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Finite sample bounds for barycenter estimation in geodesi...
[Submitted on 19 Feb 2025 (v1), last revised 6 Jul 2026 (this ve · 2025-02-20 · via stat.ML updates on arXiv.org

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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.

Submission history

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)