Gromov--Hausdorff Convergence of Discrete Transportation 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 $\mathcal{W}_N$ on the $d$-dimensional discrete torus $\mathbf{T}_N^d$ with mesh size $\frac1N$ 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 $\mathcal{W}$. The result shows the compatibility between these metrics and the well-established $2$-Wasserstein metric.