Gaussian-Smoothed Optimal Transport: Metric Structure and Statistical Efficiency
Ziv Goldfeld, Kristjan Greenewald
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 validates the theoretical results is provided, promoting Gaussian-smoothed OT as a powerful alternative to entropic OT.
BibTeX
@InProceedings{pmlr-v108-goldfeld20a,
title = {Gaussian-Smoothed Optimal Transport: Metric Structure and Statistical Efficiency},
author = {Goldfeld, Ziv and Greenewald, Kristjan},
booktitle = {Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics},
pages = {3327--3337},
year = {2020},
editor = {Chiappa, Silvia and Calandra, Roberto},
volume = {108},
series = {Proceedings of Machine Learning Research},
month = {26--28 Aug},
publisher = {PMLR},
pdf = {http://proceedings.mlr.press/v108/goldfeld20a/goldfeld20a.pdf},
url = {https://proceedings.mlr.press/v108/goldfeld20a.html},
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 validates the theoretical results is provided, promoting Gaussian-smoothed OT as a powerful alternative to entropic OT.}
}