Central limit theorem for the Sliced 1-Wasserstein distance and the max-Sliced 1-Wasserstein distance

Abstract: The Wasserstein distance has been an attractive tool in many fields. But due to its high computational complexity and the phenomenon of the curse of dimensionality in empirical estimation, various extensions of the Wasserstein distance have been proposed to overcome the shortcomings such as the Sliced Wasserstein distance. It enjoys a low computational cost and dimension-free sample complexity, but there are few distributional limit results of it. In this paper, we focus on Sliced 1-Wasserstein distance and its variant max-Sliced 1-Wasserstein distance. We utilize the central limit theorem in Banach space to derive the limit distribution for the Sliced 1-Wasserstein distance. Through viewing the empirical max-Sliced 1-Wasserstein distance as a supremum of an empirical process indexed by some function class, we prove that the function class is P-Donsker under mild moment assumption. Moreover, for computing Sliced p-Wasserstein distance based on Monte Carlo method, we explore that how many random projections that can make sure the error small in high probability. We also provide upper bound of the expected max-Sliced 1-Wasserstein between the true and the empirical probability measures under different conditions and the concentration inequalities for max-Sliced 1-Wasserstein distance are also presented. As applications of the theory, we utilize them for two-sample testing problem.

• The paper establishes central limit theorems for the Sliced 1-Wasserstein and max-Sliced 1-Wasserstein distances, showing empirical convergence to Gaussian processes under mild moment assumptions.
• It provides a detailed analysis of convergence rates and computational trade-offs by quantifying the Monte Carlo projections needed for accurate approximations.
• The results enhance non-parametric two-sample testing by linking empirical measures to robust hypothesis testing frameworks applicable in high-dimensional data analysis.

Central Limit Theorem for the Sliced 1-Wasserstein Distance and the max-Sliced 1-Wasserstein Distance

Introduction

The paper "Central limit theorem for the Sliced 1-Wasserstein distance and the max-Sliced 1-Wasserstein distance" (2205.14624) presents significant theoretical advancements in understanding the asymptotic behavior of the Sliced Wasserstein Distance (SWD) and its variant, the max-Sliced Wasserstein Distance (max-SWD). These metrics are computationally attractive alternatives to the traditional Wasserstein distance due to their reduced complexity and dimension-free sample complexity. Despite these advantages, their theoretical properties, particularly their distributional limits, have been less explored.

Sliced Wasserstein Distance (SWD)

SWD measures the distance between two probability distributions by slicing them along random one-dimensional projections and computing the Wasserstein distance along each slice. The paper establishes a central limit theorem (CLT) for the Sliced 1-Wasserstein Distance (SW1), detailing the distributional convergence of its empirical estimates under certain assumptions. The authors utilize the CLT in Banach space to derive the limit distributions, leveraging uniform random projections on the unit sphere.

Key Results

1. Central Limit Theorem for SW1: The paper proves that under mild conditions on the distribution's moments, the empirical SW1 converges in distribution to a Gaussian process. This result is crucial for statistical inference tasks involving SW1.
2. Convergence Rate: The authors provide a detailed analysis showing that the expected SW1 converges to zero at a rate dependent on the sample size and the underlying distribution's moment conditions.
3. Computational Considerations: The paper discusses the number of Monte Carlo projections needed to achieve a small approximation error with high probability, highlighting the trade-off between computational efficiency and statistical accuracy.

max-Sliced Wasserstein Distance (max-SWD)

The max-SWD takes the maximum distance over one-dimensional projections rather than averaging them like SWD. This metric provides a more robust measure of distributional discrepancy, especially useful in two-sample testing problems.

Key Results

1. Limit Distribution for max-SWD: The authors show that the max-SWD between empirical and true distributions can be linked to a supremum of an empirical process indexed by a function class. By proving that this function class is $P$-Donsker under certain conditions, they derive the associated Gaussian distribution using the continuous mapping theorem.
2. Comparison with Integral Probability Metrics: The methodology employed extends to derive similar asymptotic properties for other integral probability metrics, demonstrating the versatility of the theoretical framework.

Practical Applications

The theoretical contributions of the paper have direct applications in various statistical tasks, such as two-sample testing. By characterizing the limiting distributions of SW1 and max-SWD, the authors enable more robust hypothesis testing frameworks that are less sensitive to dimensionality and sample size.

Two-Sample Testing

The paper proposes test statistics based on SW1 and max-SWD for non-parametric two-sample testing. The asymptotic results ensure that these tests maintain their level under the null hypothesis and achieve consistency under alternatives.

Conclusion

This paper provides a rigorous theoretical foundation for SWD and max-SWD, enhancing their applicability in statistical and machine learning tasks. The derived central limit theorems and computational insights bridge the gap between theory and practice, enabling efficient implementation of these metrics in high-dimensional data analysis. Future work may explore further generalizations to other classes of Wasserstein metrics, fostering broader adoption in empirical research.