Gromov–Wasserstein distances between Gaussian distributions

• Published in: Journal of applied probability Vol. 59; no. 4; pp. 1178 - 1198
• Main Authors: Delon, Julie, Desolneux, Agnes, Salmona, Antoine
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.12.2022; Cambridge University press
• Subjects: Computer Science; Euclidean geometry; Image Processing; Mathematics; Optimization; Original Article; Probability; Probability distribution; Random variables; Upper bounds; Gromov–Wasserstein distance; Gaussian distributions; 60E99; 68T09; Wasserstein distance; 49Q22; Optimal transport; 62H25; Gromov-Wasserstein distance; optimal transport
• ISSN: 0021-9002; 1475-6072
• DOI: 10.1017/jpr.2022.16

Gromov–Wasserstein distances were proposed a few years ago to compare distributions which do not lie in the same space. In particular, they offer an interesting alternative to the Wasserstein distances for comparing probability measures living on Euclidean spaces of different dimensions. We focus on the Gromov–Wasserstein distance with a ground cost defined as the squared Euclidean distance, and we study the form of the optimal plan between Gaussian distributions. We show that when the optimal plan is restricted to Gaussian distributions, the problem has a very simple linear solution, which is also a solution of the linear Gromov–Monge problem. We also study the problem without restriction on the optimal plan, and provide lower and upper bounds for the value of the Gromov–Wasserstein distance between Gaussian distributions.