  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 (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 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), which can be
viewed as a Gromovized version of MW. 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). This distance turns out to be closely related to
several recent alternatives to Gromov-Wasserstein. We show that EW 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.
