The Observable Wasserstein Distance

2026-05-10

AI Summary

This work addresses the computational challenges of evaluating Wasserstein distances in large-scale non-Euclidean spaces by proposing a projection-based approach grounded in 1-Lipschitz observables. The method pushes forward probability measures onto the real line, computes their one-dimensional Wasserstein distances, and constructs a hierarchy of pseudometrics over nested subspaces to approximate the original distance. This hierarchical framework balances accuracy and efficiency through tunable parameters and establishes a theoretical link between the metric covering dimension of the support set and the order required for unique measure recovery—providing an analogue of the Cramér–Wold theorem in non-Euclidean settings. Theoretical analysis confirms that measures can be uniquely recovered at specific hierarchy levels, and numerical experiments on finite discrete grids demonstrate the method’s effectiveness and practicality.

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 Cramér-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.

Problem

• Wasserstein distance
• computational intractability
• non-Euclidean datasets
• probability measures
• optimal transport

Innovation

• observable Wasserstein distance
• optimal transport
• 1-Lipschitz observables
• metric covering dimension
• Cramér-Wold analogue