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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.
Cited literature [64 references]
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
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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1806.02740.pdf
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