Scaling Wasserstein distances to high dimensions via smoothing

Wasserstein distances has recently seen a surge of applications in statistics and machine learning. This stems from many advantageous properties they possess, such as metric and topological structure of Wasserstein spaces, robustness to support mismatch, compatibility to gradient-based optimization, and rich geometric properties. In practice, we rarely have access to the actual distribution and only get data from it, which necessitates estimating the distance from samples. A central issue is that such estimators suffer from the curse of dimensionality: their empirical convergence rate scales as n^{-1/d} for d-dimensional distributions. This rate deteriorates exponentially fast with dimension, making it impossible to obtain meaningful accuracy guarantees, especially given the dimensionality of real-world data.

This talk will present a novel framework of smooth p-Wasserstein distances, that inherits the properties of their classic counterparts while alleviating the empirical curse of dimensionality. Specifically, we will show that the empirical approximation error of the smooth distance decays as n^{-1/2}, in all dimensions. For the special case of the smooth 1-Wasserstein distance, we will also derive a high-dimensional limit distribution, further highlighting the favorable statistical behavior of the smooth framework. The derivation of statistical efficiency for the general pth order distance employs a new comparison result to the smooth dual Sobolev norm. Applications to implicit generative modeling will be considered, leveraging the parametric empirical convergence rates to establish n^{-1/2} generalization bounds in any dimension.

Bio:
Ziv Goldfeld is an assistant professor in the School of Electrical and Computer Engineering, and a graduate field member in Computer Science and the Center of Applied Mathematics, at Cornell University. Before joining Cornell, he was a postdoctoral research fellow in LIDS at MIT, hosted by Yury Polyanskiy. Ziv graduated with a B.Sc., M.Sc., and Ph.D. (all summa cum laude) in Electrical and Computer Engineering from Ben Gurion University, Israel. His graduate advisor was Haim Permuter. Ziv's research interests include optimal transport theory, statistical learning theory, information theory, and high-dimensional and nonparametric statistics. He seeks to understand and design inference and information processing systems by formulating and solving mathematical models.

Honors include the NSF CAREER Award, NSF CRII Award, the IBM University Award, and the Rothschild Postdoctoral Fellowship.