Finite sample bounds for barycenter estimation in geodesic spaces
Statistics Theory
2025-02-25 v2 Probability
Machine Learning
Statistics Theory
Abstract
We study the problem of estimating the barycenter of a distribution given i.i.d. data in a geodesic space. Assuming an upper curvature bound in Alexandrov's sense and a support condition ensuring the strong geodesic convexity of the barycenter problem, we establish finite-sample error bounds in expectation and with high probability. Our results generalize Hoeffding- and Bernstein-type concentration inequalities from Euclidean to geodesic spaces. Building on these concentration inequalities, we derive statistical guarantees for two efficient algorithms for the computation of barycenters.
Cite
@article{arxiv.2502.14069,
title = {Finite sample bounds for barycenter estimation in geodesic spaces},
author = {Victor-Emmanuel Brunel and Jordan Serres},
journal= {arXiv preprint arXiv:2502.14069},
year = {2025}
}