Some properties of Fr\'echet medians in Riemannian manifolds

TL;DR

This paper investigates properties of Fréchet medians in Riemannian manifolds, proving their consistency, estimating median positions under certain conditions, and establishing uniqueness for generic data in compact manifolds.

Contribution

It provides new theoretical results on the consistency, localization, and uniqueness of Fréchet medians in Riemannian geometry.

Findings

Fréchet medians are consistent in proper metric spaces.

Median positions can be estimated under specific measure conditions.

Unique medians are guaranteed for generic data in compact 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.