Projected Langevin Dynamics And A Gradient Flow For Entropic Optimal Transport

The classical Langevin diffusion provides a natural algorithm for sampling from its invariant measure, which uniquely minimizes an energy functional over the space of probability measures. We introduce analogous diffusion process that samples from an entropy-regularized optimal transport (a.k.a. Schrodinger bridge), which uniquely minimizes the same energy functional but constrained to the set of couplings of two given marginal probability measures. The law of the solution remains a coupling at each time if initialized as such. In addition, we show by means of a new logarithmic Sobolev inequality that the convergence holds exponentially fast, for sufficiently large regularization parameter and for (asymptotically) log-concave marginals. The dynamics are related, in the spirit of Otto calculus, to a gradient flow on the space of couplings, viewed as a submanifold of Wasserstein space. Based on join work with Giovanni Conforti and Soumik Pal.