This talk has two parts. First we present a possible extension of the Gromov-Wasserstein problem to the setting of metric measures spaces, whose total mass is not necessarily equal to 1. We propose a true distance and a lower bound which is more friendly for computations. Second, we study the existence of Monge maps as optimizer of the standard Gromov-Wasserstein problem for two different costs in euclidean spaces. The first cost for which we show existence of Monge maps is the scalar product, the second cost is the quadratic cost between the squared distances for which we show the structure of a bi-map. We present numerical evidence that the last result is sharp.

This talk has two parts. First we present a possible extension of the Gromov-Wasserstein problem to the setting of metric measures spaces, whose total mass is not necessarily equal to 1. We propose a true distance and a lower bound which is more friendly for computations. Second, we study the existence of Monge maps as optimizer of the standard Gromov-Wasserstein problem for two different costs in euclidean spaces. The first cost for which we show existence of Monge maps is the scalar product, the second cost is the quadratic cost between the squared distances for which we show the structure of a bi-map. We present numerical evidence that the last result is sharp.

We show how to break the curse of dimension for the estimation of optimal transport distance between two smooth distributions for the Euclidean squared distance. The approach relies on essentially one tool: represent inequality constraints in the dual formulation of OT by equality constraints with a sum of squares in reproducing kernel Hilbert space. By showing this representation is tight in the variational formulation, one can then leverage smoothness to break the curse. (*) However, the constants associated with the algorithm a priori scale exponentially with the dimension.