Convergence of the Riemannian Langevin Algorithm
This provides a method for sampling from constrained distributions, which is incremental as it extends existing Langevin algorithms to manifolds.
The paper tackles the problem of sampling from non-smooth constrained densities in Euclidean space by reducing it to sampling smooth densities on manifolds using the Riemannian Langevin Algorithm, proving rapid convergence under a log-Sobolev inequality for Hessian manifolds.
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 ${\bf R}^n$ 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.