The theory of optimal transport (OT) comes into focus, among all maps that can transform one measure of probability into another, in those that are the “cheapest”, that is, that the average cost between and his image be as small as possible. Many computational approaches have been proposed to estimate such Monge maps when is he distance, for example, 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 generalization of the entropic map suitable for and highlight a surprising link that links him to the bregmann divergence centroids generated by and the proximal operator of . We show that the choice of a scarcity-inducing norm for results in maps that apply occam‘s knife to carry, in the sense that the displacement vectors they induce are scarce, with a pattern of scarcity that varies depending on . We show the ability of our method to estimate significant OT maps for high-dimensional single-cell transcript data, in the – gene Count Space For Cells, without using dimensionality reduction, thus retaining the ability to interpret all displacements at the genetic level.
