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

Titouan Vayer

TL;DR

The paper investigates how Gromov-Wasserstein (GW) optimal plans relate to classical linear OT properties. By introducing a conditionally negative semi-definite (CND) tensor framework for the GW loss, it shows that concavity of the GW objective on the coupling polytope implies the existence of sparse, extreme-point optima and tight coupling relaxations, mirroring Monge–Kantorovich equivalence under suitable conditions. It provides a detailed characterization for separable losses and gives concrete examples, notably the squared-distance and KL-divergence cases, where the CND conditions hold via Schoenberg-type results and infinite divisibility, respectively. The work also discusses a monotonicity-type property for GW arising from a linearization at the GW optimum and clarifies the limits of these properties beyond the CND regime. Overall, CND energies offer a practical lens to understand and exploit GW structure in algorithms, while noting that such properties are not universal but are commonly encountered in practice.

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.

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

TL;DR

The paper investigates how Gromov-Wasserstein (GW) optimal plans relate to classical linear OT properties. By introducing a conditionally negative semi-definite (CND) tensor framework for the GW loss, it shows that concavity of the GW objective on the coupling polytope implies the existence of sparse, extreme-point optima and tight coupling relaxations, mirroring Monge–Kantorovich equivalence under suitable conditions. It provides a detailed characterization for separable losses and gives concrete examples, notably the squared-distance and KL-divergence cases, where the CND conditions hold via Schoenberg-type results and infinite divisibility, respectively. The work also discusses a monotonicity-type property for GW arising from a linearization at the GW optimum and clarifies the limits of these properties beyond the CND regime. Overall, CND energies offer a practical lens to understand and exploit GW structure in algorithms, while noting that such properties are not universal but are commonly encountered in practice.

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.
Paper Structure (16 sections, 14 theorems, 39 equations, 1 figure)

This paper contains 16 sections, 14 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Linear and quadratic OT
  3. Some important properties of linear OT
  4. Cyclical monotonicity
  5. Sparsity of some optimal plans
  6. Tightness of the coupling relaxation
  7. What about GW optimal transport plans ?
  8. Conditionally negative semi-definite tensor
  9. First consequence: sparsity of some optimal plans
  10. Second consequence: tightness of the coupling relaxation
  11. Third consequence: as small detour around the bilinear relaxation
  12. When is the tensor CND ? The case of separable losses
  13. Example 1: Squared case $\mathcal{L} = \mathcal{L}_2$.
  14. Example 2: Kullback-Leibler case $\mathcal{L} = \mathcal{L}_{\operatorname{KL}}$.
  15. What about cyclical monotonicity ?
  16. ...and 1 more sections

Key Result

Theorem 2.1

For any costs $\mathbf{C}$, a coupling $\mathbf{P} \in \Pi(\mathbf{a}, \mathbf{b})$ is optimal for eq:linear_ot if and only if for any $N \in \mathbb{N}^{*}, (i_1, j_1),\cdots, (i_N, j_N) \in {\operatorname{supp}}(\mathbf{P})^N$ and permutation $\sigma \in \mathfrak{S}_N$,

Figures (1)

  • Figure 1: (Left) Bipartite graph $G(\mathbf{P})$ induced by $\mathbf{P}$. Weights on the edges are the values $P_{ij}$. (Right) It contains a 3-cycle $i_1, j_1, i_2, j_2, i_3, j_3, i_1$. The forward edges $i \to j$ are marked with a $+\varepsilon$ perturbation, the backward with a $-\varepsilon$.

Theorems & Definitions (27)

  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Theorem 2.4: Birkhoff
  • proof
  • Corollary 2.5
  • proof
  • Definition 3.1
  • ...and 17 more