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arXiv · 2609.05700

Sandwich Slice: Optimal Transport Potentials Are Optimal Generalized Slicers

Rocio Diaz Martin·Xinran Liu·Matthew Thorpe·Soheil Kolouri
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Abstract

Sliced optimal transport replaces a transport problem in Rd by one-dimensional problems along scalar slices, and has been used both to lower bound the Wasserstein distance W1 (max-sliced distances) and, more recently, to upper bound it by lifting a one-dimensional optimal plan back to the ambient supports (min-sliced plans). We show that for 1-Lipschitz slicers these two directions are two sides of a single sandwich inequality and we characterize when it collapses to equality. If the slicer class contains an optimal Kantorovich-Rubinstein potential, then the same slicer simultaneously maximizes the max-sliced lower objective and minimizes the min-sliced upper objective, and both equal W1. We define the upper objective through a tie-broken transport problem, which removes the dependence on the one-dimensional ordering and yields the identities without any uniqueness assumption on the slice-optimal coupling. The result requires the slicer to be nonlinear in general, consistent with the known failure of exact linear max-sliced/W1 equivalence in dimension greater than or equal to 2, and identifies Kantorovich-Rubinstein potentials as optimal generalized slicers.

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Rocio Diaz Martin, Xinran Liu, Matthew Thorpe, Soheil Kolouri. 2026-09-04. Sandwich Slice: Optimal Transport Potentials Are Optimal Generalized Slicers. https://arxiv.org/abs/2609.05700

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