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

Gaussian-Smooth Optimal Transport: Metric Structure and Statistical Efficiency