Some properties of Fr\'echet medians in Riemannian manifolds

Abstract

The consistency of Fr\'echet medians is proved for probability measures in proper metric spaces. In the context of Riemannian manifolds, assuming that the probability measure has more than a half mass lying in a convex ball and verifies some concentration conditions, the positions of its Fr\'echet medians are estimated. It is also shown that, in compact Riemannian manifolds, the Fr\'echet sample medians of generic data points are always unique.