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Interpolating between Optimal Transport and MMD using Sinkhorn Divergences

Feydy, Jean; Séjourné, Thibault; Vialard, François-Xavier; Amari, Shun-ichi; Trouvé, Alain; Peyré, Gabriel (2018), Interpolating between Optimal Transport and MMD using Sinkhorn Divergences. https://basepub.dauphine.fr/handle/123456789/18450

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sinkhorn_divergences.pdf (1.020Mb)
Type
Document de travail / Working paper
External document link
https://hal.archives-ouvertes.fr/hal-01898858
Date
2018
Series title
Cahier de recherche CEREMADE, Université Paris-Dauphine
Pages
15
Metadata
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Author(s)
Feydy, Jean
Centre de Mathématiques et de Leurs Applications [CMLA]
Séjourné, Thibault
Département de Mathématiques et Applications - ENS Paris [DMA]
Vialard, François-Xavier
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Amari, Shun-ichi
RIKEN Center for Brain Science [Wako] [RIKEN CBS]
Trouvé, Alain
Centre de Mathématiques et de Leurs Applications [CMLA]
Peyré, Gabriel
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
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.
Subjects / Keywords
Optimal Transport; MMD; Sinkhorn Divergences

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