Mirror Langevin diffusions: Convergence rates and Markov chain approximations
Given a strongly convex function $u$, equip $R^d$ with a Riemannian metric given by the Hessian $\nabla^2 u$. This is a so-called Hessian manifold. Given a probability density $\mu$ one may run a Langevin diffusion intrinsic to the manifold with stationary distribution $\mu$. Such (Hessian) manifold-valued Langevin diffusions are called Mirror Langevin diffusions (MLD) which have recently become popular. One of the questions we explore is whether, given $\mu$, one can choose $u$ to get an exponential convergence to equilibrium for the MLD, especially if $\mu$ is not strongly log-concave. Our results are based on Lyapunov function methods and give sufficient conditions for a Poincar\'e or a log-Sobolev inequality to hold for the MLD. These, in turn, imply exponential convergence. We also introduce a Markov chain approximation to the MLD given by a two step Gibbs sampler with stationary distribution $\mu$. This Markov chain is a variant of the Sinkhorn Markov chain introduced in arXiv:2307.16421 that is conjectured to converge to a time-inhomogeneous generalization of the MLD. Under suitable assumptions, we prove that the Markov chain has a guaranteed convergence rate in $\chi^2$ that is consistent with the diffusion time scale. Our proofs are based on ideas from entropic optimal transport and strong data processing inequalities.