[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81491-en":3,"doc-seo-81491-105":30,"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},81491,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","A Lie Group Approach to Riemannian Batch Normalization","Manifold-valued measurements are common in computer vision and machine learning, motivating neural networks operating on non-Euclidean spaces. While Riemannian batch normalization has been adapted to manifolds, existing methods are often ad hoc and tailored to particular manifolds. This paper presents a unified Riemannian Batch Normalization framework on Lie groups, providing theoretical control of both Riemannian mean and variance. It is instantiated for symmetric positive definite (SPD) manifolds via deformation-based parameterized Lie group families, and validated on radar recognition, action recognition, and EEG classification with released code.","arXiv :2403 . 11261v1 [ cs .LG] 17 Mar 2024  \nPublished as a conference paper at ICLR 2024   \nA LIE GROUP APPROACH TO RIEMANNIAN BATCH NORMALIZATION  \nZiheng Chen1 , Yue Song 1 ∗, Yunmei Liu2 & Nicu Sebe 1  \n1 University of Trento, 2 University of Louisville ziheng  [ch@163.com](ch@163.com), [yue.song@unitn.it](yue.song@unitn.it)  \nABSTRACT  \nManifold-valued measurements exist in numerous applications within computer vision and machine learning. Recent studies have extended Deep Neural Networks (DNNs) to manifolds, and concomitantly, normalization techniques have also been adapted to several manifolds, referred to as Riemannian normalization. Nonetheless, most of the existing Riemannian normalization methods have been derived in an ad hoc manner and only apply to speci􀀂c manifolds. This paper establishes a uni􀀂ed framework for Riemannian Batch Normalization (RBN) techniques on Lie groups. Our framework offers the theoretical guarantee of controlling both the Riemannian mean and variance. Empirically, we focus on Symmetric Positive De􀀂nite (SPD) manifolds, which possess three distinct types of Lie group structures. Using the deformation concept, we generalize the existing Lie groups on SPD manifolds into three families of parameterized Lie groups. Speci􀀂c normalization layers induced by these Lie groups are then proposed for SPD neural networks. We demonstrate the effectiveness of our approach through three sets of experiments: radar recognition, human action recognition, and electroencephalography (EEG) classi􀀂cation. The code is available at [https://github.com/GitZH-Chen/LieBN.git](https://github.com/GitZH-Chen/LieBN.git).  \n1 INTRODUCTION  \nOver the past decade or so, Deep Neural Networks (DNNs) have achieved remarkable progress across various scienti􀀂c 􀀂elds (Hochreiter & Schmidhuber, 1997; Krizhevsky et al., 2012; He et al., 2016; Vaswani et al., 2017) . Conventionally, DNNs have been developed with the underlying assumption of the Euclidean geometry inherent to input data. Nonetheless, there exists a plethora of applications wherein the latent spaces are de􀀂ned by non-Euclidean structures such as manifolds (Bronstein et al., 2017) . To address this issue, researchers have attempted to extend various types of DNNs to manifolds based on the theories of Riemannian geometry (Huang & Van Gool, 2017; Huang et al., 2017; 2018; Ganea et al., 2018; Chakraborty et al., 2018; Brooks et al., 2019a;e;c;d; Brooks, 2020; Brooks et al., 2020; Chen et al., 2020; Chakraborty et al., 2020; Chakraborty, 2020; Chen et al., 2021; Wang et al., 2022b;a; Nguyen, 2022a;b; Nguyen & Yang, 2023; Chen et al., 2023c;e;b; Wang et al., 2024) .  \nMotivated by the great success of normalization techniques within DNNs (Ioffe & Szegedy, 2015; Ba et al., 2016; Ulyanov et al., 2016; Wu & He, 2018; Chen et al., 2023a), researchers have sought to devise normalization layers tailored for manifold-valued data. Brooks et al. (2019b) introduced Riemannian Batch Normalization (RBN) speci􀀂cally designed for SPD manifolds, with the ability to regulate the Riemannian mean. This approach was further re􀀂ned in Kobler et al. (2022b) to extend the control over the Riemannian variance. However, the above methods are constrained within the af􀀂ne-invariant metric (AIM) on SPD manifolds, limiting their applicability and generality. On the other hand, Chakraborty (2020) proposed two distinct Riemannian normalization frameworks, one tailored for Riemannian homogeneous spaces and the other catering to matrix Lie groups. Nonetheless, the normalization designed for Riemannian homogeneous spaces cannot regulate mean nor variance, while the normalization approach for matrix Lie groups is con􀀂ned to a speci􀀂c type of distance (Chakraborty, 2020, Sec. 3.2). A principled Riemannian normalization framework capable of controlling both Riemannian mean and variance remains unexplored.  \n􀀃 Corresponding author  \nPublished as a conference paper at ICLR 2024   \nGiven that Batch Normalization (","cbCaiqqDPym5qIIY","https://ap.wps.com/l/cbCaiqqDPym5qIIY","pdf",432730,2,1,25,"English","en",105,"# Abstract\n# Introduction\n# Preliminaries\n## Lie group background\n## Geometry of SPD manifolds","[{\"question\":\"What problem does the paper address in existing Riemannian batch normalization methods?\",\"answer\":\"Most existing Riemannian normalization approaches are derived ad hoc and restricted to specific manifolds, with limited ability to control statistical moments in a principled way.\"},{\"question\":\"How does the proposed framework ensure control over Riemannian statistics?\",\"answer\":\"The paper introduces a unified Riemannian Batch Normalization framework on Lie groups with theoretical guarantees to control both the Riemannian mean and the Riemannian variance.\"},{\"question\":\"Why are symmetric positive definite (SPD) manifolds used for experiments, and how are the Lie groups constructed there?\",\"answer\":\"SPD manifolds are chosen because they admit three distinct Lie group structures. The paper generalizes these into three families of parameterized Lie groups using a deformation concept, then derives normalization layers for SPD neural networks.\"}]",1784173791,63,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"a-lie-group-approach-to-riemannian-batch-normalization","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/a-lie-group-approach-to-riemannian-batch-normalization/81491/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","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 the paper address in existing Riemannian batch normalization methods?","Question",{"text":75,"@type":76},"Most existing Riemannian normalization approaches are derived ad hoc and restricted to specific manifolds, with limited ability to control statistical moments in a principled way.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed framework ensure control over Riemannian statistics?",{"text":80,"@type":76},"The paper introduces a unified Riemannian Batch Normalization framework on Lie groups with theoretical guarantees to control both the Riemannian mean and the Riemannian variance.",{"name":82,"@type":73,"acceptedAnswer":83},"Why are symmetric positive definite (SPD) manifolds used for experiments, and how are the Lie groups constructed there?",{"text":84,"@type":76},"SPD manifolds are chosen because they admit three distinct Lie group structures. The paper generalizes these into three families of parameterized Lie groups using a deformation concept, then derives normalization layers for SPD neural networks.","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":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]