Bounds on the geodesic distances on the Stiefel manifold for a family of Riemannian metrics

TL;DR

This paper establishes bounds on geodesic distances on the Stiefel manifold for a family of Riemannian metrics, enhancing computational efficiency and theoretical understanding of these distances.

Contribution

It provides explicit bounds and Lipschitz constants for geodesic distances induced by a family of metrics on the Stiefel manifold, improving algorithms and theoretical insights.

Findings

Derived bounds on geodesic distances using Frobenius distance

Identified pairs of matrices where bounds are tight

Provided Lipschitz constants between different metrics

Abstract

We give bounds on geodesic distances on the Stiefel manifold, derived from new geometric insights. The considered geodesic distances are induced by the one-parameter family of Riemannian metrics introduced by H\"uper et al. (2021), which contains the well-known Euclidean and canonical metrics. First, we give the best Lipschitz constants between the distances induced by any two members of the family of metrics. Then, we give a lower and an upper bound on the geodesic distance by the easily computable Frobenius distance. We give explicit families of pairs of matrices that depend on the parameter of the metric and the dimensions of the manifold, where the lower and the upper bound are attained. These bounds aim at improving the theoretical guarantees and performance of minimal geodesic computation algorithms by reducing the initial velocity search space. In addition, these findings contribute to advancing the understanding of geodesic distances on the Stiefel manifold and their applications.