Bounds on the geodesic distances on the Stiefel manifold for a family of Riemannian metrics

TL;DR

The paper bounds geodesic distances on the Stiefel manifold under a one-parameter family of Riemannian metrics that includes the Euclidean and canonical metrics. It proves bilipschitz equivalence across all $eta>0$, and derives computable lower and upper bounds for $d_eta$ in terms of the Frobenius distance, with explicit constructions showing when these bounds are tight. The work also characterizes the Euclidean diameter, provides linear and nonlinear bounds for nearby matrices, and extends the theory to all $eta$-distances via universal bounds. These results offer principled ways to constrain initial velocity searches in minimal geodesic computation, improving numerical guarantees and informing metric selection for applications on Stiefel manifolds.

Abstract

We give bounds on geodesic distances on the Stiefel manifold, derived from new geometric insights. The considered geodesic distances are induced by the one-parameter family of Riemannian metrics introduced by Hüper et al. (2021), which contains the well-known Euclidean and canonical metrics. First, we give the best Lipschitz constants between the distances induced by any two members of the family of metrics. Then, we give a lower and an upper bound on the geodesic distance by the easily computable Frobenius distance. We give explicit families of pairs of matrices that depend on the parameter of the metric and the dimensions of the manifold, where the lower and the upper bound are attained. These bounds aim at improving the theoretical guarantees and performance of minimal geodesic computation algorithms by reducing the initial velocity search space. In addition, these findings contribute to advancing the understanding of geodesic distances on the Stiefel manifold and their applications.

Figures (11)

• Figure 1: An artist view of the Stiefel manifold $\mathrm{St}(n,p)$, the tangent space $T_U\mathrm{St}(n,p)$, the Riemannian exponential and logarithm.
• Figure 1: Bounds between the canonical and the Euclidean distance on the Stiefel manifold $\mathrm{St}(4,2)$: $\frac{\sqrt{2}}{2} d_{\mathrm{E}}(U,\widetilde{U})\leq d_{\mathrm{c}}(U,\widetilde{U})\leq d_{\mathrm{E}}(U,\widetilde{U})$ for all $U,\widetilde{U}\in \mathrm{St}(4,2)$ (see \ref{['cor:special']}). The plot shows the canonical distances $d_\mathrm{c}$ ($\beta = \frac{1}{2}$) and the Euclidean distances $d_\mathrm{E}$ ($\beta=1$) estimated for 10000 randomly generated pairs $(U,\widetilde{U})$ on $\mathrm{St}(4,2)$.
• Figure 1: Bounds between the Frobenius distance and, on the left, the Euclidean ($\beta=1$) geodesic distance on $\mathrm{St}(5,4)$ and on the right, the canonical ($\beta=\frac{1}{2}$) geodesic distance on $\mathrm{St}(8,4)$: $\widehat{m}_\beta(\delta)\leq d_{\beta}(U,\widetilde{U})\leq \widehat{M}_\beta(\delta)$ for all $U,\widetilde{U}\in \mathrm{St}(n,p)$ with $\|\widetilde{U}-U\|_\mathrm{F} = \delta$ (see \ref{['thm:generalizedbounds']}). The plots show the Frobenius distances and the geodesic distances $d_\mathrm{\beta}$ estimated for 20 000, respectively 1 000 000, randomly generated pairs $(U,\widetilde{U})$ on $\mathrm{St}(5,4)$, respectively $\mathrm{St}(8,4)$.
• Figure 1: \ref{['prop:vecdist']} is obtained in the triangle ${\Delta}$. It can be deduced from the definition of the sine of the angle in the bottom left corner, which is expressed as the ratio of the length of the side opposite to that angle to the length of the hypotenuse.
• Figure 1: Relative error between the theoretical upper bound (considering either only \ref{['thm:linearbound']} or including \ref{['thm:first_upper_bound']}) and the largest Euclidean geodesic distance obtained by numerical experiments on $\mathrm{St}(4,3)$. Let $\delta\in[0,2\sqrt{p}]$, $M_\mathrm{E}(\delta) := \max_{\|U-\widetilde{U}\|_\mathrm{F}=\delta}d_\mathrm{E}(U,\widetilde{U})$ and $\widehat{M}_\mathrm{E}(\delta)$ be the upper bound. Then, the relative gap is defined by $\omega(\delta) :=\frac{\widehat{M}_\mathrm{E} - M_\mathrm{E}(\delta)}{M_\mathrm{E}(\delta)}$. The value $M_\mathrm{E}(\delta)$ is estimated from random samples $U_i,\widetilde{U}_i$ by $M_\mathrm{E}(\delta)\approx \max_{\|U_i-\widetilde{U}_i\|_\mathrm{F}\in[\delta+\frac{\sqrt{p}}{100})}d_\mathrm{E}(U_i,\widetilde{U}_i)$ using ZimmermannRalf22 to estimate $d_\mathrm{E}(U_i,\widetilde{U}_i)$.

Theorems & Definitions (57)

• Theorem 2.1
• Lemma 3.1
• Proof 1
• Theorem 3.2
• Proof 2
• Corollary 3.3
• Proof 3
• Corollary 3.4
• Proof 4
• Theorem 3.5