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

TL;DR

This paper explores key properties of Gromov-Wasserstein optimal transport plans, including sparsity, support on permutations, and cyclical monotonicity, comparing them with standard linear optimal transport.

Contribution

It provides a self-contained overview of conditions under which GW plans are sparse, supported on permutations, and exhibit monotonicity properties, highlighting the role of negative semi-definite conditions.

Findings

GW optimal plans can be sparse and supported on permutations under certain conditions

The negative semi-definite property influences the structure of GW plans

GW plans may satisfy cyclical monotonicity under specific assumptions

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.