[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-203638-105":59,"doc-detail-203638-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","gaussian-rbf-centered-kernel-alignment-cka-in-the-large-bandwidth-limit","Gaussian RBF Centered Kernel Alignment (CKA) - in the Large-Bandwidth Limit","","Centered kernel alignment (CKA), also called centered kernel-target alignment, serves as a similarity measure between kernels and between feature representations via kernel-based similarity matrices. The paper proves that CKA computed from a Gaussian RBF kernel converges to linear CKA in the large-bandwidth limit. The analysis depends on mean-centering of feature maps and Hilbert-Schmidt Independence Criterion (HSIC) identities, and shows convergence onset depends on representation geometry through an eccentricity parameter ρ. Experiments indicate selecting Gaussian bandwidth below ρ to retain nonlinear modeling.",{"@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/gaussian-rbf-centered-kernel-alignment-cka-in-the-large-bandwidth-limit/203638/",{"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/gaussian-rbf-centered-kernel-alignment-cka-in-the-large-bandwidth-limit/203638.png","ImageObject",300,407,{"name":92,"@type":93},"Terk","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-28","2026-09-04",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",14,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What does the paper show about Gaussian CKA in the large-bandwidth limit?","Question",{"text":113,"@type":114},"It proves that Gaussian RBF CKA converges to linear CKA as the Gaussian bandwidth approaches infinity for all representations.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"Which assumptions or identities does the convergence result rely on?",{"text":118,"@type":114},"The result relies on mean-centering of the feature maps and on Hilbert-Schmidt Independence Criterion (HSIC) identities.",{"name":120,"@type":111,"acceptedAnswer":121},"How does representation geometry affect when Gaussian CKA differs from linear CKA?",{"text":122,"@type":114},"Convergence onset is sensitive to the geometry of the feature representations, captured by a representation eccentricity parameter ρ, which constrains the bandwidth range where differences remain noticeable.","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},203638,1788562348,{"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":130,"read_time":41,"preview_page_count":4},1099525198933,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","IEEE TRANS. PAMI  \n© 20xx IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.  \n1  \nGaussian RBF Centered Kernel Alignment (CKA)  \nin the Large-Bandwidth Limit  \nSergio A. Alvarez  \nAbstract—Centered kernel alignment (CKA), also known as centered kernel-target alignment, is useful as a similarity measure between kernels and as a kernel-based similarity measure between feature representations. We prove that CKA based on a Gaussian RBF kernel converges to linear CKA in the large-bandwidth limit. The result relies on mean-centering of the feature maps and on a Hilbert-Schmidt Independence Criterion (HSIC) identity. We show that convergence onset is sensitive to the geometry of the feature representations, and that a notion of representation eccentricity, ρ, constrains the bandwidth range for which Gaussian CKA can differ noticeably from linear CKA. Our experimental results suggest that Gaussian bandwidths less than ρ should be selected in order to enable nonlinear modeling.  \nIndex Terms—Nonlinear kernels, neural networks, representations, similarity.  \n~~ ~~ ✦ ~~ ~~  \n1 INTRODUCTION  \nCENTERED Kernel Alignment (CKA) was first proposed  \nas a measure of similarity between kernels in the context of kernel learning [1], [2], building on prior work on (non-centered) kernel-target alignment [3] . Versions for functional data of CKA and the associated Hilbert-Schmidt Independence Criterion (HSIC) were proposed later [4] . CKA has been used for multiple kernel learning (e.g., [5], [6]) . CKA also enables measuring similarity between two feature representations of a set of data examples (e.g., two sets of neural network activation vectors), by comparing kernel-based similarity matrices (Gram matrices) of these representations [7] . Used in the latter manner, CKA and alternatives such as canonical correlation analysis (CCA) and orthogonal Procrustes distance [8] can provide insight into the relationship between architectural features of a network such as width and depth, and the network’s learned representations [7],[9] . CKA has been used to assess representation similarity in cross-lingual language models [10] andrecurrent neural network dynamics [11], to identify potential drug side-effects [12], to differentiate among brain activity patterns [13], and to gauge the representational effects of watermarking [14], among others.  \nAny positive-definite symmetric kernel can be used asthe base kernel for CKA, including linear, polynomial, and Gaussian radial basis function (RBF) kernels. Gaussian RBF kernels are of special interest due to their universality properties [15], which can lead to superior modeling of arbitrary nonlinearities. Understanding the behavior of CKA with Gaussian kernels is, therefore, relevant to its use asa representational similarity metric. Some prior work [7] reports finding little difference empirically between CKA similarity values based on Gaussian and linear kernels. In contrast, hyperparameter tuning experiments for [16] suggest that a noticeable difference can occur between CKA values for Gaussian and linear kernels, but also that the difference becomes negligible as Gaussian bandwidth  \n• S. A. Alvarez is with the Department of Computer Science, Boston College, Chestnut Hill, MA 02467 USA.  \nE-mail: [alvarez@bc.edu](alvarez@bc.edu)  \ngrows. We establish the latter phenomenon theoretically in the present paper, proving that Gaussian CKA converges to linear CKA as bandwidth approaches infinity, for all representations. We describe the convergence rate, as well.  \nWe are not aware of published results on the largebandwidth asymptotics of Gaussian CKA. A limit result for Gaussian kernel SVM c","cbCaiikTLVfoW9Mw","https://ap.wps.com/l/cbCaiikTLVfoW9Mw","pdf",890513,12,"English","# 1 Introduction\n# 2 Background and Notation\n## 2.1 Feature representations\n## 2.2 Kernel similarity","[{\"question\":\"What does the paper show about Gaussian CKA in the large-bandwidth limit?\",\"answer\":\"It proves that Gaussian RBF CKA converges to linear CKA as the Gaussian bandwidth approaches infinity for all representations.\"},{\"question\":\"Which assumptions or identities does the convergence result rely on?\",\"answer\":\"The result relies on mean-centering of the feature maps and on Hilbert-Schmidt Independence Criterion (HSIC) identities.\"},{\"question\":\"How does representation geometry affect when Gaussian CKA differs from linear CKA?\",\"answer\":\"Convergence onset is sensitive to the geometry of the feature representations, captured by a representation eccentricity parameter ρ, which constrains the bandwidth range where differences remain noticeable.\"}]","Gaussian RBF Centered Kernel Alignment (CKA) - in the Large-Bandwidth Limit | PDF"]