[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82131-en":3,"doc-seo-82131-105":29,"detail-sidebar-cat-0-en-105":90},{"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":4,"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},82131,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","Group Invariant Spectral Embedding","Spectral embedding methods for dimensionality reduction and clustering rely on graph Laplacians built from pairwise affinities, but they typically ignore known symmetries, incorrectly treating symmetry-related samples as distinct. This work proposes G-invariant affinity kernels that directly encode symmetry into the similarity measure. For data on a Riemannian manifold M with symmetries from a compact Lie group G, three families of invariant kernels yield graph Laplacians converging to explicit second-order differential operators on the quotient space M/G. The effective dimension decreases with dim(G), improving convergence rates and sample efficiency, validated on SO(2) and SO(3) symmetric datasets.","arXiv :2607 .08987v 1 [ cs .LG] 9 Jul 2026  \nGroup Invariant Spectral Embedding  \nYeari Vigder 1,*,†, Paulina Hoyos2,*,†, David Thong3 , Joakim And´en3 , Joe Kileel2 , and Amit  \nMoscovich 1  \n1 Department of Statistics and Operations Research, Tel Aviv University, Tel Aviv, Israel [mosco@tauex. tau.ac. il](mosco@tauex. tau.ac. il) (A. Moscovich), [vigderyeari@mail. tau.ac. il](vigderyeari@mail. tau.ac. il) (Y. Vigder)  \n2 Department of Mathematics, University of Texas at Austin, Austin, TX, USA [paulinah@utexas. edu](paulinah@utexas. edu) (P. Hoyos), [jkileel@math.utexas. edu](jkileel@math.utexas. edu) (J. Kileel)  \n3 Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden [dthong@kth.se](dthong@kth.se) (D. Thong), [janden@kth.se](janden@kth.se) (J. And´en)  \n*  \nEqual contribution  \n† Corresponding author  \nJuly 13, 2026  \nAbstract  \nSpectral embedding methods are widely used for dimensionality reduction and clustering of highdimensional datasets with intrinsic low-dimensional structures. Although many datasets of practical interest exhibit invariance under symmetries such as rotations, standard spectral embedding methods do not account for this, treating symmetry-related data points as unrelated. Our approach to this problem is to incorporate the symmetries directly into the affinity kernels used for spectral embedding. We analyze the case of a Riemannian data manifold M with symmetries given by a compact Lie group G and prove that, under suitable conditions, graph Laplacians constructed from three types of invariant kernels converge pointwise to explicit second-order differential operators on the quotient space M/G. Our analysis implies improved convergence rates, as the effective dimension drops according to the dimension of the group. We validate our approach on datasets with SO(2) or SO(3) symmetry, and show that G-invariant spectral embedding recovers the intrinsic geometry of the data, in contrast to standard spectral embedding, which fails to do so even in the limit of infinite data.  \nKeywords: dimensionality reduction, manifold learning, graph Laplacian, data symmetries, quotient manifold, sample complexity  \nMSC 2020: 62R07, 62R30, 58J70, 58J50, 35R02  \n1 Introduction  \nSpectral embedding methods are powerful tools for analyzing high-dimensional data with intrinsic low-dimensional structure. Their basic operating principle is to construct a graph from pairwise affinities between data points and then use the eigenvectors of the graph Laplacian as low-dimensional representations. These methods are widely used for tasks such as dimensionality reduction, clustering, semi-supervised learning, and data denoising. In this work, we consider datasets that exhibit invariance under known symmetry transformations. For example, in single-particle cryo-electron microscopy (cryo-EM), molecular projection images are subject to random in-plane rotations. The 2D rotations are typically viewed as nuisance parameters, which computational methods need to account for (Singer and Sigworth, 2020) . As an example of a discrete symmetry, consider set-structured data such as unlabeled graphs, stored using an adjacency matrix. Despite the vertices having no natural ordering, they are nonetheless assigned arbitrary row/column indices. However applying the same permutation on the rows and columns of the matrix results in an identical graph (Zaheer et al. , 2017; Maronet al. , 2019) .  \nb  \na  \n\n| x x′ = g · x\u003Cbr> |  |  |\n| --- | --- | --- |\n|  | y |  |\n\nb  \na  \nM = T2  \ndN ([x], [y ])  \n[x] = [x′]  \n[y]  \nN = M/G  S1 endpoints identified  \nFigure 1: An illustration of group orbits and the quotient manifold. The flat torus M = T 2 is drawn as a fundamental square with opposite edges identified. The group G = S 1 acts by translations in the a-direction, so each horizontal orbit is one equivalence class: x and x′ = g · x represent the same quotient point, while y lies on a distinct orbit. The quotient map π : M → N = M/G c","cbCaimEGcezsJp9D","https://ap.wps.com/l/cbCaimEGcezsJp9D","pdf",3953183,1,28,"English","en",105,"# Introduction\n## Our Contributions","[{\"question\":\"What problem does the paper address in standard spectral embedding?\",\"answer\":\"Standard spectral embedding builds affinities without accounting for known symmetries, so symmetry-related data points are treated as unrelated observations.\"},{\"question\":\"How does the proposed method incorporate symmetries?\",\"answer\":\"It uses G-invariant affinity kernels in the spectral embedding pipeline, effectively treating each data point as a representative of its group orbit (equivalence class).\"},{\"question\":\"What theoretical result is proved about the convergence of graph Laplacians?\",\"answer\":\"For a Riemannian manifold M with symmetries from a compact Lie group G, graph Laplacians constructed from three types of invariant kernels converge pointwise to explicit second-order differential operators on the quotient space M/G, with improved rates as the effective dimension drops by dim(G).\"}]",1784178372,71,{"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":85,"head_meta":87,"extra_data":89,"updated_unix":27},"group-invariant-spectral-embedding","",{"@graph":35,"@context":84},[36,53,67],{"@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-invariant-spectral-embedding/82131/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What problem does the paper address in standard spectral embedding?","Question",{"text":74,"@type":75},"Standard spectral embedding builds affinities without accounting for known symmetries, so symmetry-related data points are treated as unrelated observations.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the proposed method incorporate symmetries?",{"text":79,"@type":75},"It uses G-invariant affinity kernels in the spectral embedding pipeline, effectively treating each data point as a representative of its group orbit (equivalence class).",{"name":81,"@type":72,"acceptedAnswer":82},"What theoretical result is proved about the convergence of graph Laplacians?",{"text":83,"@type":75},"For a Riemannian manifold M with symmetries from a compact Lie group G, graph Laplacians constructed from three types of invariant kernels converge pointwise to explicit second-order differential operators on the quotient space M/G, with improved rates as the effective dimension drops by 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