Sharp Concentration Bounds for Bundle-Valued Statistics on Manifolds
2026-07-12 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study how to average data points that live on curved spaces, like directions attached to different points on a sphere, rather than in regular flat spaces. They show that to average such data, one must move them to a common point, but this movement introduces errors due to the space's curvature and the paths taken. The authors develop mathematical bounds that describe these errors precisely and prove that some bias caused by curvature cannot be removed by collecting more data. They also propose a robust averaging method that works well even with unusual data and confirm their results with experiments on a sphere.
manifoldfiber bundleparallel transportcurvatureholonomyHoeffding inequalityBernstein inequalitymedian-of-means estimatorminimax lower boundscentral limit theorem
Authors
Swagatam Das, Vaclav Snasel
Abstract
Many 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 Hoeffding- and 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 at the 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 the central limit theorem in the reference fiber. Controlled experiments on the sphere validate all theoretical predictions.