Approximating covering and minimum enclosing balls in hyperbolic geometry

Abstract

We generalize the Formula Presented -time (1 + ∈)-approximation algorithm for the smallest enclosing Euclidean ball [2,10] to point sets in hyperbolic geometry of arbitrary dimension. We guarantee a O (1/∈2) convergence time by using a closed-form formula to compute the geodesic α-midpoint between any two points. Those results allow us to apply the hyperbolic k-center clustering for statistical location-scale families or for multivariate spherical normal distributions by using their Fisher information matrix as the underlying Riemannian hyperbolic metric.