Title: Limit theorems for smooth Wasserstein distances
Authors: Kengo Kato - Cornell University (United States) [presenting]
Ziv Goldfeld - Cornell University (United States)
Sloan Nietert - Cornell University (United States)
Gabriel Rioux - Cornell University (United States)
Abstract: The Wasserstein distance is a metric on a space of probability measures that have seen a surge of applications in statistics, machine learning, and applied mathematics. However, statistical aspects of Wasserstein distances are bottlenecked by the curse of dimensionality, whereby the number of data points needed to be estimated accurately grows exponentially with dimension. Gaussian smoothing was recently introduced as a means to alleviate the curse of dimensionality, giving rise to a parametric convergence rate in any dimension, while preserving the Wasserstein metric and topological structure. To facilitate valid statistical inference, we develop a comprehensive limit distribution theory for the empirical smooth Wasserstein distance. The limit distribution results leverage the functional delta method after embedding the domain of the Wasserstein distance into a certain dual Sobolev space, characterizing its Hadamard directional derivative for the dual Sobolev norm, and establishing weak convergence of the smooth empirical process in the dual space. To estimate the distributional limits, we also establish the consistency of the nonparametric bootstrap.