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.