Sharp Concentration Bounds for Bundle-Valued Statistics on Manifolds

Many geometric statistics and manifold learning pipelines routinely produce observationssuch 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 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 a central limit theorem in the reference fiber. Controlled experiments on the sphere validate all theoretical predictions.