On the Structure of Optimal Transportation Plans between Discrete Measures

TL;DR

This paper proves a structural theorem for discrete optimal transportation plans, showing they can be decomposed into two deterministic plans, and applies this to estimate Wasserstein distances.

Contribution

It introduces a novel structure theorem for optimal transportation plans between discrete measures, enabling new distance estimation techniques.

Findings

Existence of a decomposition of optimal plans into two deterministic plans.

Derived bounds for infinity-Wasserstein distance using p-Wasserstein distance.

Provides a theoretical foundation for analyzing discrete optimal transport plans.

Abstract

In this paper, we prove a structure theorem for discrete optimal transportation plans. We show that, given any pair of discrete probability measures and a cost function, there exists an optimal transportation plan that can be expressed as the sum of two deterministic plans. As an application, we estimate the infinity-Wasserstein distance between two discrete probability measures $\mu$ and $\nu$ with the $p$-Wasserstein distance, times a constant depending on $\mu$, $\nu$, and the fixed cost function.