Gaussian-Smooth Optimal Transport: Metric Structure and Statistical Efficiency
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 -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.
Cite
@article{arxiv.2001.09206,
title = {Gaussian-Smooth Optimal Transport: Metric Structure and Statistical Efficiency},
author = {Ziv Goldfeld and Kristjan Greenewald},
journal= {arXiv preprint arXiv:2001.09206},
year = {2020}
}