Shooting methods for computing geodesics on the Stiefel manifold

TL;DR

This paper adapts the shooting method to compute geodesic distances on the Stiefel manifold, providing explicit Jacobian formulas, and demonstrates its effectiveness through numerical experiments and applications.

Contribution

It introduces explicit Jacobian formulas for shooting methods on the Stiefel manifold, enabling efficient computation of geodesic distances for the first time.

Findings

Accurate computation of geodesic distances demonstrated

Explicit Jacobian formulas derived and validated

Applications in statistics, shape analysis, and model reduction

Abstract

This paper shows how to use the shooting method, a classical numerical algorithm for solving boundary value problems, to compute the Riemannian distance on the Stiefel manifold $\mathrm{St}(n,p)$, the set of $ n \times p $ matrices with orthonormal columns. The main feature is that we provide neat, explicit expressions for the Jacobians. To the author's knowledge, this is the first time some explicit formulas are given for the Jacobians involved in the shooting methods to find the distance between two given points on the Stiefel manifold. This allows us to perform a preliminary analysis for the single shooting method. Numerical experiments demonstrate the algorithms in terms of accuracy and performance. Finally, we showcase three example applications in summary statistics, shape analysis, and model order reduction.