# The Observable Wasserstein Distance

arXiv: https://arxiv.org/abs/2605.09916 · HTML: https://mlsanity.com/paper/2605.09916/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 47 references that are in the map.
- First posted on arXiv 11 May 2026.

## Builds on (most cited first, 18 of 18)

- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [A Point Set Generation Network for 3D Object Reconstruction from a Single Image](https://mlsanity.com/paper/1612.00603/) (2016)
- [PCN: Point Completion Network](https://mlsanity.com/paper/1808.00671/) (2018)
- [PPF-FoldNet: Unsupervised Learning of Rotation Invariant 3D Local Descriptors](https://mlsanity.com/paper/1808.10322/) (2018)
- [Density-aware Chamfer Distance as a Comprehensive Metric for Point Cloud Completion](https://mlsanity.com/paper/2111.12702/) (2021)
- [Low-Rank Sinkhorn Factorization](https://mlsanity.com/paper/2103.04737/) (2021)
- [On Efficient Optimal Transport: An Analysis of Greedy and Accelerated Mirror Descent Algorithms](https://mlsanity.com/paper/1901.06482/) (2019)
- [Total Denoising: Unsupervised Learning of 3D Point Cloud Cleaning](https://mlsanity.com/paper/1904.07615/) (2019)
- [Point-set Distances for Learning Representations of 3D Point Clouds](https://mlsanity.com/paper/2102.04014/) (2021)
- [Statistical, Robustness, and Computational Guarantees for Sliced Wasserstein Distances](https://mlsanity.com/paper/2210.09160/) (2022)
- [On Projection Robust Optimal Transport: Sample Complexity and Model Misspecification](https://mlsanity.com/paper/2006.12301/) (2020)
- [Outlier-Robust Optimal Transport: Duality, Structure, and Statistical Analysis](https://mlsanity.com/paper/2111.01361/) (2021)
- [Minimax Confidence Intervals for the Sliced Wasserstein Distance](https://mlsanity.com/paper/1909.07862/) (2019)
- [3D Point Cloud Denoising via Deep Neural Network based Local Surface Estimation](https://mlsanity.com/paper/1904.04427/) (2019)
- [A Near-Linear Time Algorithm for the Chamfer Distance](https://mlsanity.com/paper/2307.03043/) (2023)
- [Sharp comparisons between sliced and standard 1-Wasserstein distances](https://mlsanity.com/paper/2510.16465/) (2025)
- [Metrics for Parametric Families of Networks](https://mlsanity.com/paper/2509.22549/) (2025)

## Abstract (arXiv)

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 ℝd, 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'er-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.