Kantorovich distance on a finite metric space

TL;DR

This paper studies the Kantorovich (1-Wasserstein) distance on finite metric spaces, providing new formulas and bounds, especially for graph-based metrics, and extending results to $ ext{l}_1$-embeddable distances.

Contribution

It derives a new characterization of Kantorovich distance for arbitrary weighted graphs via spanning trees and introduces related norms for $ ext{l}_1$-embeddable metrics.

Findings

K-distance equals the minimum over spanning trees for weighted graphs.

New dual LP formulation using Arens-Eells norm.

Partial extension to $ ext{l}_1$-embeddable distances.

Abstract

Kantorovich distance (or 1-Wasserstein distance) on the probability simplex of a finite metric space is the value of a Linear Programming problem for which a closed-form expression is known in some cases. When the ground distance is defined by a graph, a few examples have already been studied. In the present paper, after re-deriving, with different tools, the result for trees, we prove that, for an arbitrary weighted graph, the K-distance is the minimum of the K-distances over all the spanning trees associated with the graph. We work in the dual LP-problem by using Arens-Eells norm associated with the metric space. Finally, we introduce new norms that are naturally related to $\ell_1$-embeddable distances and allows for a partial extension of our results to this new setting.