The Observable Wasserstein Distance
This work addresses the computational bottleneck of optimal transport for large-scale, non-Euclidean datasets by providing a practical approximation framework.
The paper introduces the observable Wasserstein distance, a framework for efficiently computing lower bounds on the Wasserstein distance between probability measures on Polish metric spaces, bypassing the computational intractability of exact optimal transport. The method projects measures onto the real line via 1-Lipschitz observables and provides a tunable trade-off between bound sharpness and computational efficiency, validated through numerical experiments.
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.