Gaussian-Smooth Optimal Transport: Metric Structure and Statistical Efficiency

TL;DR

This paper introduces Gaussian-smoothed optimal transport (GOT), a new framework that maintains Wasserstein metric properties while reducing high-dimensional approximation issues, bridging the gap between classical and entropic OT.

Contribution

It proposes GOT, which preserves Wasserstein metric structure and improves statistical efficiency, converging to classical OT as smoothing diminishes.

Findings

GOT maintains the 1-Wasserstein metric structure.

GOT alleviates the curse of dimensionality in empirical approximation.

GOT converges to classical OT as smoothing parameter approaches zero.

Abstract

Optimal transport (OT), and in particular the Wasserstein distance, has seen a surge of interest and applications in machine learning. However, empirical approximation under Wasserstein distances suffers from a severe curse of dimensionality, rendering them impractical in high dimensions. As a result, entropically regularized OT has become a popular workaround. However, while it enjoys fast algorithms and better statistical properties, it looses the metric structure that Wasserstein distances enjoy. This work proposes a novel Gaussian-smoothed OT (GOT) framework, that achieves the best of both worlds: preserving the 1-Wasserstein metric structure while alleviating the empirical approximation curse of dimensionality. Furthermore, as the Gaussian-smoothing parameter shrinks to zero, GOT $\Gamma$-converges towards classic OT (with convergence of optimizers), thus serving as a natural extension. An empirical study that supports the theoretical results is provided, promoting Gaussian-smoothed OT as a powerful alternative to entropic OT.