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Shortest Geodesic Loops, Sectional Curvature, and Injectivity Radius of the Stiefel Manifold

Jakob Stoye, Simon Mataigne, P. -A. Absil, Ralf Zimmermann

TL;DR

This work determines the length of the shortest nontrivial geodesic loops on the Stiefel manifold under the one-parameter beta-metric family and uses these loop lengths together with sharp sectional-curvature bounds to compute the injectivity radius across broad beta-ranges. The beta-length is shown to be $\ell_\beta = \min\{\sqrt{2\beta},1\}\,2\pi$, unifying the canonical and Euclidean metrics as special cases and resolving the $\beta\le2$ regime while settling $\beta>2$ via a matrix-exponential-inverse analysis. Sharp curvature bounds close a prior gap, enabling exact injectivity radii for $\beta\in(0,1/3]\cup[2/3,1]$ and tight bounds in $\beta\in(1/3,2/3)$, with conjectural sharpness supported by numerical evidence. The results have implications for optimization and statistical methods on Stiefel manifolds by clarifying geodesic-minimality regions and stability of distance-based procedures across the beta-family.

Abstract

We determine the length of the shortest nontrivial geodesic loops on the Stiefel manifold endowed with any member of the one-parameter family of Riemannian metrics introduced by Hüper et al. (2021). This family includes, in particular, the canonical and Euclidean metrics. By combining existing and new bounds on the sectional curvature, we determine the exact value of the injectivity radius of the Stiefel manifold under a wide range of members of the metric family.

Shortest Geodesic Loops, Sectional Curvature, and Injectivity Radius of the Stiefel Manifold

TL;DR

This work determines the length of the shortest nontrivial geodesic loops on the Stiefel manifold under the one-parameter beta-metric family and uses these loop lengths together with sharp sectional-curvature bounds to compute the injectivity radius across broad beta-ranges. The beta-length is shown to be , unifying the canonical and Euclidean metrics as special cases and resolving the regime while settling via a matrix-exponential-inverse analysis. Sharp curvature bounds close a prior gap, enabling exact injectivity radii for and tight bounds in , with conjectural sharpness supported by numerical evidence. The results have implications for optimization and statistical methods on Stiefel manifolds by clarifying geodesic-minimality regions and stability of distance-based procedures across the beta-family.

Abstract

We determine the length of the shortest nontrivial geodesic loops on the Stiefel manifold endowed with any member of the one-parameter family of Riemannian metrics introduced by Hüper et al. (2021). This family includes, in particular, the canonical and Euclidean metrics. By combining existing and new bounds on the sectional curvature, we determine the exact value of the injectivity radius of the Stiefel manifold under a wide range of members of the metric family.

Paper Structure

This paper contains 20 sections, 27 theorems, 156 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Organization of the paper.
  4. Preliminaries on the Stiefel manifold
  5. The Stiefel manifold
  6. Geodesics and the injectivity radius
  7. The beta-length of geodesic loops (beta in (0,2])
  8. The Euclidean length of geodesic loops
  9. The beta-length of shortest geodesic loops (beta in (0,2])
  10. The exponential inverse and invariance groups
  11. The beta-length of geodesic loops (beta in (2,oo)
  12. The structure of shortest geodesic loops
  13. The beta-length of shortest geodesic loops (beta in (2,oo))
  14. Bounds for the sectional curvature
  15. Implications for the injectivity radius
  16. ...and 5 more sections

Key Result

Lemma 3.1

Let $\gamma$ be a unit-speed loop with constant Frenet curvatures in the form of eq:curvconstcurv_even. Let $b_{\min} = \min\{|b_i|\mid b_i\neq 0\}$, i.e., the smallest non-zero $b$-coefficient in eq:curvconstcurv_even. The Euclidean length of $\gamma$ satisfies

Figures (2)

  • Figure 1: The value of $K_\beta$ for $\beta\in(0,1]$.
  • Figure 2: Graphical illustration of $\frac{\ell_\beta}{2}$ (solid red line), $\frac{\pi}{\sqrt{K_\beta}}$ (blue dashed line) and $\sqrt{2}t_\beta^\mathrm{r}$ (green dashed-dotted line) for $\beta\in(0,1]$. The injectivity radius satisfies $\min\{\sqrt{2\beta}\pi,\pi,\frac{\pi}{\sqrt{K_\beta}}\} \leq \mathrm{inj}(\mathrm{St}_\beta(n,p)) \leq \min\{\sqrt{2\beta}\pi,\pi,\sqrt{2}t_\beta^\mathrm{r}\}$. The thin blue shaded area for $\beta\in(\frac{1}{3},\frac{2}{3})$ is the region where the injectivity radius lies.

Theorems & Definitions (54)

  • Definition 2.1: Injectivity Radius
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • Theorem 3.4
  • proof
  • Lemma 4.1
  • proof
  • ...and 44 more