Geodesic slice sampling on Riemannian manifolds

Summary We propose a theoretically justified and practically applicable slice-sampling-based Markov chain Monte Carlo method for approximate sampling from probability measures on Riemannian manifolds. The latter naturally arise as posterior distributions in Bayesian inference of matrix-valued parameters, for example belonging to either the Stiefel or the Grassmann manifold. Our method, called geodesic slice sampling, generalizes hit-and-run slice sampling on ℝd to Riemannian manifolds by abstracting straight lines to geodesics. It is reversible with respect to the distribution of interest and converges to the latter in total variation distance. We demonstrate the robustness of our sampler’s performance compared to other Markov chain Monte Carlo methods dealing with manifold-valued distributions through extensive numerical experiments, on both synthetic and real data. In particular, we illustrate the sampler’s remarkable ability to cope with anisotropic target densities, without using gradient information and preconditioning.