Gromov-Wasserstein-like Distances in the Gaussian Mixture Models Space

Antoine Salmona, Agnes Desolneux, Julie Delon

Abstract: The Gromov-Wasserstein (GW) distance is frequently used in machine learning to compare distributions across distinct metric spaces. Despite its utility, it remains computationally intensive, especially for large-scale problems. Recently, a novel Wasserstein distance specifically tailored for Gaussian mixture models and known as $ MW_2 $ (mixture Wasserstein) has been introduced by several authors. In scenarios where data exhibit clustering, this approach simplifies to a small-scale discrete optimal transport problem, which complexity depends solely on the number of Gaussian components in the GMMs. This paper aims to extend $ MW_2 $ by introducing new Gromov-type distances. These distances are designed to be isometry-invariant in Euclidean spaces and are applicable for comparing GMMs across different dimensional spaces. Our first contribution is the Mixture Gromov Wasserstein distance ($MGW_2$), which can be viewed as a ’Gromovized’ version of $ MW_2 $ . This new distance has a straightforward discrete formulation, making it highly efficient for estimating distances between GMMs in practical applications. To facilitate the derivation of a transport plan between GMMs, we present a second distance, the Embedded Wasserstein distance ($ EW_2 $). This distance turns out to be closely related to several recent alternatives to Gromov-Wasserstein. We show that can be adapted to derive a distance as well as optimal transportation plans between GMMs. We demonstrate the efficiency of these newly proposed distances on medium to large-scale problems, including shape matching and hyperspectral image color transfer.

Submission Length: Long submission (more than 12 pages of main content)

Previous TMLR Submission Url: https://openreview.net/forum?id=5SmXTBHHHy&referrer=%5BAuthor%20Console%5D(%2Fgroup%3Fid%3DTMLR%2FAuthors%23your-submissions)

Changes Since Last Submission: Section Acknowledgments (which was breaking double-blind policy) has been removed.

Code: https://github.com/AntoineSalmona/MixtureGromovWasserstein

Assigned Action Editor: ~Jeff_Phillips1

Submission Number: 2525