论文标题
跨越算法作为非线性高斯 - 塞德尔
The leapfrog algorithm as nonlinear Gauss-Seidel
论文作者
论文摘要
优化，图像和信号处理中的几个应用程序与属于Stiefel歧管ST（N，P）的数据，即具有正统列的N-P矩阵集。某些应用，例如Riemannian质量中心，需要评估ST上两个任意点之间的Riemannian距离（N，P）。这可以通过明确构建这两个点连接的测量线来完成。现有的寻找大地学的方法是J. L. Noakes的跨越算法。该算法与高斯 - 西德尔方法有关，Gauss-seidel方法是一种经典的迭代方法，用于求解可以扩展到非线性系统的方程式线性系统。我们提出了跨越的趋同证明，作为非线性高斯 - seidel方法。我们的讨论仅限于Stiefel歧管的情况，但是，它可能会推广到其他嵌入式子手机。我们讨论了Leapfrog的其他方面，并提出了一些数值实验。
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.
