The Observable Wasserstein Distance

TL;DR

The paper introduces the observable Wasserstein distance, a scalable framework for lower bounding Wasserstein distances on complex metric spaces by projecting measures onto the real line using Lipschitz observables.

Contribution

It develops a hierarchy of pseudo-metrics based on observable projections, establishing an injectivity result linking measure support complexity to metric recovery.

Findings

Hierarchy provides a tunable trade-off between bound sharpness and efficiency

Numerical experiments validate the effectiveness of the proposed approximations

Framework generalizes sliced Wasserstein to non-Euclidean spaces

Abstract

We introduce the observable Wasserstein distance, a framework for deriving lower bounds on the Wasserstein distance between probability measures on Polish metric spaces, designed to bypass the computational intractability of exact optimal transport in large-scale, non-Euclidean datasets. Analogous to the sliced Wasserstein distance in $\mathbb{R}^d$, our approach projects measures onto the real line via 1-Lipschitz observables and computes the Wasserstein distances between the resulting pushforward distributions. We define a hierarchy of pseudo-metrics by restricting observables to a nested chain of subspaces. A central theoretical contribution is an injectivity result linking the metric covering dimension of the support of a measure to the specific order in the hierarchy that guarantees unique recovery. This serves as a metric-space analogue to the Cram\'{e}r-Wold Device for Euclidean distributions. We demonstrate that this hierarchy offers a tunable trade-off between sharpness as a lower bound on the Wasserstein distance and computational efficiency. We also present a discrete computational model for finite grids and numerical experiments validating the efficacy and utility of these approximations.