The Observable Wasserstein Distance

• URL: http://arxiv.org/abs/2605.09916v1
• Date: Mon, 11 May 2026 03:10:33 GMT
• Title: The Observable Wasserstein Distance
• Abstract summary: We introduce the observable Wasserstein distance, a framework for deriving lower bounds on the Wasserstein distance between probability measures on Polish metric spaces.<n>Our approach projects measures onto the real distributions line via 1-Lipschitz observables and computes the Wasserstein distances between the resulting pushforward.<n>We demonstrate that this hierarchy offers a tunable trade-off between sharpness as a lower bound on the Wasserstein distance and computational efficiency.
• Score: 2.533665398856814
• License: http://creativecommons.org/publicdomain/zero/1.0/
• 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.

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