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Abstract:
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Statistical distances, i.e., measures of discrepancy 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 statistical behavior of the Gaussian-smoothed $p$-Wasserstein distance (SWp) for arbitrary $p > 1$. To that end, we leverage the Benamou-Brenier dynamic formulation of optimal transport to prove that SWp is controlled by a related, $p$th order smooth dual Sobolev norm. We derive the limit distribution of the smooth empirical process in the dual Sobolev space under a sub-Gaussian condition, which implies a parametric empirical convergence rate of SWp. We also discuss applications to two-sample testing and minimum distance estimation.
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