From Smooth Wasserstein Distance to Dual Sobolev Norm: Empirical Approximation and Statistical Applications

Statistical distances, i.e., discrepancy measures between probability distributions, 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 the smooth framework to high dimensions, we conduct an in-depth study of the structural and statistical behavior of the Gaussian-smoothed $p$-Wasserstein distance $\mathsf{W}_p^{(\sigma)}$, for arbitrary $p\geq 1$. We start by showing that $\mathsf{W}_p^{(\sigma)}$ admits a metric structure that is topologically equivalent to classic $\mathsf{W}_p$ and is stable with respect to perturbations in $\sigma$. Moving to statistical questions, we explore the asymptotic properties of $\mathsf{W}_p^{(\sigma)}(\hat{\mu}_n,\mu)$, where $\hat{\mu}_n$ is the empirical distribution of $n$ i.i.d. samples from $\mu$. To that end, we prove that $\mathsf{W}_p^{(\sigma)}$ is controlled by a $p$th order smooth dual Sobolev norm $\mathsf{d}_p^{(\sigma)}$. Since $\mathsf{d}_p^{(\sigma)}(\hat{\mu}_n,\mu)$ coincides with the supremum of an empirical process indexed by Gaussian-smoothed Sobolev functions, it lends itself well to analysis via empirical process theory. We derive the limit distribution of $\sqrt{n}\mathsf{d}_p^{(\sigma)}(\hat{\mu}_n,\mu)$ in all dimensions $d$, when $\mu$ is sub-Gaussian. Through the aforementioned bound, this implies a parametric empirical convergence rate of $n^{-1/2}$ for $\mathsf{W}_p^{(\sigma)}$, contrasting the $n^{-1/d}$ rate for unsmoothed $\mathsf{W}_p$ when $d \geq 3$. As applications, we provide asymptotic guarantees for two-sample testing and minimum distance estimation. When $p=2$, we further show that $\mathsf{d}_2^{(\sigma)}$ can be expressed as a maximum mean discrepancy.