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