Kengo Kato (Cornell University)

Math & Stats Department Colloquium

Friday, March 18th, 2022

Time: 2:30 p.m.  Place: Online (via Zoom)

Speaker: Kengo Kato (Cornell University)

Title: Scalable statistical theory for smooth Wasserstein distances

Abstract: The Wasserstein distance is a metric on a space of probability measures that has 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 accurately estimate them 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, in this work, 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 consistency of the nonparametric bootstrap. Finally, we use the limit distribution theory to study applications to generative modeling via minimum distance estimation with the smooth Wasserstein distance, showing asymptotic normality of optimal solutions for the quadratic cost.

Kengo Kato is a Full Professor in the Department of Statistics and Data Science at Cornell University. Prior to Cornell, he was an Associate Professor in the Graduate School of Economics at the University of Tokyo. His research interests include mathematical statistics, econometrics, quantile regression, high-dimensional/nonparametric statistics. He receives many awards, including Analysis Award (2021), Japan Academy Medal (2020), JSPS Prize (2020).