Entropic Optimal Transport has arisen in the literature as a computationally efficient approximation to unregularized optimal transport problems such as approximating the Wasserstein distance between two measures. A major reason for this computational advantage is the Sinkhorn Algorithm (also know as the Iterated Proportional Fitting Procedure amongst other names) however the convergence of this algorithm has not been proven in the full range of cases covered by the unregularized theory withing the literature.

Indeed for the quadratic cost between two marginals with finite second moment the unregularized theory is well known. The equivalent results in the Entropic literature require exponential moments of the cost with respect to the marginals, in the quadratic case this require both marginals to have finite moments of every order. In this talk we show the convergence of the Sinkhorn Algorithm under the assumption that a non-negative cost is integrable with respect to the product measure - an assumption that in the case of the quadratic cost corresponds to both marginals having finite second moment.