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On the rate of convergence of empirical barycentres in metric spaces: curvature, convexity and extendible geodesics

Abstract : This paper provides rates of convergence for empirical (generalised) barycenters on compact geodesic metric spaces under general conditions using empirical processes techniques. Our main assumption is termed a variance inequality and provides a strong connection between usual assumptions in the field of empirical processes and central concepts of metric geometry. We study the validity of variance inequalities in spaces of non-positive and non-negative Aleksandrov curvature. In this last scenario, we show that variance inequalities hold provided geodesics, emanating from a barycenter, can be extended by a constant factor. We also relate variance inequalities to strong geodesic convexity. While not restricted to this setting, our results are largely discussed in the context of the $2$-Wasserstein space.
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https://hal.archives-ouvertes.fr/hal-01810530
Contributor : Thibaut Le Gouic Connect in order to contact the contributor
Submitted on : Monday, June 17, 2019 - 3:17:41 PM
Last modification on : Friday, January 7, 2022 - 9:52:02 AM

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  • HAL Id : hal-01810530, version 3
  • ARXIV : 1806.02740

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Adil Ahidar-Coutrix, Thibaut Le Gouic, Quentin Paris. On the rate of convergence of empirical barycentres in metric spaces: curvature, convexity and extendible geodesics. 2019. ⟨hal-01810530v3⟩

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