Sliced optimal transport, which is basically a Radon transform followed by one-dimensional optimal transport, became popular in various applications due to its efficient computation. In this paper, we deal with sliced optimal transport on the sphere $\mathbb{S}^{d-1}$ and on the rotation group SO(3). We propose a parallel slicing procedure of the sphere which requires again only optimal transforms on the line. We analyze the properties of the corresponding parallelly sliced optimal transport, which provides in particular a rotationally invariant metric on the spherical probability measures. For SO(3), we introduce a new two-dimensional Radon transform and develop its singular value decomposition. Based on this, we propose a sliced optimal transport on SO(3). As Wasserstein distances were extensively used in barycenter computations, we derive algorithms to compute the barycenters with respect to our new sliced Wasserstein distances and provide synthetic numerical examples on the 2-sphere that demonstrate their behavior both the free and fixed support setting of discrete spherical measures. In terms of computational speed, they outperform the existing methods for semicircular slicing as well as the regularized Wasserstein barycenters.

翻译：切片最优传输本质上是Radon变换后接一维最优传输，因其计算高效而在各类应用中广受欢迎。本文处理球面$\mathbb{S}^{d-1}$和旋转群SO(3)上的切片最优传输问题。我们提出球面上的并行切片方法，该方法同样仅需在直线上进行最优变换。我们分析了相应并行切片最优传输的性质，特别地，它为球面概率测度提供了旋转不变的度量。针对SO(3)，我们引入一种新的二维Radon变换并发展其奇异值分解。基于此，我们提出SO(3)上的切片最优传输。鉴于Wasserstein距离在重心计算中的广泛使用，我们推导了基于新型切片Wasserstein距离的重心计算算法，并在二维球面上给出合成数值示例，展示了离散球面测度在自由支撑和固定支撑两种设定下的行为。在计算速度方面，这些方法优于已有的半圆切片方法以及正则化Wasserstein重心方法。