The Gromov-Wasserstein problem is an optimal transport (OT)-like optimization problem over the set of transport plans between two probability measures, possibly supported on different spaces. It induces a distance over the set of probability measures that is invariant by isometries. Just like in the OT case, it is natural to ask for conditions guaranteeing some structure on the optimal transport plans, for instance if these are induced by a (Monge) map, i.e. if every point of the source measure is sent on one point of the target measure (there is no splitting of mass). In [1], 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. After a short introduction to OT theory with a focus on the existence of optimal maps, I will go over the results of [1]. Depending on time, I may also talk about some numerics for (ii), in the simplest case, dimension one, where there is still much to be understood.

[1] https://arxiv.org/abs/2210.11945