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

Ralf Zimmermann, Jakob Stoye

TL;DR

This work provides a complete account of sectional curvature bounds for Grassmann and Stiefel manifolds under canonical and Euclidean metrics, proving $0\le K_c^{Gr}\le 2$ and $0\le K_c^{St}\le 5/4$ (canonical) with sharpness across admissible dimensions. It further establishes $-\tfrac{1}{2} \le K_e^{St} \le 1$ (Euclidean), identifies rank-based extremizers (rank-2 for canonical Grassmann/Stiefel, rank-1 for Euclidean), and demonstrates the practical link between high curvature and low-rank tangent structures. The paper strengthens the theoretical foundation of Riemannian computing by providing refined matrix-norm inequalities, explicit maximizers, and injectivity-radius implications, complemented by numerical experiments and detailed appendices on Lie-group essentials. Overall, the results offer precise curvature controls that inform the behavior and analysis of manifold-based algorithms in applications requiring Grassmann/Stiefel geometry.

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.

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

TL;DR

This work provides a complete account of sectional curvature bounds for Grassmann and Stiefel manifolds under canonical and Euclidean metrics, proving and (canonical) with sharpness across admissible dimensions. It further establishes (Euclidean), identifies rank-based extremizers (rank-2 for canonical Grassmann/Stiefel, rank-1 for Euclidean), and demonstrates the practical link between high curvature and low-rank tangent structures. The paper strengthens the theoretical foundation of Riemannian computing by providing refined matrix-norm inequalities, explicit maximizers, and injectivity-radius implications, complemented by numerical experiments and detailed appendices on Lie-group essentials. Overall, the results offer precise curvature controls that inform the behavior and analysis of manifold-based algorithms in applications requiring Grassmann/Stiefel geometry.

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.
Paper Structure (26 sections, 9 theorems, 111 equations, 5 figures)

This paper contains 26 sections, 9 theorems, 111 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Organisation of the paper
  3. Notational specifics
  4. Curvature of Lie groups and quotients of Lie groups
  5. Lie group curvature formulae
  6. Transport to arbitrary locations
  7. Canonical curvature formulae on Grassmann and Stiefel
  8. Grassmann sectional curvature
  9. Stiefel sectional curvature
  10. Matrix norm inequalities and curvature estimates
  11. The classical Grassmann sectional curvature bounds
  12. Bounds on the canonical sectional curvature on Stiefel
  13. Bounds on the Euclidean sectional curvature on Stiefel
  14. The lower bound on the Euclidean Stiefel curvature
  15. The upper bound on the Euclidean Stiefel curvature
  16. ...and 11 more sections

Key Result

Theorem 2

\newlabelthm:quotcurv0 Let $\mathcal{G}$ be a connected Lie group such that the Lie algebra $\mathfrak g$ admits an $\operatorname{Ad}(\mathcal{G})$-invariant inner product $\langle \cdot,\cdot\rangle_I$. Let $\mathcal{G}/\mathcal{H}$ be a homogeneous space as above and let $\mathfrak m = \mathfr

Figures (5)

  • Figure 1: (Corresponding to \ref{['sec:Exp1']}.) Sectional curvature on $SO(20)$, $St(20,10)$, $Gr(20,10)$ for the tangent sections defined by \ref{['eq:exp1_matrices']}. For $SO(20)$, $Gr(20,10)$, and $St(20,10)$ under the canonical metric, the respective global sectional curvature maximum is attained for the matrices with subblocks $B_1$, $B_2$ as stated in \ref{['thm:StiefelCurvBound']}. For $St(n,p)$ under the Euclidean metric, the extreme curvature cases occur for the rank-one matrices from \ref{['eq:St_eucl_curvmax_matrices']} (max) and \ref{['eq:St_eucl_curvmin_matrices']} (min).
  • Figure 2: (Corresponding to \ref{['sec:Exp2']}.) Averaged sectional curvature for random tangent sections $X=X(A_1,B_1,C_1), Y=Y(A_2,B_2,C_2)\in \operatorname{skew}(2p)$. The dimension of the sub-blocks is $p$. For Grassmann and Stiefel (canonical metric), $X,Y$ are projected onto the respective horizontal space. For Stiefel under the Euclidean metric, $X= A_1B_1, Y=A_2B_2$ are formed. In all cases, the tangent vectors are orthonormalized according to the respective metric.
  • Figure 3: Sectional curvature on Stiefel for the tangent sections spanned by the matrix blocks of \ref{['sec:numex_3']}. Left: canonical metric, Right: Euclidean metric.
  • Figure 4: Sectional curvature on Stiefel when the weight in the spanning normalized tangent matrices is shifted from the $B$-blocks to the $A$-blocks.
  • Figure 5: Plot of the function $\tilde{f}$ from \ref{['eq:globmax']} that bounds the Stiefel sectional curvature in the range $[-\sqrt{2},\sqrt{2}]^2$.

Theorems & Definitions (19)

  • Remark 1
  • Theorem 2: gallier2020differential, Prop. 23.29
  • Lemma 3
  • Proof 1
  • Lemma 4
  • Lemma 5
  • Remark 6
  • Proof 2: \ref{['lem:B1B2B1B2_term']}
  • Theorem 7
  • Proof 3
  • ...and 9 more