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Discrepancy measures between probability distributions, often termedstatistical distances, are ubiquitous in probability theory, statistics andmachine learning. To combat the curse of dimensionality when estimating thesedistances from data, recent work has proposed smoothing out localirregularities in the measured distributions via convolution with a Gaussiankernel. Motivated by the scalability of this framework to high dimensions, weinvestigate the structural and statistical behavior of the Gaussian-smoothed$p$-Wasserstein distance $\mathsf{W}_p^{(\sigma)}$, for arbitrary $p\geq 1$.After establishing basic metric and topological properties of$\mathsf{W}_p^{(\sigma)}$, we explore the asymptotic statistical behavior of$\mathsf{W}_p^{(\sigma)}(\hat{\mu}_n,\mu)$, where $\hat{\mu}_n$ is theempirical distribution of $n$ independent observations from $\mu$. We provethat $\mathsf{W}_p^{(\sigma)}$ enjoys a parametric empirical convergence rateof $n^{-1/2}$, which contrasts the $n^{-1/d}$ rate for unsmoothed$\mathsf{W}_p$ when $d \geq 3$. Our proof relies on controlling$\mathsf{W}_p^{(\sigma)}$ by a $p$th-order smooth Sobolev distance$\mathsf{d}_p^{(\sigma)}$ and deriving the limit distribution of$\sqrt{n}\,\mathsf{d}_p^{(\sigma)}(\hat{\mu}_n,\mu)$, for all dimensions $d$.As applications, we provide asymptotic guarantees for two-sample testing andminimum distance estimation using $\mathsf{W}_p^{(\sigma)}$, with experimentsfor $p=2$ using a maximum mean discrepancy formulation of$\mathsf{d}_2^{(\sigma)}$.

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