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--Mérigot 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 Hölder-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.
| Comments: | Comments are welcome! |
| Subjects: | Metric Geometry (math.MG); Functional Analysis (math.FA); Probability (math.PR) |
| MSC classes: | 49Q22, 53C23, 60B10 |
| Cite as: | arXiv:2605.25448 [math.MG] |
| (or arXiv:2605.25448v1 [math.MG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.25448 arXiv-issued DOI via DataCite (pending registration) |
From: Bangxian Han [view email]
[v1]
Mon, 25 May 2026 05:58:49 UTC (18 KB)
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