Convergence of the Riemannian Langevin Algorithm

We study the Riemannian Langevin Algorithm for the problem of sampling from a distribution with density nu with respect to the natural measure on a manifold with metric g. We assume that the target density satisfies a log-Sobolev inequality with respect to the metric and prove that the manifold generalization of the Unadjusted Langevin Algorithm converges rapidly to nu for Hessian manifolds. This allows us to reduce the problem of sampling non-smooth (constrained) densities in bf Rn to sampling smooth densities over appropriate manifolds, while needing access only to the gradient of the log-density, and this, in turn, to sampling from the natural Brownian motion on the manifold. Our main analytic tools are (1) an extension of self-concordance to manifolds, and (2) a stochastic approach to bounding smoothness on manifolds. A special case of our approach is sampling isoperimetric densities restricted to polytopes by using the metric defined by the logarithmic barrier.