We consider the Wasserstein metric on the Gaussian mixture models (GMMs), which is defined as the pullback of the full Wasserstein metric on the space of smooth probability distributions with finite second moment. It derives a class of Wasserstein metrics on probability simplices over one-dimensional bounded homogeneous lattices via a scaling limit of the Wasserstein metric on GMMs. Specifically, for a sequence of GMMs whose variances tend to zero, we prove that the limit of the Wasserstein metric exists after certain renormalization. Generalizations of this metric in general GMMs are established, including inhomogeneous lattice models whose lattice gaps are not the same, extended GMMs whose mean parameters of Gaussian components can also change, and the second-order metric containing high-order information of the scaling limit. We further study the Wasserstein gradient flows on GMMs for three typical functionals: potential, internal, and interaction energies. Numerical examples demonstrate the effectiveness of the proposed GMM models for approximating Wasserstein gradient flows.

翻译：我们考虑高斯混合模型（GMM）上的Wasserstein度量，该度量定义为具有有限二阶矩的光滑概率分布空间上完整Wasserstein度量的拉回。通过GMM上Wasserstein度量的一种标度极限，导出了一维有界齐次格上概率单纯形上的一类Wasserstein度量。具体而言，对于方差趋于零的一列GMM，我们证明在经过特定重标度后，Wasserstein度量的极限存在。我们建立了该度量在一般GMM中的推广形式，包括格间距不同的非齐次格模型、高斯分量均值参数也可变化的扩展GMM，以及包含标度极限高阶信息的二阶度量。我们进一步研究了三种典型泛函（势能、内能和相互作用能）在GMM上的Wasserstein梯度流。数值算例验证了所提出的GMM模型在近似Wasserstein梯度流方面的有效性。