Quantitative uniform stability of the iterative proportional fitting procedure

We establish that the iterates of the iterative proportional fitting procedure, also known as Sinkhorn's algorithm and commonly used to solve entropy-regularised optimal transport problems, are stable w.r.t. perturbations of the marginals, uniformly in time. Our result is quantitative and stated in terms of the 1-Wasserstein metric. As a corollary we establish a quantitative stability result for Schrodinger bridges.