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

arXiv: https://arxiv.org/abs/2602.16265 · HTML: https://mlsanity.com/paper/2602.16265/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 18 references that are in the map.
- First posted on arXiv 18 Feb 2026.

## Builds on (most cited first, 10 of 10)

- [The Unbalanced Gromov Wasserstein Distance: Conic Formulation and Relaxation](https://mlsanity.com/paper/2009.04266/) (2020)
- [CO-Optimal Transport](https://mlsanity.com/paper/2002.03731/) (2020)
- [Online Graph Dictionary Learning](https://mlsanity.com/paper/2102.06555/) (2021)
- [On the existence of Monge maps for the Gromov-Wasserstein problem](https://mlsanity.com/paper/2210.11945/) (2022)
- [Distributional Reduction: Unifying Dimensionality Reduction and Clustering with Gromov-Wasserstein](https://mlsanity.com/paper/2402.02239/) (2024)
- [Optimal Transport for Machine Learners](https://mlsanity.com/paper/2505.06589/) (2025)
- [Multi-Marginal Gromov-Wasserstein Transport and Barycenters](https://mlsanity.com/paper/2205.06725/) (2022)
- [(Probably) Concave Graph Matching](https://mlsanity.com/paper/1807.09722/) (2018)
- [On Probabilistic Embeddings in Optimal Dimension Reduction](https://mlsanity.com/paper/2408.02433/) (2024)
- [Gromov-Wasserstein at Scale, Beyond Squared Norms](https://mlsanity.com/paper/2602.06658/) (2026)

## 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.