Gaussian-Smoothed Optimal Transport: Metric Structure and Statistical Efficiency



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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.

Please cite our work using the BibTeX below.

  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 = 	 {},
  url = 	 {},
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