Computations of optimal transport distance with Fisher information regularization

We propose a fast algorithm to approximate the optimal transport distance. The main idea is to add a Fisher information regularization into the dynamical setting of the problem, originated by Benamou and Brenier. The regularized problem is shown to be smooth and strictly convex, thus many classical fast algorithms are available. In this paper, we adopt Newton's method, which converges to the minimizer with a quadratic rate. Several numerical examples are provided.