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

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

Problem Statement
The Stiefel manifold, denoted $St(n, p)$, is a fundamental matrix manifold appearing in optimization, numerical methods for differential equations, and statistics. A central problem in Riemannian geometry on this manifold is determining the injectivity radius, which provides a sufficient condition for the minimality of geodesics connecting two points. The injectivity radius, $inj(M)$, is defined as the largest radius $r$ such that every geodesic of length $\le r$ is the unique shortest curve between its endpoints.

For a complete Riemannian manifold, the injectivity radius is determined by two competing factors: half the length of the shortest nontrivial geodesic loop ($\ell/2$) and the conjugate radius. The latter is bounded from below by $\pi/\sqrt{K}$, where $K$ is a global upper bound on the sectional curvature. While the injectivity radius is known for specific metrics (e.g., the Euclidean metric and the canonical metric), it remains an open problem for the general one-parameter family of metrics introduced by H"uper et al. (2021), parameterized by $\beta > 0$. Specifically, the exact length of the shortest geodesic loops and the sharp upper bound on sectional curvature were not fully established for all $\beta$, preventing the exact calculation of the injectivity radius across the entire parameter space.

Methodology
The authors address this problem by combining differential geometric analysis with matrix-analytic techniques. The methodology is divided into three main components:

1. Analysis of Geodesic Loop Lengths ($\ell_\beta$):

   • Case $\beta \in (0, 2]$: The authors utilize the observation that geodesics on the Stiefel manifold, when viewed as curves in the ambient Euclidean space $\mathbb{R}^{n \times p}$, possess constant Frenet curvatures. By deriving bounds on the Euclidean length of these curves and translating them to the $\beta$-metric, they establish lower bounds on the loop length.
   • Case $\beta > 2$: The Frenet curvature approach is insufficient. Instead, the authors employ a matrix-analytic proof. They analyze the structure of the matrix exponential inverse and the invariance groups of orthogonal matrices. By decomposing the initial velocity matrices into sub-blocks corresponding to specific spectral properties, they reduce the problem to analyzing geodesic loops with specific block-diagonal structures, proving that the length cannot be less than $2\pi$.

2. Sectional Curvature Bounds ($K_\beta$):
   The authors derive an explicit formula for the sectional curvature of the Stiefel manifold under the $\beta$-metric. They perform term-by-term estimates on the curvature expression using the norms of the matrix blocks ($A$ and $B$) defining the tangent vectors. This analysis identifies a gap in existing literature for $\beta \in (2/3, 1/\sqrt{2})$, which the authors close by proving that the sectional curvature is bounded above by 1 in this interval.

3. Synthesis via Klingenberg's Theorem:
   Using the calculated loop lengths $\ell_\beta$ and the curvature bounds $K_\beta$, the authors apply Klingenberg's theorem, which states that $inj(St_\beta(n, p)) = \min\{ \ell_\beta/2, \text{conj}_{I_{n \times p}}(St_\beta(n, p)) \}$. They combine this with the lower bound on the conjugate radius ($\pi/\sqrt{K_\beta}$) to determine the exact injectivity radius or tight bounds for various ranges of $\beta$.

Key Contributions and Results

• Exact Length of Shortest Geodesic Loops: The paper proves that for all $\beta > 0$, the length $\ell_\beta$ of the shortest nontrivial geodesic loop on $St_\beta(n, p)$ (for $2 \le p \le n-1$) is given by:
  $$\ell_\beta = \min\{ \sqrt{2\beta}, 1 \} 2\pi$$
  This result unifies and extends previous findings for the canonical metric ($\beta=1/2$) and the Euclidean metric ($\beta=1$), where the length was known to be $2\pi$. For $\beta > 2$, the length is shown to be exactly $2\pi$.

• Sharp Sectional Curvature Bounds: The authors provide a sharp upper bound $K_\beta$ on the sectional curvature for the range $\beta \in (2/3, 1/\sqrt{2})$, filling a gap in the existing literature. The bound is:
  $$K_\beta = \begin{cases} \frac{1}{4\beta} & 0 < \beta \le \frac{1}{6}(4-\sqrt{10}) \\ \frac{4-3\beta}{2} & \frac{1}{6}(4-\sqrt{10}) < \beta \le \frac{2}{3} \\ 1 & \frac{2}{3} \le \beta \le 1 \end{cases}$$
  (Note: The paper notes that for $\beta > 1$, the curvature bound is 1, consistent with the Euclidean case).

• Injectivity Radius Determination:

  • For $\beta \in (0, 1/3] \cup [2/3, 1]$, the authors determine the exact value of the injectivity radius:
    • For $\beta \in (0, 1/3]$: $inj = \sqrt{2\beta}\pi$.
    • For $\beta \in [2/3, 1]$: $inj = \pi$.
  • For $\beta \in (1/3, 2/3)$, the exact value is not determined due to the lack of an explicit expression for the conjugate radius. However, the paper provides tight upper and lower bounds, showing the value differs by at most 3% from the conjectured value in [33].
  • For $\beta > 1$, the injectivity radius is bounded above by $\pi$.

• Special Cases:

  • For $p=1$ (unit sphere), $inj = \pi$.
  • For $p=n$ (orthogonal group $O(n)$), the conjugate radius is explicitly calculated as $\sqrt{8\beta}\pi$ for $n=3$ and $\sqrt{4\beta}\pi$ for $n>3$, leading to an injectivity radius of $\sqrt{2\beta}\pi$.

Significance
The paper claims to resolve the problem of determining the injectivity radius for the Stiefel manifold under a wide range of the $\beta$-metrics, a crucial parameter for optimization algorithms on manifolds. By establishing the exact length of the shortest geodesic loops for all $\beta$ and closing the gap in sectional curvature bounds, the authors provide the necessary geometric data to apply Klingenberg's theorem.

The work is modest regarding the interval $\beta \in (1/3, 2/3)$, acknowledging that the exact injectivity radius remains a conjecture in this range, supported by numerical experiments in prior literature. However, the paper strengthens the conjecture for the canonical metric ($\beta=1/2$) by providing explicit expressions for Jacobi fields, confirming that the identified conjugate points are indeed the first conjugate points. The results provide a rigorous foundation for ensuring the minimality of geodesics in manifold-based optimization and data fitting applications.