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Projected Langevin dynamics and a gradient flow for entropic optimal transport
Abstract
The classical Langevin diffusion provides a natural algorithm for sampling from its invariant measure, which can be characterized as the unique minimizer of an energy functional over the space of probability measures. We introduce an 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 diffusion remains a coupling at each time if initialized as such. In addition, we show an exponential convergence rate by means of a new kind of logarithmic Sobolev inequality, in the case of sufficiently high temperature and (asymptotically) log-concave marginals. The dynamics can be viewed as a gradient flow on the space of couplings, viewed as a submanifold of Wasserstein space. Analogous constructions are possible for other constrained sampling problems, and as time permits I will discuss the surprisingly closely related example of mean field variational inference in Bayesian statistics.
