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：优化、 图像 和 信号处理 中 几个应用程序 处理 Stiefel plents St (n, p) 的数据, 即 一组 n- by- p 矩阵, 带有正态列。 一些应用程序, 如 Riemannian 质量中心, 需要评估St(n, p) 上两个任意点之间的里曼尼距离。 可以通过明确构建连接这两个点的大地测量学来完成。 现有的测地法是 J. L. Noakes 的跳蛙算法。 这个算法与高斯- 赛德尔 方法有关, 这是一种典型的迭代方法, 用来解决可扩展至非线性系统的直线性方程系统。 我们提出跳蛙的趋同证据, 作为一种非线性高斯- 赛德尔 方法。 我们的讨论限于 Stiefel 柱体的情况, 但是, 它可以推广到其他嵌入的子体。 我们讨论跳蛙的其他方面, 并提出一些数字实验 。
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