Tangent cut locus on the Stiefel manifold

Details

Supervisors

Absil, Pierre-Antoine

Faculty

Ecole polytechnique de Louvain

Degree label

Master [120] : ingénieur civil en mathématiques appliquées, à finalité spécialisée

Abstract

enMany applications in statistics on manifolds require geodesic distance, which is closely related to the problem of finding a minimizing geodesic between two points. On the Stiefel manifold, no method currently exists to systematically solve this problem. The cut time, when a geodesic ceases to be length minimizing in a given initial direction, cannot even be characterized. This master's thesis aims to provide new insights into the cut time for a one-parameter family of Riemannian metrics introduced by Hüper et al., (2021). To this end, we propose a method for estimating the cut time on the Stiefel manifold, based on previously established bounds on the geodesic distance. We also generalize this method to compact, connected Riemannian manifolds that admit an upper bound satisfying some conditions. In addition, we establish invariance properties of the cut time under specific orthogonal transformations, which facilitate both theoretical understanding and numerical implementation. Finally, visualizations in two-dimensional subspaces of the tangent space support the proposed method and provide geometric intuition.