Stability of Schrödinger potentials and convergence of Sinkhorn’s algorithm

We study the stability of entropically regularized optimal transport with respect to marginals. Given marginals converging weakly, we establish a strong convergence for Schrödinger potentials, describing density couplings. When converge in total variation, couplings also variation. This is applied show that Sinkhorn’s algorithm converges variation when costs are quadratic and subgaussian or, more generally, all continuous satisfying an integrability condition.