Interpolating between Optimal Transport and MMD using Sinkhorn\n Divergences

Abstract

Comparing probability distributions is a fundamental problem in data\nsciences. Simple norms and divergences such as the total variation and the\nrelative entropy only compare densities in a point-wise manner and fail to\ncapture the geometric nature of the problem. In sharp contrast, Maximum Mean\nDiscrepancies (MMD) and Optimal Transport distances (OT) are two classes of\ndistances between measures that take into account the geometry of the\nunderlying space and metrize the convergence in law.\n This paper studies the Sinkhorn divergences, a family of geometric\ndivergences that interpolates between MMD and OT. Relying on a new notion of\ngeometric entropy, we provide theoretical guarantees for these divergences:\npositivity, convexity and metrization of the convergence in law. On the\npractical side, we detail a numerical scheme that enables the large scale\napplication of these divergences for machine learning: on the GPU, gradients of\nthe Sinkhorn loss can be computed for batches of a million samples.\n