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

Optimal transport (OT) theory focuses, among all maps $T:\mathbb{R}^d\rightarrow \mathbb{R}^d$ that can morph a probability measure onto another, on those that are the “thriftiest”, i.e. such that the averaged cost $c(\mathbf{x}, T(\mathbf{x}))$ between $\mathbf{x}$ and its image $T(\mathbf{x})$ be as small as possible. Many computational approaches have been proposed to estimate such Monge maps when $c$ is the $\ell_2^2$ 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 $c(\mathbf{x},\mathbf{y}):=h(\mathbf{x}-\mathbf{y})$, where $h:=\tfrac{1}{2}\|\cdot\|_2^2+\tau$ and $\tau$ is a regularizer. We propose a generalization of the entropic map suitable for $h$, and highlight a surprising link tying it with the Bregman centroids of the divergence $D_h$ generated by $h$, and the proximal operator of $\tau$. We show that choosing a sparsity-inducing norm for $\tau$ results in maps that apply Occam’s razor to transport, in the sense that the displacement vectors $\Delta(\mathbf{x}):= T(\mathbf{x})-\mathbf{x}$ they induce are sparse, with a sparsity pattern that varies depending on $\mathbf{x}$. We showcase the ability of our method to estimate meaningful OT maps for high-dimensional single-cell transcription data, in the $34000$-$d$ space of gene counts for cells, without using dimensionality reduction, thus retaining the ability to interpret all displacements at the gene level.