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.

翻译：本文在高斯混合模型集合上引入了两种Gromov-Wasserstein型距离。第一种距离定义为高斯测度空间中两个离散分布之间的Gromov-Wasserstein距离。该距离可作为Gromov-Wasserstein距离的替代方案，适用于仅需评估分布间差异程度但无需直接推导点云间最优传输计划的场景。为构建可导出最优传输计划的方法，我们进一步定义了不可比空间中测度之间的另一种距离，该距离与Gromov-Wasserstein距离具有紧密关联。当将第二种距离中的可行传输耦合限制为高斯混合模型本身时，这定义了高斯混合模型间的另一种距离，可作为Gromov-Wasserstein的另一种替代方案，并允许推导点间最优指派。最后，我们通过类比第二种距离为第一种距离设计了传输计划，并在形状匹配与高光谱图像颜色迁移等中大规模问题上验证了其实用价值。