Sharp Concentration Bounds for Bundle-Valued Statistics on Manifolds

• URL: http://arxiv.org/abs/2607.10592v1
• Date: Sun, 12 Jul 2026 06:11:31 GMT
• Title: Sharp Concentration Bounds for Bundle-Valued Statistics on Manifolds
• Abstract summary: We develop a non-asymptotic concentration theory for transported empirical means.<n>We quantify and quantify this bias through bundle curvature and loop geometry.<n>We establish a robust median-of-means estimator achieving optimal rates under heavy tails.
• Score: 19.099044165107696
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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