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 for 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：切片最优输运本质上是一种先进行拉东变换再执行一维最优输运的方法，因其高效的计算性能而在众多应用中得到普及。本文研究球面 $\mathbb{S}^{d-1}$ 与旋转群 SO(3) 上的切片最优输运。我们提出一种球面的并行切片方法，该方法同样仅需进行直线上的最优输运变换。我们分析了相应并行切片最优输运的性质，特别地，它为球面概率测度提供了一种旋转不变度量。对于 SO(3)，我们引入一种新的二维拉东变换并推导其奇异值分解。基于此，我们提出了 SO(3) 上的切片最优输运方法。由于 Wasserstein 距离在重心计算中被广泛使用，我们推导了基于新切片 Wasserstein 距离的重心计算算法，并在二维球面上提供了合成数值算例，以展示其在离散球面测度的自由支撑与固定支撑设定下的表现。在计算速度方面，这些方法优于现有的半圆切片方法以及正则化 Wasserstein 重心计算方法。