On the Riemannian barycentre of a Markov chain

Abstract

The Riemannian barycentre is one of the most widely used statistical descriptors for probability distributions on Riemannian manifolds. At present, existing algorithms are able to compute the Riemannian barycentre of a probability distribution, only if i.i.d. samples of this distribution are readily available. However, there are many cases where i.i.d. samples are quite difficult to obtain, and have to be replaced with non-independent samples, generated by a Markov chain Monte Carlo method. To overcome this difficulty, the present paper proposes a new Markov chain Monte Carlo algorithm for computing the Riemannian barycentre of a probability distribution on a Hadamard manifold (a simply connected, complete Riemannian manifold with non-positive curvature). This algorithm relies on two original propositions, proved in the paper. The first proposition states that the recursive barycentre of samples generated from a geometrically ergodic Markov chain converges in the mean-square to the Riemannian barycentre of the stationary distribution of this chain. The second proposition provides verifiable conditions which ensure a Metropolis-Hastings Markov chain, with its values in a symmetric Hadamard manifold, is geometrically ergodic. This latter result yields a partial solution, in the context of Riemannian manifolds, to the problem of geometric ergodicity of Metropolis-Hastings chains, which has previously attracted extensive attention when considered in Euclidean space. In addition to these two propositions, the new Markov chain Monte Carlo algorithm, proposed in this paper, is applied to a problem of Bayesian inference, arising from computer vision.