A Generalized Class of Hard Thresholding Algorithms for Sparse Signal Recovery

Abstract

We introduce a whole family of hard thresholding algorithms for the recovery of sparse signals \({\mathbf {x}}\in {\mathbb C}^N\) from a limited number of linear measurements \({\mathbf y}= \mathbf{A}{\mathbf {x}}\in {\mathbb C}^m\), with \(m \ll N\). Our results generalize previous ones on hard thresholding pursuit algorithms. We show that uniform recovery of all \(s\)-sparse vectors \({\mathbf {x}}\) can be achieved under a certain restricted isometry condition. While these conditions might be unrealistic in some cases, it is shown that with high probability, our algorithms select a correct set of indices at each iteration, as long as the active support is smaller than the actual support of the vector to be recovered, with a proviso on the shape of the vector. Our theoretical findings are illustrated by numerical examples.