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Gromov–Wasserstein distances between Gaussian distributions

Part of: Multivariate analysis Distribution theory - Probability

Published online by Cambridge University Press:  18 August 2022

Julie Delon ,
Agnes Desolneux  and
Antoine Salmona
Julie Delon*
Affiliation:
Université de Paris
Agnes Desolneux*
Affiliation:
CNRS and ENS Paris-Saclay
Antoine Salmona*
Affiliation:
ENS Paris-Saclay
*
*Postal address: Université de Paris, CNRS, MAP5 UMR 8145 and Institut Universitaire de France, 45 rue des Saints-Pères, 75006 Paris, France. Email: julie.delon@parisdescartes.fr
**Postal address: ENS Paris-Saclay, CNRS, Centre Borelli, UMR 9010, 4 avenue des sciences, 91190 Gif-sur-Yvette, France. Email: agnes.desolneux@ens-paris-saclay.fr
***Postal address: ENS Paris-Saclay, CNRS, Centre Borelli, UMR 9010, 4 avenue des sciences, 91190 Gif-sur-Yvette, France. Email: antoinesalmona2@gmail.com
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Abstract

Gromov–Wasserstein distances were proposed a few years ago to compare distributions which do not lie in the same space. In particular, they offer an interesting alternative to the Wasserstein distances for comparing probability measures living on Euclidean spaces of different dimensions. We focus on the Gromov–Wasserstein distance with a ground cost defined as the squared Euclidean distance, and we study the form of the optimal plan between Gaussian distributions. We show that when the optimal plan is restricted to Gaussian distributions, the problem has a very simple linear solution, which is also a solution of the linear Gromov–Monge problem. We also study the problem without restriction on the optimal plan, and provide lower and upper bounds for the value of the Gromov–Wasserstein distance between Gaussian distributions.

Keywords

Optimal transportWasserstein distanceGromov–Wasserstein distanceGaussian distributions

MSC classification

Primary: 60E99: None of the above, but in this section
Secondary: 62H25: Factor analysis and principal components; correspondence analysis
Type
Original Article
Information
Journal of Applied Probability , Volume 59 , Issue 4 , December 2022 , pp. 1178 - 1198
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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