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The ultimate upper bound on the injectivity radius of the Stiefel manifold

P. -A. Absil, Simon Mataigne

TL;DR

This work addresses the injectivity radius of the Stiefel manifold $\mathrm{St}(n,p)$ under the family of metrics $g_\beta$ introduced by Hüper et al., which unifies the canonical and Euclidean cases. The authors derive a $\beta$-dependent upper bound on the injectivity radius by locating conjugate points along geodesics and by bounding geodesic loop lengths, yielding explicit expressions such as $\mathrm{inj}_{St_\beta(n,p)}\le \min\{ \sqrt{2\beta}\,\pi, \pi, t^{\mathrm r}_\beta\sqrt2\}$ for $2\le p\le n-2$, with $t^{\mathrm r}_\beta$ the smallest positive root of $\frac{\sin t}{t}+(1+2\alpha)\cos t=0$ and $\alpha=\tfrac{1}{2\beta}-1$. For canonical and Euclidean metrics the bounds specialize to $0.913...\pi$ and $\pi$ respectively, and numerical experiments strongly support the conjecture that these bounds are tight for all $\beta$. Recent independent results confirm the Euclidean case, validating the conjecture in that instance. The findings have implications for optimization and geometric analysis on Stiefel manifolds, informing both theoretical understanding and numerical procedures on these spaces.

Abstract

We exhibit conjugate points on the Stiefel manifold endowed with any member of the family of Riemannian metrics introduced by Hüper et al. (2021). This family contains the well-known canonical and Euclidean metrics. An upper bound on the injectivity radius of the Stiefel manifold in the considered metric is then obtained as the minimum between the length of the geodesic along which the points are conjugate and the length of certain geodesic loops. Numerical experiments support the conjecture that the obtained upper bound is in fact equal to the injectivity radius.

The ultimate upper bound on the injectivity radius of the Stiefel manifold

TL;DR

This work addresses the injectivity radius of the Stiefel manifold under the family of metrics introduced by Hüper et al., which unifies the canonical and Euclidean cases. The authors derive a -dependent upper bound on the injectivity radius by locating conjugate points along geodesics and by bounding geodesic loop lengths, yielding explicit expressions such as for , with the smallest positive root of and . For canonical and Euclidean metrics the bounds specialize to and respectively, and numerical experiments strongly support the conjecture that these bounds are tight for all . Recent independent results confirm the Euclidean case, validating the conjecture in that instance. The findings have implications for optimization and geometric analysis on Stiefel manifolds, informing both theoretical understanding and numerical procedures on these spaces.

Abstract

We exhibit conjugate points on the Stiefel manifold endowed with any member of the family of Riemannian metrics introduced by Hüper et al. (2021). This family contains the well-known canonical and Euclidean metrics. An upper bound on the injectivity radius of the Stiefel manifold in the considered metric is then obtained as the minimum between the length of the geodesic along which the points are conjugate and the length of certain geodesic loops. Numerical experiments support the conjecture that the obtained upper bound is in fact equal to the injectivity radius.
Paper Structure (18 sections, 13 theorems, 97 equations, 2 figures, 1 algorithm)

This paper contains 18 sections, 13 theorems, 97 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. The injectivity radius of a complete Riemannian manifold
  3. Background on the Stiefel manifold
  4. A Riemannian-submersion route to the noncanonical metrics
  5. The canonical case
  6. Preparatory lemmas
  7. Conjugate points
  8. Geodesic loops
  9. Case $p=1$
  10. Case $p=n \geq 2$
  11. Case $\beta=\frac{1}{2}$ (canonical metric)
  12. Case $2 \leq p \leq n-1$ and $\beta\neq\frac{1}{2}$ (noncanonical metrics)
  13. Upper bounds on the injectivity radius
  14. Case $p=1$ or $p\geq n-1$
  15. Case $2 \leq p \leq n-2$
  16. ...and 3 more sections

Key Result

Theorem 2.1

\newlabelthm:ab0 Let $\xi \in \mathrm{U}_x\mathcal{M}$ with $t_{\mathrm{c}}(\xi)<\infty$. Then $T = t_{\mathrm{c}}(\xi)$ if and only if the following holds for $t_0 = T$ and for no smaller value of $t_0$: (a) $\gamma_\xi(t_0)$ is a conjugate point of $x$ along $\gamma_\xi$; or (nonexclusive) (b) t

Figures (2)

  • Figure 1: Numerical experiments using Algorithm \ref{['alg:semi']} for values of $\beta$ going from $0.1$ to $1.5$, spaced by $0.05$. For each value of $\beta$, we ran an experiment for $\rho = \hat{\imath}_\beta$ and $\rho = \hat{\imath}_\beta + 0.05$, where $\hat{\imath}_\beta$ is the upper bound defined in \ref{['eq:inj']}. The plot reports a black dot if Algorithm \ref{['alg:semi']} reached a prescribed iteration limit (as large as possible, subject to the figure being generated within reasonable time), and a white dot if Algorithm \ref{['alg:semi']} returned (hence providing a certificate that $\rho > \mathrm{inj}_{\mathrm{St}_\beta(n,p)}$). This figure was produced with $(n,p)=(4,2)$. We obtained the same figure for other values of $n$ and $p$ with $2\leq p \leq n-2$.
  • Figure 2: Numerical experiments using Algorithm \ref{['alg:semi']} with $(n,p)=(4,2)$ and $\beta=0.5$ for $\rho = \hat{\imath}_\beta+\delta$ where $\delta\in\{1,0.1,0.01,0.001,0.0001\}$. The plot reports the number of iterations required by Algorithm \ref{['alg:semi']} to return, as observed in representative runs. Note that, in view of the random nature of Algorithm \ref{['alg:semi']}, the values differ between runs. The purpose of this figure is to show how a typical return time evolves as the upper dot located above $\beta=\frac{1}{2}$ in Figure \ref{['fig:beta-rho']} is brought closer to the line.

Theorems & Definitions (24)

  • Theorem 2.1: Lemma 5.2 in CE75, Proposition 4.1 in Sak96
  • Proposition 3.1
  • Proposition 3.2
  • Lemma 4.1
  • Proof 1
  • Lemma 4.2
  • Proof 2
  • Lemma 4.3
  • Proof 3
  • Theorem 5.1
  • ...and 14 more