The leapfrog algorithm as nonlinear Gauss-Seidel

TL;DR

This paper proves the convergence of the leapfrog algorithm, used for computing geodesics on the Stiefel manifold, by framing it as a nonlinear Gauss-Seidel method, with numerical validation.

Contribution

It provides the first convergence proof of the leapfrog algorithm as a nonlinear Gauss-Seidel method for the Stiefel manifold.

Findings

Convergence of leapfrog algorithm established.

Numerical experiments support theoretical results.

Potential for generalization to other manifolds.

Abstract

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.