High curvature means low-rank: On the sectional curvature of Grassmann and Stiefel manifolds and the underlying matrix trace inequalities

TL;DR

This paper investigates the sectional curvature of Stiefel and Grassmann manifolds, providing refined bounds and revealing that high curvature correlates with low-rank matrices, which impacts Riemannian computing methods.

Contribution

It offers the first complete proof of the curvature bounds for the Stiefel manifold under different metrics and links curvature maxima to low-rank matrix structures.

Findings

Global curvature bounds for Stiefel manifold confirmed.

Maximum curvature occurs at rank-two matrices under the canonical metric.

Extreme curvature cases for Euclidean Stiefel occur at rank-one matrices.

Abstract

Methods and algorithms that work with data on nonlinear manifolds are collectively summarized under the term `Riemannian computing'. In practice, curvature can be a key limiting factor for the performance of Riemannian computing methods. Yet, curvature can also be a powerful tool in the theoretical analysis of Riemannian algorithms. In this work, we investigate the sectional curvature of the Stiefel and Grassmann manifold. On the Grassmannian, tight curvature bounds are known since the late 1960ies. On the Stiefel manifold under the canonical metric, it was believed that the sectional curvature does not exceed 5/4. Under the Euclidean metric, the maximum was conjectured to be at 1. For both manifolds, the sectional curvature is given by the Frobenius norm of certain structured commutator brackets of skew-symmetric matrices. We provide refined inequalities for such terms and pay special attention to the maximizers of the curvature bounds. In this way, we prove for the Stiefel manifold that the global bounds of 5/4 (canonical metric) and 1 (Euclidean metric) hold indeed. With this addition, a complete account of the curvature bounds in all admissible dimensions is obtained. We observe that `high curvature means low-rank', more precisely, for the Stiefel and Grassmann manifolds under the canonical metric, the global curvature maximum is attained at tangent plane sections that are spanned by rank-two matrices, while the extreme curvature cases of the Euclidean Stiefel manifold occur for rank-one matrices. Numerical examples are included for illustration purposes.