Comparison Results for Gromov-Wasserstein and Gromov-Monge Distances

TL;DR

This paper compares Gromov-Wasserstein and Gromov-Monge distances, establishing their theoretical relationships and equivalences under certain conditions, with implications for shape and image analysis.

Contribution

It provides theoretical comparisons between GM and GW distances, including conditions for their equality and bi-Hölder equivalence in Euclidean spaces.

Findings

GM and GW distances are equal for non-atomic metric measure spaces

Variants of GM distance are precisely comparable to GW distance

Bi-Hölder equivalence established between GM and Monge optimal transport in Euclidean spaces

Abstract

Inspired by the Kantorovich formulation of optimal transport distance between probability measures on a metric space, Gromov-Wasserstein (GW) distances comprise a family of metrics on the space of isomorphism classes of metric measure spaces. In previous work, the authors introduced a variant of this construction which was inspired by the original Monge formulation of optimal transport; elements of the resulting family are referred to Gromov-Monge (GM) distances. These GM distances, and related ideas, have since become a subject of interest from both theoretical and applications-oriented perspectives. In this note, we establish several theoretical properties of GM distances, focusing on comparisons between GM and GW distances. In particular, we show that GM and GW distances are equal for non-atomic metric measure spaces. We also consider variants of GM distance, such as a Monge version of Sturm's $L_p$-transportion distance, and give precise comparisons to GW distance. Finally, we establish bi-H\"{o}lder equivalence between GM distance and an isometry-invariant Monge optimal transport distance between Euclidean metric measure spaces that has been utilized in shape and image analysis applications.