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

2026-02-18

AI Summary

This study investigates the structural properties of Gromov–Wasserstein (GW) optimal transport plans, focusing on sparsity, permutation-supported structure, and cyclical monotonicity. By introducing a conditional negative semi-definiteness assumption and integrating tools from optimal transport theory, GW distance analysis, and combinatorial optimization, the authors prove that under this condition, GW optimal transport plans simultaneously exhibit sparsity and permutation-supported structure. This work provides the first rigorous theoretical guarantees for such structural characteristics within the nonlinear optimal transport framework, thereby deepening the understanding of the geometric nature of GW distances and highlighting fundamental differences between GW transport and classical linear optimal transport.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

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.

Problem

• sparsity
• extremal structure
• monotonicity
• Gromov-Wasserstein
• optimal transport

Innovation

• Gromov-Wasserstein distance
• sparsity
• permutation support
• conditionally negative semi-definite
• cyclical monotonicity