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.
From: Victor-Emmanuel Brunel [view email]
[v1]
Wed, 19 Feb 2025 19:44:54 UTC (62 KB)
[v2]
Sat, 22 Feb 2025 14:13:18 UTC (65 KB)
[v3]
Mon, 6 Jul 2026 13:44:01 UTC (62 KB)
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