In this paper, we introduce two Gromov-Wasserstein-type distances on the set
of Gaussian mixture models. The first one takes the form of a
Gromov-Wasserstein distance between two discrete distributionson the space of
Gaussian measures. This distance can be used as an alternative to
Gromov-Wasserstein for applications which only require to evaluate how far the
distributions are from each other but does not allow to derive directly an
optimal transportation plan between clouds of points. To design a way to define
such a transportation plan, we introduce another distance between measures
living in incomparable spaces that turns out to be closely related to
Gromov-Wasserstein. When restricting the set of admissible transportation
couplings to be themselves Gaussian mixture models in this latter, this defines
another distance between Gaussian mixture models that can be used as another
alternative to Gromov-Wasserstein and which allows to derive an optimal
assignment between points. Finally, we design a transportation plan associated
with the first distance by analogy with the second, and we illustrate their
practical uses on medium-to-large scale problems such as shape matching and
hyperspectral image color transfer.

该研究论文介绍了两种 Gromov-Wasserstein 类型的距离，用于高斯混合模型集合。这些距离可作为 Gromov-Wasserstein 的替代品，用于评估两个分布间的差异，并且为点云之间的最优传输计划提供了一种定义方式。同时，该研究还提供了实际应用案例，如形状匹配和高光谱图像颜色转换。