# Scaling limits of discrete optimal transport

arXiv: https://arxiv.org/abs/1809.01092 · HTML: https://mlsanity.com/paper/1809.01092/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Sep 2018.

## Built on by (most cited first, 2 of 2)

- [Static and Dynamic Approaches to Computing Barycenters of Probability Measures on Graphs](https://mlsanity.com/paper/2603.26940/) (2026)
- [Scaling Limits of the Wasserstein information matrix on Gaussian Mixture Models](https://mlsanity.com/paper/2309.12997/) (2023)

## Abstract (arXiv)

We consider dynamical transport metrics for probability measures on discretisations of a bounded convex domain in ℝd. These metrics are natural discrete counterparts to the Kantorovich metric mathbbW2, defined using a Benamou-Brenier type formula. Under mild assumptions we prove an asymptotic upper bound for the discrete transport metric WT in terms of mathbbW2, as the size of the mesh 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.