README.md

InjectivityStiefel.jl

This repository is associated to the publication "P.-A. Absil, S. Mataigne, The ultimate upper bound on the injectivity radius of the Stiefel manifold, 2024".

Installation

In order to use this repository, the user should have a Julia installation where the following packages are installed: LinearAlgebra, SkewLinearAlgebra, Plots, LaTeXStrings, Colors. These packages are easily obtained from the package installation environment as follows. In Julia REPL, press ] to access the installation environment and for each package, do

(@v1.6) pkg> add Name_of_Package

Use

This repository contains de following folders: src, plots and figures.

• The folder src contains the algorithms described in the paper, namely Algorithm 8.1 under the name checkradius. These are the routines needed to produce the results.
  • skewlog.jl contains a routine to compute the skew-symmetric matrix logarithm of an orthogonal matrix.
  • check_injectivity_radius.jl contains the algorithm corroborating the injectivity radius on the Stiefel manifold $\mathrm{St}(n,p)$ and subroutines (Algorithm 8.1).
• The folder plots contains files to perform numerical experiments with the algorithms from the folder src.
  • plot_inj_beta.jl runs an experiment on $\mathrm{St}(n,p)$ using the routine check_radius. The routine is called for various values of parameter $\beta>0$, where $\beta$ is the parameter of the metric on $\mathrm{St}(n,p)$ (see paper). For each $\beta$, the value $\rho$ is set to the conjectured injectivity radius $î_\beta$ (see Theorem 7.1) and $î_\beta+\delta$. For any $\delta>0$, check_radius should stop in a finite number of iteration with probability $1$ (see Section 8). When $\rho=î_\beta$, check_radius should never return if Conjecture 8.1 is true (i.e., should reach itermax). plot_inj_beta.jl returns a plot with, for each pair $(\beta,\rho)$, a white dot if the algorithm returned in finite time and a black dot if the algorithm reached itermax.
  • plot_iteration_count.jl also runs an experiment on $\mathrm{St}(n,p)$ using the routine check_radius. Given $\beta>0$, and various values $\delta>0$, plot_iteration_count.jl displays how many iterations are needed to return when check_radius is called with $\rho=î_\beta+\delta$. As $\delta\rightarrow 0$, we observe that the number of iterations needed tends to infinity.
• The folder figures contains output figures of the file plot_inj_beta.jl set with $(n,p) \in {(4,2),(4,3),(5,3)}$. On all figures, check_radius reached itermax for $\rho = î_\beta$ --- that is, could never contradict the conjecture $î_\beta=\mathrm{inj}_{\mathrm{St}(n,p)}$.

Bibtex

If you use the content of this repository, please cite:

@misc{absil2024ultimate,
      title={The ultimate upper bound on the injectivity radius of the Stiefel manifold}, 
      author={P. -A. Absil and Simon Mataigne},
      year={2024},
      eprint={2403.02079},
      archivePrefix={arXiv},
      primaryClass={math.DG}
}