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.

翻译：比较球面概率分布在多个领域具有重要价值，包括地质学、医学领域、计算机视觉和深度表示学习。基于最优传输的距离度量（如Wasserstein距离）在比较概率测度方面的效用，推动了开发适用于球面概率测度的高效计算变体的研究。本文提出一种利用立体投影和广义Radon变换的高速可并行化球面测度比较距离，我们称之为立体球面切片Wasserstein（S3W）距离。我们严谨处理了立体投影造成的距离畸变问题，并针对所提出的度量及其旋转不变变体进行了详尽的理论分析。最后，通过包含梯度流和自监督学习在内的广泛数值研究，我们在速度和精度两个维度上评估了所提度量的性能，并与近期基线方法进行了比较。