The G\"unt\"urk-Thao theorem revisited: polyhedral cones and limiting examples

TL;DR

This paper extends a recent theorem on the convergence of random projections in Hilbert spaces from regular subspaces to polyhedral cones, providing new insights and tightness examples for the limits of such extensions.

Contribution

It generalizes the G"unt"urk-Thao theorem to polyhedral cones and demonstrates the limits of this extension with specific counterexamples.

Findings

The theorem holds for polyhedral cones in Hilbert spaces.

Counterexamples show the theorem fails for certain non-polyhedral sets.

The extension is tight and cannot be further generalized without exceptions.

Abstract

In 2023, G\"unt\"urk and Thao proved that the sequence $(x^{(n)})_{n\in\mathbb{N}}$ generated by random (relaxed) projections drawn from a finite collection of innately regular closed subspaces in a real Hilbert space satisfies $\sum_{n\in\mathbb{N}} \|x^{(n)}-x^{(n+1)}\|^\gamma <+\infty$ for all $\gamma>0$. We extend their result to a finite collection of polyhedral cones. Moreover, we construct examples showing the tightness of our extension: indeed, the result fails for a line and a convex set in $\mathbb{R}^2$, and for a plane and a non-polyhedral cone in $\mathbb{R}^3$.