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.
Beschreibung
Interpolating between Optimal Transport and MMD using Sinkhorn Divergences
%0 Conference Paper
%1 pmlr-v89-feydy19a
%A Feydy, Jean
%A Séjourné, Thibault
%A Vialard, Francois-Xavier
%A Amari, Shun-ichi
%A Trouve, Alain
%A Peyré, Gabriel
%B Proceedings of Machine Learning Research
%D 2019
%E Chaudhuri, Kamalika
%E Sugiyama, Masashi
%I PMLR
%K divergences information optimal-transport readings theory
%P 2681--2690
%T Interpolating between Optimal Transport and MMD using Sinkhorn Divergences
%U http://proceedings.mlr.press/v89/feydy19a.html
%V 89
%X 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.
@inproceedings{pmlr-v89-feydy19a,
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.},
added-at = {2019-12-11T13:48:11.000+0100},
author = {Feydy, Jean and S\'{e}journ\'{e}, Thibault and Vialard, Fran\c{c}ois-Xavier and Amari, Shun-ichi and Trouve, Alain and Peyr\'{e}, Gabriel},
biburl = {https://www.bibsonomy.org/bibtex/270e330990d8508ff3923db3e616be796/kirk86},
booktitle = {Proceedings of Machine Learning Research},
description = {Interpolating between Optimal Transport and MMD using Sinkhorn Divergences},
editor = {Chaudhuri, Kamalika and Sugiyama, Masashi},
interhash = {357f3eda78e510891e297e44b80d2a91},
intrahash = {70e330990d8508ff3923db3e616be796},
keywords = {divergences information optimal-transport readings theory},
month = {16--18 Apr},
pages = {2681--2690},
pdf = {http://proceedings.mlr.press/v89/feydy19a/feydy19a.pdf},
publisher = {PMLR},
series = {Proceedings of Machine Learning Research},
timestamp = {2019-12-11T13:48:11.000+0100},
title = {Interpolating between Optimal Transport and MMD using Sinkhorn Divergences},
url = {http://proceedings.mlr.press/v89/feydy19a.html},
volume = 89,
year = 2019
}