Interpolating between Optimal Transport and MMD using Sinkhorn Divergences

TL;DR

This paper introduces Sinkhorn divergences, a new family of geometric measures that interpolate between MMD and OT, with theoretical guarantees and scalable GPU-based computation for large datasets.

Contribution

It proposes Sinkhorn divergences, providing a theoretical framework and practical algorithms for large-scale geometric distribution comparison.

Findings

Proves positivity, convexity, and convergence properties of Sinkhorn divergences.

Develops a GPU-accelerated numerical scheme for large sample sizes.

Enables efficient computation of gradients for machine learning applications.

Abstract

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.