On the existence of Monge maps for the Gromov-Wasserstein problem

TL;DR

This paper investigates conditions under which optimal transportation plans for the Gromov-Wasserstein problem are induced by maps or 2-maps in Euclidean spaces, providing existence results and numerical insights into their structure.

Contribution

It establishes the existence of Monge maps for certain cost functions and measures, and demonstrates the potential for 2-maps to be optimal, highlighting limitations of map-based solutions.

Findings

Existence of optimal maps for inner product costs under absolute continuity.

Existence of optimal 2-maps for squared distance costs under absolute continuity.

Numerical evidence of 2-maps not being representable as single maps in dimension one.

Abstract

The Gromov--Wasserstein problem is a non-convex optimization problem over the polytope of transportation plans between two probability measures supported on two spaces, each equipped with a cost function evaluating similarities between points. Akin to the standard optimal transportation problem, it is natural to ask for conditions guaranteeing some structure on the optimizers, for instance if these are induced by a (Monge) map. We study this question in Euclidean spaces when the cost functions are either given by (i) inner products or (ii) squared distances, two standard choices in the literature. We establish the existence of an optimal map in case (i) and of an optimal 2-map (the union of the graphs of two maps) in case (ii), both under an absolute continuity condition on the source measure. Additionally, in case (ii) and in dimension one, we numerically design situations where optimizers of the Gromov--Wasserstein problem are 2-maps but are not maps. This suggests that our result cannot be improved in general for this cost. Still in dimension one, we additionally establish the optimality of monotone maps under some conditions on the measures, thereby giving insight on why such maps often appear to be optimal in numerical experiments.