Gromov-Hausdorff convergence of discrete transportation metrics

TL;DR

This paper proves that discrete transportation metrics on a d-dimensional discrete torus converge to the continuous 2-Wasserstein distance on the torus as the mesh size decreases, establishing a key link between discrete and continuous optimal transport.

Contribution

It provides the first convergence proof of discrete transportation metrics to the classical Wasserstein distance, bridging discrete and continuous optimal transport frameworks.

Findings

Discrete transportation metrics converge to W_2 as mesh size decreases

First convergence result for these discrete metrics

Shows compatibility between discrete and continuous transport metrics

Abstract

This paper continues the investigation of `Wasserstein-like' transportation distances for probability measures on discrete sets. We prove that the discrete transportation metrics on the d-dimensional discrete torus with mesh size 1/N converge, when $N\to\infty$, to the standard 2-Wasserstein distance W_2 on the continuous torus in the sense of Gromov-Hausdorff. This is the first convergence result for the recently developed discrete transportation metrics. The result shows the compatibility between these metrics and the well-established 2-Wasserstein metric.