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arXiv · 2602.16265

On sparsity, extremal structure, and monotonicity properties of Wasserstein and Gromov-Wasserstein optimal transport plans

Titouan Vayer
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Abstract

This note gives a self-contained overview of some important properties of the Gromov-Wasserstein (GW) distance, compared with the standard linear optimal transport (OT) framework. More specifically, I explore the following questions: are GW optimal transport plans sparse? Under what conditions are they supported on a permutation? Do they satisfy a form of cyclical monotonicity? In particular, I present the conditionally negative semi-definite property and show that, when it holds, there are GW optimal plans that are sparse and supported on a permutation.

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Titouan Vayer. 2026-02-18. On sparsity, extremal structure, and monotonicity properties of Wasserstein and Gromov-Wasserstein optimal transport plans. https://arxiv.org/abs/2602.16265

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