[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85978-en":3,"doc-seo-85978-105":30,"detail-sidebar-cat-0-en-105":83},{"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},85978,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","Sharp Concentration Bounds for Bundle-Valued Statistics on Manifolds","Many geometric statistics and manifold learning pipelines generate observations—such as tangent vectors or local frames—living in fibers attached to points of a base manifold rather than in a single vector space. Transporting these observations to a common reference fiber induces curvature- and holonomy-driven effects absent from classical concentration theory. The work develops non-asymptotic, finite-sample, dimension-free Hoeffding and Bernstein-type concentration bounds via sharp Hilbert-space inequalities, including deterministic holonomy bias and an unavoidable bias–variance error floor. It further provides minimax lower bounds, a median-of-means estimator for heavy tails, and a central limit theorem, with sphere experiments validating results.","arXiv :2607 . 10592v 1 [ cs .LG] 12 Jul 2026  \nSharp Concentration Bounds for Bundle-Valued Statistics on Manifolds  \nSwagatam Das*  \nElectronics and Communication Sciences Unit  \nIndian Statistical Institute  \nKolkata 700108, India  \nVclav Snel VSB Technical University of Ostrava  \nOstrava, Czech Republic  \nAbstract  \nMany geometric statistics and manifold learning pipelines routinely produce observations—such as tangent vectors or local frames—whose natural home is a varying family of fibers attached to different points of a base manifold, rather than a single shared vector space. Forming empirical averages requires transporting these observations to a common reference fiber, thereby introducing curvature-and holonomy-driven effects that are absent from classical concentration theory. We develop a non-asymptotic concentration theory for such transported empirical means, deriving finite-sample, dimension-free Hoeffdingand Bernstein-type bounds via sharp Hilbert-space inequalities. When shortest paths to the reference point are non-unique, transport becomes path-dependent and introduces a deterministic holonomy bias; we isolate and quantify this bias through bundle curvature and loop geometry, with sharp closed-form formulas for the tangent bundle of a round sphere. The resulting bias–variance decomposition separates the stochastic fluctuation decaying atthe classical n−1/2 rate in sample size n, from a curvature-driven error floor that no amount of additional data can eliminate; minimax lower bounds confirm both terms are unavoidable. We further establish a robust median-of-means estimator achieving optimal rates under heavy tails, and a central limit theorem in the reference fiber. Controlled experiments on the sphere validate all theoretical predictions.  \nKeywords: Concentration inequalities, Riemannian statistics, vector bundles, parallel transport, holonomy, geometric machine learning.  \n1 Introduction  \nConsider the problem of representing a global wind map. At every point on the Earth’s surface, we must attach a small, flat coordinate system to describe the local wind’s speed and direction.  \n* Corresponding author: [swagatam.das@isical.ac.in](swagatam.das@isical.ac.in)  \nIn the language of geometry, this collection of “data spaces attached to points” is a vector bundle. While we can easily draw a flat grid on a local map of a single city, the Earth’s intrinsic curvature makes it impossible to “comb” these local coordinate systems into one single, consistent grid for the entire planet without creating a topological “twist” or a seam.  \nThis geometric tension has a precise algebraic counterpart. A vector bundle (E,π, M) formalizes exactly this picture: M is the base manifold (the Earth’s surface, in our example), E is the total space collecting all local coordinate systems together, and π : E → M is a smooth surjection—a projection map—that tells you which point of M each local coordinate system is attached to. At each point x ∈ M , the preimage Ex := π −1(x)  Rk is a vector space fiber, playing the role of the local coordinate system for wind direction and speed at x. A wind map is then a section of the bundle—a smooth assignment s : M → E satisfying π (s(x)) = x, so that s(x) ∈ Ex at every point, meaning each location on Earth is assigned a wind vector living in its own local coordinate system. The impossibility of a globally consistent grid reflects the fact that E may be globally “twisted,” precluding a single coordinate system even though it locally resembles a product space. This is the natural language for geometric ML pipelines encompassing manifolds, graphs, and Lie groups, where one wishes to capture coordinate-invariant features such as directions or local frames [15] . It also mirrors the structure of gauge fields in physics, where curvature governs both field strength and, as we shall see, the quantification of statistical uncertainty.  \nWhile concentration inequalities are central to non-asymptotic learnin","cbCaibBr5KbTFSio","https://ap.wps.com/l/cbCaibBr5KbTFSio","pdf",601409,3,1,65,"English","en",105,"# Introduction\n## Bundle-valued observations and vector bundles\n## Transport, holonomy, and limitations of Euclidean concentration\n## Problem setup and contributions","[{\"question\":\"What robustness and asymptotic results are provided beyond concentration?\",\"answer\":\"A robust median-of-means estimator achieves optimal rates under heavy tails, and a central limit theorem holds in the reference fiber; sphere experiments confirm the theory.\"}]",1784207525,164,{"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":78,"head_meta":80,"extra_data":82,"updated_unix":28},"sharp-concentration-bounds-for-bundle-valued-statistics-on-manifolds","",{"@graph":36,"@context":77},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/sharp-concentration-bounds-for-bundle-valued-statistics-on-manifolds/85978/",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","2026-07-26","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"What robustness and asymptotic results are provided beyond concentration?","Question",{"text":75,"@type":76},"A robust median-of-means estimator achieves optimal rates under heavy tails, and a central limit theorem holds in the reference fiber; sphere experiments confirm the theory.","Answer","https://schema.org",{"og:url":51,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,115,120,123,127],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":98,"slug":130},19,"General","general"]