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arXiv · 2502.14069

Finite sample bounds for barycenter estimation in geodesic spaces

Victor-Emmanuel Brunel·Jordan Serres
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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.

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Victor-Emmanuel Brunel, Jordan Serres. 2025-02-19. Finite sample bounds for barycenter estimation in geodesic spaces. https://arxiv.org/abs/2502.14069

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