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[2602.16265] On sparsity, extremal structure, and monotonicity properties of Wasserstein and Gromov-Wasserstein optimal transport plans
Summary
This article explores the properties of Gromov-Wasserstein (GW) distance in optimal transport, focusing on sparsity, extremal structures, and monotonicity, providing insights into conditions that influence these characteristics.
Why It Matters
Understanding the properties of GW distance is crucial for advancements in machine learning, particularly in applications involving optimal transport. This research sheds light on the conditions under which GW plans exhibit sparsity and cyclic monotonicity, which can enhance algorithm efficiency and theoretical foundations in the field.
Key Takeaways
- GW distance offers a framework for analyzing optimal transport plans.
- Sparsity in GW plans can be achieved under specific conditions.
- Cyclical monotonicity is a significant property of GW optimal plans.
- The conditionally negative semi-definite property is crucial for understanding GW plans.
- Insights from this research can improve algorithmic efficiency in machine learning.
Statistics > Machine Learning arXiv:2602.16265 (stat) [Submitted on 18 Feb 2026] Title:On sparsity, extremal structure, and monotonicity properties of Wasserstein and Gromov-Wasserstein optimal transport plans Authors:Titouan Vayer (COMPACT) View a PDF of the paper titled On sparsity, extremal structure, and monotonicity properties of Wasserstein and Gromov-Wasserstein optimal transport plans, by Titouan Vayer (COMPACT) View PDF 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. Subjects: Machine Learning (stat.ML); Machine Learning (cs.LG) Cite as: arXiv:2602.16265 [stat.ML] (or arXiv:2602.16265v1 [stat.ML] for this version) https://doi.org/10.48550/arXiv.2602.16265 Focus to learn more arXiv-issued DOI via DataCite (pending registration) Submission history From: Titouan Vayer [view email] [via CCSD proxy] [v1] Wed, 18 Feb 2026 08:35:36 UTC (52 KB) Full-text links: Access Paper: View a PDF of the paper titled On sparsity, extremal structure, and monotonicity p...
