球面上的切片最优传输
Sliced Optimal Transport on the Sphere
AI总结：
本文针对2-球面上的最优传输问题，提出垂直切片变换和半圆变换两种球面切片方法，分别导出适用于偶概率测度和任意概率测度的切片Wasserstein距离，并通过插值与分类实验验证了其有效性。
摘要：
切片最优传输将多维域上的最优传输约化为直线上的传输。更准确地说，切片最优传输是众所周知的Radon变换与累积密度变换的串联，它解析地给出了约化传输问题的解。受这一概念的启发，我们针对2-球面上的最优传输提出了两种改进方案。首先，作为Radon变换的对应物，我们引入了垂直切片变换，它沿所有正交于给定方向的圆进行积分。其次，我们引入了一种半圆变换，它沿所有半大圆并以适当的权函数进行积分。这两种变换都被推广到球面上的任意测度。垂直切片变换可以与区间上的最优传输相结合，并导出一个限制在偶概率测度上的切片Wasserstein距离；而半圆变换则与圆上的最优传输相关，并为任意概率测度导出一个不同的切片Wasserstein距离。这两种新的球面切片最优传输概念的适用性通过处理球面概率测度的插值和分类的概念验证示例得到了展示。数值实现依赖于这两种变换的奇异值分解和快速Fourier技术。对于关于概率测度的反演，我们提出最小化一个熵正则化的Kullback--Leibler散度，该最小化可以通过原始-对偶邻近分裂算法在数值上实现。
英文摘要：
Sliced optimal transport reduces optimal transport on multi-dimensional domains to transport on the line. More precisely, sliced optimal transport is the concatenation of the well-known Radon transform and the cumulative density transform, which analytically yields the solutions of the reduced transport problems. Inspired by this concept, we propose two adaptions for optimal transport on the 2-sphere. Firstly, as counterpart to the Radon transform, we introduce the vertical slice transform, which integrates along all circles orthogonal to a given direction. Secondly, we introduce a semicircle transform, which integrates along all half great circles with an appropriate weight function. Both transforms are generalized to arbitrary measures on the sphere. While the vertical slice transform can be combined with optimal transport on the interval and leads to a sliced Wasserstein distance restricted to even probability measures, the semicircle transform is related to optimal transport on the circle and results in a different sliced Wasserstein distance for arbitrary probability measures. The applicability of both novel sliced optimal transport concepts on the sphere is demonstrated by proof-of-concept examples dealing with the interpolation and classification of spherical probability measures. The numerical implementation relies on the singular value decompositions of both transforms and fast Fourier techniques. For the inversion with respect to probability measures, we propose the minimization of an entropy-regularized Kullback--Leibler divergence, which can be numerically realized using a primal-dual proximal splitting algorithm.
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