Gaussian-Smoothed Optimal Transport: Metric Structure and Statistical Efficiency

Optimal transport (OT), in particular the Wasserstein distance, has seen a surge of interest and applications in machine learning. However, empirical approximation under Wasserstein distances su↵ers from a severe curse of dimensionality, rendering them impractical in high dimensions. As a result, entropically regularized OT has become a popular workaround. While it enjoys fast algorithms and better statistical properties, it however loses 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.