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

Methods and algorithms that work with data on nonlinear manifolds are collectively summarised 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 from the quotient space view point. 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. For both of these 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 that the global bound of 5/4 for Stiefel holds 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, the global curvature maximum is attained at tangent plane sections that are spanned by rank-two matrices. Numerical examples are included for illustration purposes.