Entropic Partial Optimal Transport and Partial Gromov--Wasserstein Distance between Gaussian Mixtures

2026-08-10

AI Summary

This work addresses the limitations of classical optimal transport and Gromov–Wasserstein distance, which enforce global mass conservation and thus lack robustness in the presence of outliers, missing data, or partial domain overlap. The authors introduce, for the first time, a framework of partial optimal transport and partial Gromov–Wasserstein distance tailored to Gaussian mixture models. By incorporating entropy regularization and penalty terms to model partial matching, they establish the existence and uniqueness of solutions and develop continuous transport plans, displacement interpolation, and barycentric projection maps, verifying their metric properties. Theoretical analysis combined with experiments demonstrates that the proposed approach achieves superior robustness on synthetic Gaussian mixtures and point cloud data, clearly elucidating the influence of the penalty parameter and entropy regularization on matching performance.

Abstract

Optimal transport and Gromov--Wasserstein distances are useful tools for comparing probability measures and metric measure spaces, but their balanced formulations force all mass to be matched. This constraint is often too strong for data with outliers, missing parts, or only partial overlap. In this paper, we develop entropic partial optimal transport for Gaussian mixture models and define a partial mixture Gromov--Wasserstein distance. For the finite entropic partial optimal transport problem, we prove the existence and uniqueness of the minimizer and establish quantitative large-penalty estimates. Moreover, the resulting entropic partial component couplings induce continuous partial transport plans through Gaussian optimal maps. We analyze their large-penalty and subsequent zero-entropy limits and construct the associated displacement interpolations and barycentric projection maps. In addition, by identifying each Gaussian mixture with a finite metric measure space of Gaussian components, we establish the metric property and large-penalty limit of the partial mixture Gromov--Wasserstein distance. Finally, numerical experiments on synthetic Gaussian mixtures and point clouds illustrate the effects of the penalty and entropic regularization and the robustness of partial matching to outliers.

Problem

• optimal transport
• Gromov–Wasserstein distance
• partial matching
• outliers
• Gaussian mixtures

Innovation

• entropic partial optimal transport
• Gaussian mixture models
• partial Gromov–Wasserstein distance
• metric measure spaces
• robust partial matching