  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 supports the theoretical results is provided, promoting
Gaussian-smoothed OT as a powerful alternative to entropic OT.
