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

arXiv 2602.16265 ↗First posted 18 Feb 2026stat.ML cs.LGcited by 0

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Abstract (arXiv)

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.

Key facts. No citing papers in the map yet (data through 7 Oct 2026). Builds on 10 of its 18 references that are in the map. First posted on arXiv 18 Feb 2026.

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10 of 18 references.

References · 10

  1. 2020The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxationcited by 55
  2. 2020CO-Optimal Transportcited by 42
  3. 2021Online Graph Dictionary Learningcited by 37
  4. 2022On the existence of Monge maps for the Gromov-Wasserstein problemcited by 16
  5. 2024Distributional Reduction: Unifying Dimensionality Reduction and Clustering with Gromov-Wassersteincited by 12
  6. 2022Multi-Marginal Gromov-Wasserstein Transport and Barycenterscited by 9
  7. 2025Optimal Transport for Machine Learnerscited by 8
  8. 2018(Probably) Concave Graph Matchingcited by 7
  9. 2024On Probabilistic Embeddings in Optimal Dimension Reductioncited by 5
  10. 2026Gromov-Wasserstein at Scale, Beyond Squared Normscited by 4

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