Bounds on the geodesic distances on the Stiefel manifold

Details

Supervisors

Absil, Pierre-Antoine ; Miolane, Nina

Faculty

Ecole polytechnique de Louvain

Degree label

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

Abstract

The past few years have seen the emergence of many applications for statistics on manifolds. The manifold-based statistical tools often rely on the notion of mean of two points, i.e., the midpoint of a minimizing geodesic between these two points. An important manifold for which we cannot guarantee the computation of a minimizing geodesic is the Stiefel manifold, which is the set of orthogonal p-frames in R^n. The goal of this master’s thesis is to provide new insights into the Stiefel manifold to enhance the performances of minimizing geodesic computation algorithms. To this aim, we provide new results on the bilipschitz equivalence of a previously proposed one-parameter family of Riemannian metrics. We also show that the geodesic distances induced by this family of Riemannian metrics are equivalent to the easy-to compute Frobenius distance. Subsequently, we obtain tight bounds between the geodesic and the Frobenius distances. Many operations on the Stiefel manifold, and in particular the computation of a minimizing geodesic, require the computation of the matrix exponential of skew-symmetric matrices. To conclude this work, we present the SkewLinearAlgebra.jl package, a Julia library specialized in the efficient computation of the eigenvalue decomposition and the exponential of skew-symmetric matrices.