On Approximating the Riemannian 1-Center

TL;DR

This paper extends a Euclidean 1-center approximation algorithm to Riemannian geometries, analyzing its convergence and applying it to hyperbolic space and positive definite matrices.

Contribution

It generalizes a known Euclidean algorithm to Riemannian settings and demonstrates its application to specific geometries.

Findings

Algorithm converges efficiently in hyperbolic space

Effective approximation method for positive definite matrices

Provides convergence rate analysis for Riemannian cases

Abstract

In this paper, we generalize the simple Euclidean 1-center approximation algorithm of Badoiu and Clarkson (2003) to Riemannian geometries and study accordingly the convergence rate. We then show how to instantiate this generic algorithm to two particular cases: (1) hyperbolic geometry, and (2) Riemannian manifold of symmetric positive definite matrices.