2023/09/22 by Wuchen Li, Li, Wuchen, Jiaxi Zhao +1 · 1 citation
Computer Science · Mathematics · Medicine · #41A60 #62B11 #FOS: Computer and information sciences #FOS: Mathematics #Geometric Analysis and Curvature Flows #Hormonal and reproductive studies #Machine Learning (stat.ML) #Numerical Analysis (math.NA) #Probability (math.PR) #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2309.12997
openalex publication_date 2023/09/22 · openalex created_date 2023/09/26 · openalex updated_date 2026/08/01
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.