揭示了广义沃尔什距离最优传输方案的稀疏性与排列支持特性
On sparsity, extremal structure, and monotonicity properties of Wasserstein and Gromov-Wasserstein optimal transport plans
- 基于条件负半定性质，推导出最优传输计划的稀疏结构
- 在特定条件下，最优传输方案支持在排列上且具有稀疏性
- 为理解非线性传输机制提供理论支撑，适合几何学习研究者
本文系统阐述了广义沃尔什（Gromov-Wasserstein, GW）距离相较于标准线性最优传输（OT）框架的重要性质。具体探讨以下问题：GW最优传输方案是否稀疏？在何种条件下其支撑集为排列？是否满足某种循环单调性？特别地，本文提出条件负半定性，并证明当该性质成立时，存在稀疏且支撑于排列上的最优传输方案。
原文摘要 · Abstract (English)
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.
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