# Limit Distribution Theory for the Smooth 1-Wasserstein Distance with Applications

arXiv: https://arxiv.org/abs/2107.13494 · HTML: https://mlsanity.com/paper/2107.13494/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 128 references that are in the map.
- First posted on arXiv 28 Jul 2021.

## Builds on (most cited first, 25 of 25)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [Generalization and Equilibrium in Generative Adversarial Nets (GANs)](https://mlsanity.com/paper/1703.00573/) (2017)
- [Gromov-Wasserstein Alignment of Word Embedding Spaces](https://mlsanity.com/paper/1809.00013/) (2018)
- [Unsupervised Alignment of Embeddings with Wasserstein Procrustes](https://mlsanity.com/paper/1805.11222/) (2018)
- [Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem](https://mlsanity.com/paper/1905.11882/) (2019)
- [Statistical and Topological Properties of Sliced Probability Divergences](https://mlsanity.com/paper/2003.05783/) (2020)
- [Plugin Estimation of Smooth Optimal Transport Maps](https://mlsanity.com/paper/2107.12364/) (2021)
- [Hierarchical Optimal Transport for Document Representation](https://mlsanity.com/paper/1906.10827/) (2019)
- [Rates of Estimation of Optimal Transport Maps using Plug-in Estimators via Barycentric Projections](https://mlsanity.com/paper/2107.01718/) (2021)
- [Multilevel Clustering via Wasserstein Means](https://mlsanity.com/paper/1706.03883/) (2017)
- [On the Discrimination-Generalization Tradeoff in GANs](https://mlsanity.com/paper/1711.02771/) (2017)
- [Asymptotic Guarantees for Learning Generative Models with the Sliced-Wasserstein Distance](https://mlsanity.com/paper/1906.04516/) (2019)
- [Convergence and Concentration of Empirical Measures under Wasserstein Distance in Unbounded Functional Spaces](https://mlsanity.com/paper/1804.10556/) (2018)
- [On Projection Robust Optimal Transport: Sample Complexity and Model Misspecification](https://mlsanity.com/paper/2006.12301/) (2020)
- [Inference for Empirical Wasserstein Distances on Finite Spaces](https://mlsanity.com/paper/1610.03287/) (2016)
- [Entropic Optimal Transport: Geometry and Large Deviations](https://mlsanity.com/paper/2102.04397/) (2021)
- [Smooth p-Wasserstein Distance: Structure, Empirical Approximation, and Statistical Applications](https://mlsanity.com/paper/2101.04039/) (2021)
- [Multivariate goodness-of-Fit tests based on Wasserstein distance](https://mlsanity.com/paper/2003.06684/) (2020)
- [Wasserstein Barycenter Model Ensembling](https://mlsanity.com/paper/1902.04999/) (2019)
- [Convergence of Gaussian-smoothed optimal transport distance with sub-gamma distributions and dependent samples](https://mlsanity.com/paper/2103.00394/) (2021)

## Built on by (most cited first, 1 of 1)

- [Nonparametric mixture MLEs under Gaussian-smoothed optimal transport distance](https://mlsanity.com/paper/2112.02421/) (2021)

## Abstract (arXiv)

The smooth 1-Wasserstein distance (SWD) W1σ was recently proposed as a means to mitigate the curse of dimensionality in empirical approximation while preserving the Wasserstein structure. Indeed, SWD exhibits parametric convergence rates and inherits the metric and topological structure of the classic Wasserstein distance. Motivated by the above, this work conducts a thorough statistical study of the SWD, including a high-dimensional limit distribution result for empirical W1σ, bootstrap consistency, concentration inequalities, and Berry-Esseen type bounds. The derived nondegenerate limit stands in sharp contrast with the classic empirical W1, for which a similar result is known only in the one-dimensional case. We also explore asymptotics and characterize the limit distribution when the smoothing parameter σ is scaled with n, converging to 0 at a sufficiently slow rate. The dimensionality of the sampled distribution enters empirical SWD convergence bounds only through the prefactor (i.e., the constant). We provide a sharp characterization of this prefactor's dependence on the smoothing parameter and the intrinsic dimension. This result is then used to derive new empirical convergence rates for classic W1 in terms of the intrinsic dimension. As applications of the limit distribution theory, we study two-sample testing and minimum distance estimation (MDE) under W1σ. We establish asymptotic validity of SWD testing, while for MDE, we prove measurability, almost sure convergence, and limit distributions for optimal estimators and their corresponding W1σ error. Our results suggest that the SWD is well suited for high-dimensional statistical learning and inference.