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

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 (written GMMs in the paper for the sake of brevity) and known as M W2 (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 incorporate invariance properties into M W2. This is done by introducing new Gromov-type distances, designed to be isometry-invariant in Euclidean spaces and applicable for comparing GMMs across different dimensional spaces. Our first contribution is the Mixture Gromov Wasserstein distance (M GW2), which can be viewed as a ’Gromovized’ version of M W2. 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 (EW2). This distance turns out to be closely related to several recent alternatives to Gromov-Wasserstein. We show that EW2 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.