# Smooth p-Wasserstein Distance: Structure, Empirical Approximation, and Statistical Applications

arXiv: https://arxiv.org/abs/2101.04039 · HTML: https://mlsanity.com/paper/2101.04039/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 73 references that are in the map.
- First posted on arXiv 11 Jan 2021.

## Builds on (most cited first, 21 of 21)

- [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)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [Wasserstein Auto-Encoders](https://mlsanity.com/paper/1711.01558/) (2017)
- [Data-driven Distributionally Robust Optimization Using the Wasserstein Metric: Performance Guarantees and Tractable Reformulations](https://mlsanity.com/paper/1505.05116/) (2015)
- [Near-linear time approximation algorithms for optimal transport via Sinkhorn iteration](https://mlsanity.com/paper/1705.09634/) (2017)
- [Statistical Aspects of Wasserstein Distances](https://mlsanity.com/paper/1806.05500/) (2018)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [Distributionally Robust Stochastic Optimization with Wasserstein Distance](https://mlsanity.com/paper/1604.02199/) (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)
- [Max-Sliced Wasserstein Distance and its use for GANs](https://mlsanity.com/paper/1904.05877/) (2019)
- [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)
- [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)
- [Optimal Transport Based Distributionally Robust Optimization: Structural Properties and Iterative Schemes](https://mlsanity.com/paper/1810.02403/) (2018)
- [Inference for Empirical Wasserstein Distances on Finite Spaces](https://mlsanity.com/paper/1610.03287/) (2016)
- [Pufferfish Privacy Mechanisms for Correlated Data](https://mlsanity.com/paper/1603.03977/) (2016)
- [Multivariate goodness-of-Fit tests based on Wasserstein distance](https://mlsanity.com/paper/2003.06684/) (2020)

## Built on by (most cited first, 15 of 15)

- [Controlling Wasserstein Distances by Kernel Norms with Application to Compressive Statistical Learning](https://mlsanity.com/paper/2112.00423/) (2021)
- [Sample Complexity of Probability Divergences under Group Symmetry](https://mlsanity.com/paper/2302.01915/) (2023)
- [Statistical and Topological Properties of Gaussian Smoothed Sliced Probability Divergences](https://mlsanity.com/paper/2110.10524/) (2021)
- [Algorithm- and Data-Dependent Generalization Bounds for Diffusion Models](https://mlsanity.com/paper/2506.03849/) (2025)
- [Information Processing Equalities and the Information-Risk Bridge](https://mlsanity.com/paper/2207.11987/) (2022)
- [Multivariate Stochastic Dominance via Optimal Transport and Applications to Models Benchmarking](https://mlsanity.com/paper/2406.06425/) (2024)
- [Differentially Private Wasserstein Barycenters](https://mlsanity.com/paper/2510.03021/) (2025)
- [Limit Distribution Theory for the Smooth 1-Wasserstein Distance with Applications](https://mlsanity.com/paper/2107.13494/) (2021)
- [A View on Out-of-Distribution Identification from a Statistical Testing Theory Perspective](https://mlsanity.com/paper/2405.03052/) (2024)
- [Generative Conditional Distributions by Neural (Entropic) Optimal Transport](https://mlsanity.com/paper/2406.02317/) (2024)
- [Scalable Sobolev IPM for Probability Measures on a Graph](https://mlsanity.com/paper/2502.00737/) (2025)
- [Quantitative convergence of trained single layer neural networks to Gaussian processes](https://mlsanity.com/paper/2509.24544/) (2025)
- [Gaussian-Smoothed Sliced Probability Divergences](https://mlsanity.com/paper/2404.03273/) (2024)
- [Bounds in Wasserstein Distance for Locally Stationary Functional Time Series](https://mlsanity.com/paper/2504.06453/) (2025)
- [E-ROBOT: a dimension-free method for robust statistics and machine learning via Schr"odinger bridge](https://mlsanity.com/paper/2509.11532/) (2025)

## Abstract (arXiv)

Discrepancy measures between probability distributions, often termed statistical distances, are ubiquitous in probability theory, statistics and machine learning. To combat the curse of dimensionality when estimating these distances from data, recent work has proposed smoothing out local irregularities in the measured distributions via convolution with a Gaussian kernel. Motivated by the scalability of this framework to high dimensions, we investigate the structural and statistical behavior of the Gaussian-smoothed p-Wasserstein distance mathsfWp(σ), for arbitrary p≥ 1. After establishing basic metric and topological properties of mathsfWp(σ), we explore the asymptotic statistical behavior of mathsfWp(σ)(hatμn,μ), where hatμn is the empirical distribution of n independent observations from μ. We prove that mathsfWp(σ) enjoys a parametric empirical convergence rate of n-1/2, which contrasts the n-1/d rate for unsmoothed mathsfWp when d ≥ 3. Our proof relies on controlling mathsfWp(σ) by a pth-order smooth Sobolev distance mathsfdp(σ) and deriving the limit distribution of √n,mathsfdp(σ)(hatμn,μ), for all dimensions d. As applications, we provide asymptotic guarantees for two-sample testing and minimum distance estimation using mathsfWp(σ), with experiments for p=2 using a maximum mean discrepancy formulation of mathsfd2(σ).