Stability of Schrödinger bridges and Sinkhorn semigroups for log-concave models

Abstract: In this article we obtain several new results and developments in the study of entropic optimal transport problems (a.k.a. Schr\"odinger problems) with general reference distributions and log-concave target marginal measures. Our approach combines transportation cost inequalities with the theory of Riccati matrix difference equations arising in filtering and optimal control theory. This methodology is partly based on a novel entropic stability of Schr\"odinger bridges and closed form expressions of a class of discrete time algebraic Riccati equations. In the context of regularized entropic transport these techniques provide new sharp entropic map estimates. When applied to the stability of Sinkhorn semigroups, they also yield a series of novel contraction estimates in terms of the fixed point of Riccati equations. The strength of our approach is that it is applicable to a large class of models arising in machine learning and artificial intelligence algorithms. We illustrate the impact of our results in the context of regularized entropic transport, proximal samplers and diffusion generative models as well as diffusion flow matching models