Sandwich Slice: Optimal Transport Potentials Are Optimal Generalized Slicers
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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BibTeX
@misc{martin2026sandwich,
title = {{Sandwich Slice: Optimal Transport Potentials Are Optimal Generalized Slicers}},
author = {Rocio Diaz Martin and Xinran Liu and Matthew Thorpe and Soheil Kolouri},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05700},
eprint = {2609.05700},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05700},
}
RIS
TY - UNPB
TI - Sandwich Slice: Optimal Transport Potentials Are Optimal Generalized Slicers
AU - Rocio Diaz Martin
AU - Xinran Liu
AU - Matthew Thorpe
AU - Soheil Kolouri
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05700
UR - https://arxiv.org/abs/2609.05700
AB - 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.
ER -