Sharp Concentration Bounds for Bundle-Valued Statistics on Manifolds

In many geometric machine learning pipelines, data (such as tangent vectors or local frames) reside in varying fibers of a vector bundle rather than a single vector space. This paper addresses the statistical challenge of aggregating these observations: how can we form a reliable empirical mean when the underlying manifold's curvature makes parallel transport path-dependent and potentially ambiguous?

The authors propose a framework where bundle-valued observations are transported to a fixed reference fiber using a measurable transport rule. By reducing these observations to a single Hilbert space, they apply sharp concentration inequalities (Hoeffding and Bernstein types) to bound the deviation of the transported mean. Crucially, they introduce a bias-variance decomposition that separates the stochastic fluctuation—which vanishes as the sample size n increases—from a deterministic holonomy bias, which is governed by the bundle's curvature and the geometry of the paths chosen for transport.

The study establishes that the transported empirical mean achieves the optimal n^-1/2 convergence rate in the stochastic component, matching Euclidean expectations. However, it identifies an unavoidable 'error floor' caused by holonomy when shortest paths are non-unique. The authors provide explicit, dimension-free concentration bounds and derive sharp formulas for this bias on the tangent bundle of a round sphere. They also introduce a robust median-of-means estimator to handle heavy-tailed distributions and confirm the optimality of their bounds through minimax lower bounds.

This work provides a rigorous statistical foundation for geometric deep learning and manifold statistics. By explicitly quantifying the trade-off between sampling variability and curvature-driven ambiguity, the paper offers practitioners a clear guide for designing estimators in gauge-equivariant neural networks, intrinsic regression, and other tasks involving data on curved domains.