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arXiv · 2605.25448

Quantitative Stability of Wasserstein Barycenters over Alexandrov Spaces with Lower Curvature Bounds

Bang-Xian Han·Zhuo-Nan Zhu
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Abstract

We prove quantitative stability estimates for Wasserstein barycenters on Alexandrov spaces with curvature bounded from below. The proof combines the variational strategy of Carlier--Delalande--M\'erigot with heat-kernel regularization, which supplies the regularity needed for dual convexity arguments in this non-smooth curved setting. The main result is an explicit strong-convexity modulus for the barycentric variance functional. As a consequence, barycenters depend H\"older-continuously on the underlying distributions with respect to the $1$-Wasserstein distance on the space of probability measures. We derive empirical-barycenter consistency and entropy-based sample-complexity bounds. Our proof does not rely on linear structure; in particular, the resulting estimates appear to be new even on smooth compact Riemannian manifolds.

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Bang-Xian Han, Zhuo-Nan Zhu. 2026-05-25. Quantitative Stability of Wasserstein Barycenters over Alexandrov Spaces with Lower Curvature Bounds. https://arxiv.org/abs/2605.25448

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