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 (GMMs) and known as MW2 (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 MW2. 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 (MGW2), which can be viewed as a "Gromovized" version of MW2. 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.

翻译：Gromov-Wasserstein (GW) 距离在机器学习中常被用于比较不同度量空间中的分布。尽管其具有实用性，但计算量仍然很大，尤其在大规模问题上。最近，多位学者提出了一种专门针对高斯混合模型 (GMMs) 的新型Wasserstein距离，称为 MW2（混合Wasserstein距离）。在数据呈现聚类特性的场景中，该方法可简化为一个小规模的离散最优传输问题，其复杂度仅取决于GMM中高斯分量的数量。本文旨在将不变性属性融入 MW2。这是通过引入新的Gromov型距离来实现的，这些距离被设计为在欧几里得空间中具有等距不变性，并适用于比较不同维度空间中的GMMs。我们的第一个贡献是混合Gromov Wasserstein距离 (MGW2)，它可以被视为 MW2 的“Gromov化”版本。这种新距离具有简洁的离散形式，使其在实际应用中能高效估计GMMs之间的距离。为了便于推导GMMs之间的传输方案，我们提出了第二种距离，即嵌入Wasserstein距离 (EW2)。该距离被证明与最近提出的几种Gromov-Wasserstein替代方案密切相关。我们证明了EW2可用于推导GMMs之间的距离以及最优传输方案。我们在中到大规模问题上（包括形状匹配和高光谱图像颜色迁移）验证了这些新提出距离的高效性。