Computing the Riemannian Logarithm on the Stiefel Manifold: Metrics, Methods, and Performance
Ralf Zimmermann, Knut Hüper
Abstract
We address the problem of computing Riemannian normal coordinates on the real, compact Stiefel manifold of orthonormal frames. The Riemannian normal coordinates are based on the so-called Riemannian exponential and the associated Riemannian logarithm map and enable one to transfer almost any computational procedure to the realm of the Stiefel manifold. To compute the Riemannian logarithm is to solve the (local) geodesic endpoint problem. Instead of restricting the consideration to geodesics with respect to a single selected metric, we consider a family of Riemannian metrics introduced by Hüper, Markina, and Silva Leite that includes the Euclidean and the canonical metric as prominent examples. As main contributions, we provide (1) a unified, structured, reduced formula for the Stiefel geodesics. The formula is unified in the sense that it works for the full family of metrics under consideration. It is structured in the sense that it relies on matrix exponentials of skew-symmetric matrices exclusively. It is reduced in relation to the dimension of the matrices of which matrix exponentials have to be calculated. We provide (2) a unified method to tackle the geodesic endpoint problem numerically, and (3) we improve the existing Riemannian log algorithm under the canonical metric in terms of the computational efficiency. The findings are illustrated by means of numerical examples, where the novel algorithms prove to be the most efficient methods known to this date.