Riemannian Median and Its Estimation

TL;DR

This paper introduces the Riemannian median for probability measures on manifolds, characterizes its properties, and proposes a convergent subgradient algorithm extending Euclidean methods to Riemannian spaces.

Contribution

It defines the Riemannian median, provides conditions for its uniqueness, and develops a practical subgradient algorithm with convergence guarantees.

Findings

Proved the convergence of the proposed subgradient algorithm.

Established error bounds and convergence rates.

Extended the classical Weiszfeld algorithm to Riemannian manifolds.

Abstract

In this paper, we define the geometric median of a probability measure on a Riemannian manifold, give its characterization and a natural condition to ensure its uniqueness. In order to calculate the median in practical cases, we also propose a subgradient algorithm and prove its convergence as well as estimating the error of approximation and the rate of convergence. The convergence property of this subgradient algorithm, which is a generalization of the classical Weiszfeld algorithm in Euclidean spaces to the context of Riemannian manifolds, also answers a recent question in P. T. Fletcher et al. [13]