On the Existence of Monge Maps for the Gromov-Wasserstein Problem

Théo Lacombe, Théo Dumont, François-Xavier Vialard

Abstract:

In this work, we study the structure of minimizers of the quadratic Gromov–Wasserstein (GW)
problem on Euclidean spaces for two different costs. The first one is the scalar product for which we
prove that it is always possible to find optimizers as Monge maps and we detail the structure of such
optimal maps. The second cost is the squared Euclidean distance for which we show that the worst
case scenario is the existence of a bi-map structure. Both results are direct and indirect consequences
of an existence result of optimal maps in the standard optimal transportation problem for costs
that are defined by submersions. In dimension one for the squared Euclidean distance, we show
numerical evidence for a negative answer to the existence of a Monge map under the conditions of
Brenier’s theorem, suggesting that our result cannot be improved in general. In addition, we show
that a monotone map is optimal in some non-symmetric situations, thereby giving insight on why
such a map often appears to be optimal in numerical experiments.