Sharp Concentration Bounds for Bundle-Valued Statistics on Manifolds

2026-07-12

AI Summary

This work addresses the non-negligible bias in empirical averages on manifold fiber bundles—such as tangent bundles—where parallel transport is distorted by curvature and holonomy, rendering classical concentration inequalities invalid. By parallel-translating observations to a common reference fiber, the authors establish a non-asymptotic concentration theory that, for the first time, quantifies the holonomy-induced bias arising from curvature and loop geometry. They introduce a bias–variance decomposition wherein the variance decays at the standard $n^{-1/2}$ rate, while the bias constitutes an irreducible lower bound. Furthermore, they construct a median-of-means estimator that achieves optimal convergence rates under heavy-tailed distributions. The theory yields a closed-form expression for the bias on the tangent bundle of the unit sphere, and experiments confirm both the inevitability of this bias and the robustness of the proposed estimator.

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.

Problem

• manifold
• bundle-valued statistics
• holonomy
• curvature
• concentration bounds

Innovation

• bundle-valued statistics
• non-asymptotic concentration
• holonomy bias
• curvature-driven error
• median-of-means estimator