On Approximating the Riemannian 1-Center

Marc Arnaudon (1) , Frank Nielsen (2, 3)

1 LMA [Poitiers] - Laboratoire de mathématiques et applications [UMR 6086]
2 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
3 Sony Computer Science Laboratories [Tokyo, Japan]

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.

Mots clés: 1-center; minimax; circumcenter; Riemannian geometry; core-set; approximation

Domaines: Géométrie algorithmique [cs.CG]; Géométrie différentielle [math.DG]

Marc Arnaudon, Frank Nielsen. On Approximating the Riemannian 1-Center. 2011. ⟨hal-00560187v1⟩