Endpoint Geodesics on the Stiefel Manifold Embedded in Euclidean Space

Abstract

The Stiefel manifold has no known analytical formula for endpoint geodesics, i.e., locally shortest length curves between two given points. In this work, we consider the Stiefel manifold as a submanifold of Euclidean space, and its geometry and metric are inherited from this ambient Hilbert space. With this geometric interpretation, we derive two numerical schemes to compute endpoint geodesics on the Stiefel manifold: the shooting method and path-straightening. We perform a variety of numerical experiments to showcase and compare the performance of these two algorithms under various conditions. Additionally, we show an application of the Stiefel geodesic and Karcher mean calculation in the context of affine-invariant shape analysis.