Monge, Bregman and Occam: Interpretable Optimal Transport in High-Dimensions with Feature-Sparse Maps

Optimal transport (OT) theory focuses, among all maps that can morph a probability measure onto another, on those that are the “thriftiest”, i.e. such that the averaged cost between and its image be as small as possible. Many computational approaches have been proposed to estimate such Monge maps when is the distance, e.g., using entropic maps (Pooladian and Niles-Weed, 2021), or neural networks (Makkuva et al., 2020;
Korotin et al., 2020). We propose a new model for transport maps, built on a family of translation invariant costs , where and is a regularizer. We propose a…