Obstructions to extension of Wasserstein distances for variable masses

TL;DR

This paper investigates the challenges in extending Wasserstein distances to all measures, revealing that natural conditions restrict the existence of such distances on unbounded spaces and zero measures, but exceptions exist on bounded spaces.

Contribution

The paper proves that under natural conditions, Wasserstein distance extensions on unbounded spaces must have a constant scaling factor, and no such extension exists when including the zero measure.

Findings

Scaling factor must be constant on unbounded spaces

No distance can include the zero measure on unbounded spaces

Examples of non-constant scaling factors exist on bounded spaces

Abstract

We study the possibility of defining a distance on the whole space of measures, with the property that the distance between two measures having the same mass is the Wasserstein distance, up to a scaling factor. We prove that, under very weak and natural conditions, if the base space is unbounded, then the scaling factor must be constant, independently of the mass. Moreover, no such distance can exist, if we include the zero measure. Instead, we provide examples with non-constant scaling factors for the case of bounded base spaces.