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

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 W_p(σ), for arbitrary p ≥ 1. After establishing basic metric and topological properties of W_p(σ), we explore the asymptotic statistical behavior of W_p(σ)(µ_n, µ), where µ_n is the empirical distribution of n independent observations from µ. We prove that W_p(σ) enjoys a parametric empirical convergence rate of n⁻¹/², which contrasts the n^−1/d rate for unsmoothed W_p when d ≥ 3. Our proof relies on controlling W_p(σ) by a pth-order smooth Sobolev distance d(_p^σ) and deriving the limit distribution of √nd(_p^σ)(µ_n, µ), for all dimensions d. As applications, we provide asymptotic guarantees for two-sample testing and minimum distance estimation using W_p(σ), with experiments for p = 2 using a maximum mean discrepancy formulation of d(₂^σ).