# On the Structure of Optimal Transportation Plans between Discrete Measures

arXiv: https://arxiv.org/abs/2104.13106 · HTML: https://mlsanity.com/paper/2104.13106/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Apr 2021.

## Built on by (most cited first, 1 of 1)

- [Normalizing flows as approximations of optimal transport maps via linear-control neural ODEs](https://mlsanity.com/paper/2311.01404/) (2023)

## Abstract (arXiv)

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 μ and nu with the p-Wasserstein distance, times a constant depending on μ, nu, and the fixed cost function.