# Sharp Concentration Bounds for Bundle-Valued Statistics on Manifolds

arXiv: https://arxiv.org/abs/2607.10592 · HTML: https://mlsanity.com/paper/2607.10592/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 39 references that are in the map.
- First posted on arXiv 12 Jul 2026.

## Builds on (most cited first, 2 of 2)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Wasserstein Parallel Transport for Predicting the Dynamics of Statistical Systems](https://mlsanity.com/paper/2603.23736/) (2026)

## Built on by (most cited first, 1 of 1)

- [Riemannian Simultaneous Inference for Tangent Vector Field Regression](https://mlsanity.com/paper/2609.21910/) (2026)

## Abstract (arXiv)

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.