On the rate of convergence of empirical barycentres in metric spaces: curvature, convexity and extendible geodesics
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This paper provides rates of convergence for empirical barycentres of a Borel probability measure on a metric space under general conditions. Our results are given in the form of sharp oracle inequalities. Our main assumption, of geometrical nature, is shown to be satisfied at least in two meaningful scenarios. The first one is a form of weak curvature constraint of the underlying space referred to as (k, α)-convexity, compatible with a positive upper curvature bound. The second scenario considers the case of a nonnegatively curved space on which geodesics, emanating from a barycentre, can be extended.
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