Medians and means in Riemannian geometry: existence, uniqueness and computation

TL;DR

This paper explores the existence, uniqueness, and computation of medians and means in Riemannian manifolds, introducing algorithms and applications in radar target detection.

Contribution

It provides new theoretical results on medians and means in Riemannian geometry, along with practical algorithms and applications.

Findings

Existence and uniqueness of local medians established.

Convergence of subgradient algorithms proven.

Application to radar target detection demonstrated.

Abstract

This paper is a short summary of our recent work on the medians and means of probability measures in Riemannian manifolds. Firstly, the existence and uniqueness results of local medians are given. In order to compute medians in practical cases, we propose a subgradient algorithm and prove its convergence. After that, Fr\'echet medians are considered. We prove their statistical consistency and give some quantitative estimations of their robustness with the aid of upper curvature bounds. We also show that, in compact Riemannian manifolds, the Fr\'echet medians of generic data points are always unique. Stochastic and deterministic algorithms are proposed for computing Riemannian p-means. The rate of convergence and error estimates of these algorithms are also obtained. Finally, we apply the medians and the Riemannian geometry of Toeplitz covariance matrices to radar target detection.