Limit Distribution Theory for the Smooth 1-Wasserstein Distance with Applications

Understand

The smooth 1-Wasserstein distance (SWD) $W_1^\sigma$ was recently proposed as a means to mitigate the curse of dimensionality in empirical approximation while preserving the Wasserstein structure.

• Indeed, SWD exhibits parametric convergence rates and inherits the metric and topological structure of the classic Wasserstein distance.
• Motivated by the above, this work conducts a thorough statistical study of the SWD, including a high-dimensional limit distribution result for empirical $W_1^\sigma$, bootstrap consistency, concentration inequalities, and Berry-Esseen type bounds.
• The derived nondegenerate limit stands in sharp contrast with the classic empirical $W_1$, for which a similar result is known only in the one-dimensional case.