  Comparing spherical probability distributions is of great interest in various
fields, including geology, medical domains, computer vision, and deep
representation learning. The utility of optimal transport-based distances, such
as the Wasserstein distance, for comparing probability measures has spurred
active research in developing computationally efficient variations of these
distances for spherical probability measures. This paper introduces a
high-speed and highly parallelizable distance for comparing spherical measures
using the stereographic projection and the generalized Radon transform, which
we refer to as the Stereographic Spherical Sliced Wasserstein (S3W) distance.
We carefully address the distance distortion caused by the stereographic
projection and provide an extensive theoretical analysis of our proposed metric
and its rotationally invariant variation. Finally, we evaluate the performance
of the proposed metrics and compare them with recent baselines in terms of both
speed and accuracy through a wide range of numerical studies, including
gradient flows and self-supervised learning. Our code is available at
https://github.com/mint-vu/s3wd.
