Gromov–Wasserstein distances between Gaussian distributions

Abstract 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 Wasserstein for comparing probability measures living on Euclidean spaces of different dimensions. We focus distance with ground cost defined as squared distance, and we study form optimal plan between Gaussian distributions. show that when is restricted distributions, problem has very simple linear solution, also solution Gromov–Monge problem. without restriction plan, provide lower upper bounds value