uea.ac.uk)  \n2 Department of Physics and Technology, UiT The Arctic University of Norway, NO-9037, Tromsø, Norway  \n[rahul.baburajan@uit.no](rahul.baburajan@uit.no), [georgios.leontidis@uit.no](georgios.leontidis@uit.no)  \nAbstract. While recent advancements like the Poincaré ResNet have demonstrated the potential of learning visual representations directly in hyperbolic space, their optimisation remains hampered by the computationally intensive nature of Riemannian gradients and the strict boundaries of the manifold. Furthermore, standard hyperbolic networks treat spatial transformations of the same object as distinct hierarchical concepts, leading to redundant parameter usage and vanishing signals. We propose Equivariant Poincaré ResNets, combining hyperbolic geometry with discrete symmetry groups (C4 and D4 ) . We identify critical roadblocks in applying Euclidean equivariance to hyperbolic space and propose geometrically safe tensor reshaping, left-regular permutations for hyperbolic group convolutions, and joint-orientation Poincaré Midpoint Batch normalisation. Empirically, embedding equivariance drastically reduces the optimisation space, accelerating convergence while accelerating convergence while respecting the boundary constraints of the Poincaré ball and preserving spatial-group equivariance.  \nKeywords: Hyperbolic Learning · Equivariant Neural Networks  \n1 Introduction  \nDeep learning in hyperbolic space has demonstrated profound capabilities for embedding hierarchical visual data with minimal distortion [1, 20, 26] . The recent introduction of Poincaré residual networks has pushed this boundary further, enabling the learning of visual representations entirely within the Poincaré ball from the pixel level [42] . However, optimising deep convolutional networks on Riemannian manifolds introduces significant challenges. The parameter space is tightly constrained, and traversing the curvature requires computationally expensive operations, such as exponential mappings and Fréchet mean estimations.  \nA critical inefficiency in current hyperbolic visual networks is their inability to inherently recognise spatial symmetries. An object rotated by 90 degrees is treated by the network as an entirely new hierarchical concept, forcing the optimiser to require expensive Riemannian gradient steps to learn redundant representations. In Euclidean space, one approach to solve this is through groupequivariant convolutional networks [9] . Such networks have been shown to be  \n2 A. Durrant et al.  \nhighly data-efficient [3], and thus we propose that the Hyperbolic networks computational overhead can be somewhat mitigated through use of symmetric priors.  \nIn this paper, we investigate how to embed C4 (rotations) and D4 (rotationsand reflections) spatial-group equivariance over Poincaré-valued feature fields directly into the Poincaré ResNet architecture. Translating equivariance to hyperbolic space is highly non-trivial due to the non-Euclidean nature of feature concatenation and channel manipulation. We propose three primary contributions to enable the Equivariant Poincaré ResNets: (i) We introduce a geometrically safe β-scaling formulation for flattening and unflattening Poincaré tensors, allowing orientation channels to be decoupled without violating the expected norms of the manifold. (ii) We formulate hyperbolic lifting and group convolutions by projecting base tangent-space filters through discrete symmetry transformationsand left-regular permutations. (iii) We extend Poincaré Midpoint Batch normalisation to act jointly over group orientations, preventing the normalisation step from destroying the learned equivariance.  \n2 Background and Related Work  ","cbCaiuVjGeGafCa9","https://ap.wps.com/l/cbCaiuVjGeGafCa9","pdf",811180,1,19,"English","en",105,"# Introduction\n# Background and Related Work","[{\"question\":\"What optimization challenges affect hyperbolic networks in Poincaré space?\",\"answer\":\"Optimization is hindered by the computational cost of Riemannian gradients and by strict constraints from the manifold boundaries of the Poincaré ball.\"},{\"question\":\"Why do standard hyperbolic networks inefficiently handle spatial transformations?\",\"answer\":\"They do not inherently recognize spatial symmetries; rotations such as 90 degrees are learned as entirely separate hierarchical concepts, forcing redundant representation learning and weaker signals.\"},{\"question\":\"What key methods enable group equivariance in Equivariant Poincaré ResNets?\",\"answer\":\"The approach introduces geometrically safe tensor reshaping with β-scaling, uses hyperbolic lifting with discrete symmetry transformations and left-regular permutations for group convolutions, and extends Poincaré midpoint batch normalization across group orientations.\"}]",1784181261,48,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"group-equivariant-poincare-convolutional-networks","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/group-equivariant-poincare-convolutional-networks/82528/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What optimization challenges affect hyperbolic networks in Poincaré space?","Question",{"text":75,"@type":76},"Optimization is hindered by the computational cost of Riemannian gradients and by strict constraints from the manifold boundaries of the Poincaré ball.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why do standard hyperbolic networks inefficiently handle spatial transformations?",{"text":80,"@type":76},"They do not inherently recognize spatial symmetries; rotations such as 90 degrees are learned as entirely separate hierarchical concepts, forcing redundant representation learning and weaker signals.",{"name":82,"@type":73,"acceptedAnswer":83},"What key methods enable group equivariance in Equivariant Poincaré ResNets?",{"text":84,"@type":76},"The approach introduces geometrically safe tensor reshaping with β-scaling, uses hyperbolic lifting with discrete symmetry transformations and left-regular permutations for group convolutions, and extends Poincaré midpoint batch normalization across group orientations.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":21,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":106,"slug":137},"General","general"]