Robust Statistics on Riemannian Manifolds via the Geometric Median

Abstract

The geometric median is a classic robust estimator of centrality for data in Euclidean spaces. In this paper we formulate the geometric median of data on a Riemannian manifold as the minimizer of the sum of geodesic distances to the data points. We prove existence and uniqueness of the geometric median on manifolds with non-positive sectional curvature and give sufficient conditions for uniqueness on positively curved manifolds. Generalizing the Weiszfeld procedure for finding the geometric median of Euclidean data, we present an algorithm for computing the geometric median on an arbitrary manifold. We show that this algorithm converges to the unique solution when it exists. This method produces a robust central point for data lying on a manifold, and should have use in a variety of vision applications involving manifolds. We give examples of the geometric median computation and demonstrate its robustness for three types of manifold data: the 3D rotation group, tensor manifolds, and shape spaces.