Interpolating between Optimal Transport and MMD using Sinkhorn Divergences

Jean Feydy (1) , Thibault Séjourné (2) , François-Xavier Vialard (3, 4, 5) , Shun-Ichi Amari (6) , Alain Trouvé (1) , Gabriel Peyré (7)

1 CMLA - Centre de Mathématiques et de Leurs Applications
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 Université Paris Dauphine-PSL
4 MOKAPLAN - Méthodes numériques pour le problème de Monge-Kantorovich et Applications en sciences sociales
5 UPEM - Université Paris-Est Marne-la-Vallée
6 RIKEN CBS - RIKEN Center for Brain Science [Wako]
7 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Domaines: Statistiques [math.ST]

Jean Feydy, Thibault Séjourné, François-Xavier Vialard, Shun-Ichi Amari, Alain Trouvé, et al.. Interpolating between Optimal Transport and MMD using Sinkhorn Divergences. 2018. ⟨hal-01898858⟩