Convergence of the Riemannian Langevin Algorithm
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We study the Riemannian Langevin Algorithm for the problem of sampling from a distribution with density ν 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 ν for Hessian manifolds. This allows us to reduce the problem of sampling non-smooth (constrained) densities in R^n 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.
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