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

Titouan Vayer

stat.MLFeb 18, 2026

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

Authors: Titouan Vayer

Organizations: COMPACT · Inria, Rennes, France.

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.

Optimal Transport

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A Provably Convergent and Practical Algorithm for Gromov--Wasserstein Optimal Transport

Ling Liang, Lei Yang

Department of Mathematics The University of Tennessee Knoxville, TN 37916, USA · School of Computer Science and Engineering Guangdong Province Key Laboratory of Computational Science Sun Yat-sen University Guangzhou, Guangdong 510275, China

Gradient Descent Nonconvex Optimization Constrained Optimization Optimal Transport

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Gromov--Wasserstein optimal transport (GWOT) aligns metric measure spaces by matching their within-domain relational structures, but large-scale GWOT remains challenging because its objective is nonconvex and projection onto the transport polytope is often solved only approximately in practice. This leads to a gap between practical projected-gradient implementations and convergence theory, which typically assumes exact projections. For squared-loss GWOT, we propose an inexact projected-gradient framework with a verifiable feasibility-residual-based inexact condition for the projection subproblem. This condition is directly computable and avoids unknown quantities such as the exact projection point. Under this implementable condition, we prove subsequential convergence to stationary points and, with a mild tolerance-decay condition, convergence of the whole sequence. The resulting method retains the simplicity and sparsity of projected-gradient schemes while providing rigorous convergence guarantees, turning projected-gradient methods into a principled and scalable approach for GWOT with provable reliability.

Abstract

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