On approximating the Riemannian 1-center

We generalize the Euclidean 1-center approximation algorithm of Bădoiu and Clarkson (2003) [6] to arbitrary Riemannian geometries, and study the corresponding convergence rate. We then show how to instantiate this generic algorithm to two particular settings: (1) the hyperbolic geometry, and (2) the Riemannian manifold of symmetric positive definite matrices.