Imposing and learning structure in OT displacements through cost engineering

I will highlight in this work the flexibility provided by the Gangbo-McCann theorem, which provides a generic way to tie kantorovich dual potential solutions to optimal maps for the Monge problem. We show in particular how setting the ground cost to the squared-Euclidean distance + a regularizer induces displacements that have a structure that is well suited to that regularizer (e.g. sparse if that regularizer is the L1 norm). We propose an approach, in more recent work, to learn parameters of that regularizer.