Monge, Bregman and Occam: Interpretable Optimal Transport in High-Dimensions with Feature-Sparse Maps
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Optimal transport (OT) theory focuses, among all maps T:ℝ^d→ℝ^d that can morph a probability measure onto another, on those that are the “thriftiest”, i.e. such that the averaged cost c(x, T(x)) between x and its image T(x) be as small as possible. Many computational approaches have been proposed to estimate such Monge maps when c is the ℓ_2^2 distance, e.g., using entropic maps [Pooladian'22], or neural networks [Makkuva'20, Korotin'20]. We propose a new model for transport maps, built on a family of translation invariant costs c(x, y):=h(x-y), where h:=12·_2^2+τ and τ 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 τ. We show that choosing a sparsity-inducing norm for τ results in maps that apply Occam's razor to transport, in the sense that the displacement vectors Δ(x):= T(x)-x they induce are sparse, with a sparsity pattern that varies depending on 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.
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