Almost Sure Convergence of the Kaczmarz Algorithm with Random Measurements

The Kaczmarz algorithm is an iterative method for reconstructing a signal x∈ℝd from an overcomplete collection of linear measurements yn=〈x,φn〉, n≥1. We prove quantitative bounds on the rate of almost sure exponential convergence in the Kaczmarz algorithm for suitable classes of random measurement vectors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{\varphi_{n}\}_{n=1}^{\infty} \subset {\mathbb {R}}^{d}$\end{document}. Refined convergence results are given for the special case when each φn has i.i.d. Gaussian entries and, more generally, when each φn/∥φn∥ is uniformly distributed on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{S}^{d-1}$\end{document}. This work on almost sure convergence complements the mean squared error analysis of Strohmer and Vershynin for randomized versions of the Kaczmarz algorithm.