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.

翻译：本文对Gromov-Wasserstein（GW）距离相较于标准线性最优输运（OT）框架的一些重要性质进行了独立而系统的综述。具体而言，本文探讨以下问题：GW最优输运方案是否具有稀疏性？在何种条件下其支撑集为置换映射？是否满足某种形式的循环单调性？特别地，本文引入条件负半定性这一性质，并证明在该性质成立时，存在具有稀疏性且支撑于置换映射的GW最优方案。