Riemannian median and its estimation

In this paper, we define the geometric median for a probability measure on a Riemannian manifold, give its characterization and a natural condition to ensure its uniqueness. In order to compute the geometric median in practical cases, we also propose a subgradient algorithm and prove its convergence as well as estimating the error of approximation and the rate of convergence. The convergence property of this subgradient algorithm, which is a generalization of the classical Weiszfeld algorithm in Euclidean spaces to the context of Riemannian manifolds, also improves a recent result of P. T. Fletcher et al . [ NeuroImage 45 (2009) S143–S152].