[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82048-en":3,"doc-seo-82048-105":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},82048,7971461740909,"Levi","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","LieBN Batch Normalization over Lie Groups","Manifold-valued measurements frequently arise in machine learning, where deep networks are extended from Euclidean space to Riemannian manifolds and require geometry-aware normalization. Existing Riemannian batch normalization methods are often limited to specific manifolds or do not properly normalize distributions over manifold-valued samples. This work introduces LieBN, a framework for Riemannian batch normalization over Lie groups, using left- and right-invariant metrics to control Riemannian mean and variance. Experiments across nine geometries validate the approach and code is released.","LieBN: Batch Normalization over Lie Groups  \nZiheng Chen, Yue Song, Rui Wang, Xiao-Jun Wu, and Nicu Sebe  \nAbstract—Manifold-valued measurements are prevalent in various machine learning tasks. Recent advances have extended Deep Neural Networks (DNNs) to operate on manifolds, accompanied by normalization techniques tailored to different geometries, collectively referred to as Riemannian normalization. However, most existing Riemannian normalization methods are either designed for specific manifolds or fail to effectively normalize manifold-valued sample distributions. To address these limitations, we propose LieBN, a framework for Riemannian Batch Normalization (RBN) over Lie groups. Our approach leverages the theoretically convenient left-and right-invariant metrics, which naturally exist in every Lie group, and provides theoretical guarantees for controlling the Riemannian mean and variance. We instantiate LieBN across nine distinct geometries: four on the Symmetric Positive Definite (SPD) manifold, one on the group of rotation matrices, and four on the manifold of full-rank correlation matrices. Notably, among the SPD metrics, we introduce a novel right-invariant metric and extend three existing Lie group structures via matrix power deformation. Extensive experiments on different manifolds validate the effectiveness of our framework. The code is available at [https://github.com/GitZH-Chen/LieBN.git](https://github.com/GitZH-Chen/LieBN.git).  \nIndex Terms—Riemannian batch normalization, Lie groups, symmetric positive definite matrices, rotations, correlation matrices.  \n~~ ~~ ✦ ~~ ~~  \narXiv :2607 .08783v1 [ cs .LG] 13 Jun 2026  \n1 INTRODUCTION  \nOver the past decade or so, Deep Neural Networks (DNNs) have achieved significant progress across various scientific fields [4], [5], [6], [7] . Traditionally, DNNs have been developed under the assumption that the latent space of the input data is Euclidean. However, many applications involve nonEuclidean structures, such as manifolds [8] . To address this challenge, researchers have extended various types of DNNs to manifolds, based on the theories of Riemannian geometry [9], [10], [11], [12], [13], [14], [15], [16], [17], [18],[19],[20],[21],[22],[23],[24],[25],[26],[27] .  \nMotivated by the great success of normalization techniques [28], [29], [30], [31], researchers have sought to devise normalization layers tailored for manifold-valued data. Brooks et al [32] introduced Riemannian Batch Normalization (RBN) designed specifically for the Symmetric Positive Definite (SPD) manifold, with the ability to normalize the Riemannian mean. Kobler et al [33] extended this approach to further control the Riemannian variance. However, the above methods are constrained within the AffineInvariant Metric (AIM) on the SPD manifold, limiting their applicability. On the other hand, Chakraborty [34] proposed two distinct Riemannian normalization frameworks: one for Riemannian homogeneous spaces [34, Algs. 1-2] and another for matrix Lie groups [34, Algs. 3-4] . Nonetheless, the normalization designed for Riemannian homogeneous spaces cannot normalize mean nor variance, while the one for matrix Lie groups is confined to a specific type of distance [34, Sec. 3.2]. Meanwhile, Luo [35, Alg. 2] proposed an RBN layer for general geometries. However, similar to [34, Algs. 1- 2], it lacks theoretical guarantees for normalizing sample  \n• Ziheng Chen and Nicu Sebe are with the Department of Information Engineering and Computer Science, University of Trento, Trento, Italy. Yue Song is with Computing and Mathematical Sciences, Caltech, CA, USA. Rui Wang and Xiao-Jun Wu are with the School of Artificial Intelligence and Computer Science, Jiangnan University, Wuxi, China. E-mail: ziheng [ch@163.com](ch@163.com), [yuesong@caltech.edu](yuesong@caltech.edu), niculae.sebe@unitn.it,  \n{cs   wr, wu [xiaojun](xiaojun}@jiangnan.edu.cn)[}](xiaojun}@jiangnan.edu.cn)[@jiangnan.edu.cn](xiaojun}@jiangnan.edu.cn).  \nstatist","cbCaitYl7a4RTSWg","https://ap.wps.com/l/cbCaitYl7a4RTSWg","pdf",5455033,1,32,"English","en",105,"# Introduction\n## Motivation and background\n## Related work and limitations\n## Proposed LieBN framework\n## Instantiation on multiple manifolds","[{\"question\":\"What problem does LieBN address in manifold-valued machine learning?\",\"answer\":\"It targets normalization for manifold-valued sample distributions, where prior Riemannian batch normalization methods are often restricted to specific manifolds or lack effective distribution-level normalization.\"},{\"question\":\"How does LieBN achieve theoretical control of statistics?\",\"answer\":\"LieBN leverages left- and right-invariant metrics available on any Lie group, providing theoretical guarantees for controlling the Riemannian mean and variance.\"},{\"question\":\"Which manifold geometries are used to instantiate and evaluate LieBN?\",\"answer\":\"The framework is instantiated across nine geometries, including four on the SPD manifold, one on the group of rotation matrices, and four on the manifold of full-rank correlation matrices.\"}]",1784177807,81,{"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},"liebn-batch-normalization-over-lie-groups","",{"@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/liebn-batch-normalization-over-lie-groups/82048/",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 problem does LieBN address in manifold-valued machine learning?","Question",{"text":75,"@type":76},"It targets normalization for manifold-valued sample distributions, where prior Riemannian batch normalization methods are often restricted to specific manifolds or lack effective distribution-level normalization.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does LieBN achieve theoretical control of statistics?",{"text":80,"@type":76},"LieBN leverages left- and right-invariant metrics available on any Lie group, providing theoretical guarantees for controlling the Riemannian mean and variance.",{"name":82,"@type":73,"acceptedAnswer":83},"Which manifold geometries are used to instantiate and evaluate LieBN?",{"text":84,"@type":76},"The framework is instantiated across nine geometries, including four on the SPD manifold, one on the group of rotation matrices, and four on the manifold of full-rank correlation 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