  The Wasserstein distance and its variations, e.g., the sliced-Wasserstein
(SW) distance, have recently drawn attention from the machine learning
community. The SW distance, specifically, was shown to have similar properties
to the Wasserstein distance, while being much simpler to compute, and is
therefore used in various applications including generative modeling and
general supervised/unsupervised learning. In this paper, we first clarify the
mathematical connection between the SW distance and the Radon transform. We
then utilize the generalized Radon transform to define a new family of
distances for probability measures, which we call generalized
sliced-Wasserstein (GSW) distances. We also show that, similar to the SW
distance, the GSW distance can be extended to a maximum GSW (max-GSW) distance.
We then provide the conditions under which GSW and max-GSW distances are indeed
distances. Finally, we compare the numerical performance of the proposed
distances on several generative modeling tasks, including SW flows and SW
auto-encoders.
