Several applications in optimization, image, and signal processing deal with data that belong to the Stiefel manifold St(n,p), that is, the set of n-by-p matrices with orthonormal columns. Some applications, like the Riemannian center of mass, require evaluating the Riemannian distance between two arbitrary points on St(n,p). This can be done by explicitly constructing the geodesic connecting these two points. An existing method for finding geodesics is the leapfrog algorithm of J. L. Noakes. This algorithm is related to the Gauss-Seidel method, a classical iterative method for solving a linear system of equations that can be extended to nonlinear systems. We propose a convergence proof of leapfrog as a nonlinear Gauss-Seidel method. Our discussion is limited to the case of the Stiefel manifold, however, it may be generalized to other embedded submanifolds. We discuss other aspects of leapfrog and present some numerical experiments.

翻译：摘要：优化、图像和信号处理中的若干应用涉及属于施蒂费尔流形St(n,p)的数据，即由n×p正交列矩阵构成的集合。某些应用（如黎曼质心）需要计算St(n,p)上任意两点间的黎曼距离，这可通过显式构造连接这两点的测地线实现。现有的一种测地线求解方法是J.L. Noakes提出的蛙跳算法，该算法与高斯-赛德尔方法——一种可推广至非线性系统的经典线性方程组迭代求解法——存在关联。我们提出将蛙跳算法作为非线性高斯-赛德尔方法的收敛性证明。本文讨论仅限定于施蒂费尔流形情形，但该结果可推广至其他嵌入子流形。此外，我们讨论了蛙跳算法的其他特性并展示了数值实验。