Scaling limits of discrete optimal transport

TL;DR

This paper investigates how discrete transport metrics approximate the continuous Wasserstein metric on discretized domains, establishing bounds and conditions for convergence, with implications for mesh design and metric approximation.

Contribution

It provides asymptotic upper bounds for discrete transport metrics in relation to Wasserstein distance and identifies isotropy as a key condition for lower bounds and convergence.

Findings

Upper bounds for discrete transport metrics in terms of Wasserstein distance

Lower bounds may fail without mesh isotropy

Isotropy ensures Gromov-Hausdorff convergence of the metrics

Abstract

We consider dynamical transport metrics for probability measures on discretisations of a bounded convex domain in $\mathbb{R}^d$. These metrics are natural discrete counterparts to the Kantorovich metric $\mathbb{W}_2$, defined using a Benamou-Brenier type formula. Under mild assumptions we prove an asymptotic upper bound for the discrete transport metric $\mathcal{W}_{\mathcal{T}}$ in terms of $\mathbb{W}_2$, as the size of the mesh $\mathcal{T}$ tends to $0$. However, we show that the corresponding lower bound may fail in general, even on certain one-dimensional and symmetric two-dimensional meshes. In addition, we show that the asymptotic lower bound holds under an isotropy assumption on the mesh, which turns out to be essentially necessary. This assumption is satisfied, e.g., for tilings by convex regular polygons, and it implies Gromov-Hausdorff convergence of the transport metric.