An Optimal Transport approach for the Schrödinger bridge problem and convergence of Sinkhorn algorithm

Abstract: This paper exploit the equivalence between the Schr\"odinger Bridge problem and the entropy penalized optimal transport in order to find a different approach to the duality, in the spirit of optimal transport. This approach results in a priori estimates which are consistent in the limit when the regularization parameter goes to zero. In particular, we find a new proof of the existence of maximizing entropic-potentials and therefore, the existence of a solution of the Schr\"odinger system. Our method extends also when we have more than two marginals: we can provide an alternative proof of the convergence of the Sinkhorn algorithm with two marginals and we show that the Sinkhorn algorithm converges in the multi-marginal case.