Iterative Hard Thresholding with Near Optimal Projection for Signal Recovery

Recovering signals that have sparse representations under a given dictionary from a set of linear measurements got much attention in the recent decade. However, most of the work has focused on recovering the signal’s representation, forcing the dictionary to be incoherent and with no linear dependencies between small sets of its columns. A series of recent papers show that such dependencies can be allowed by aiming at recovering the signal itself. However, most of these contributions focus on the analysis framework. One exception to these is the work reported in [1], proposing a variant of the CoSaMP for the synthesis model, and showing that signal recovery is possible even in high-coherence cases. In the theoretical study of this technique the existence of an efficient near optimal projection scheme is assumed. In this paper we extend the above work, showing that under very similar assumptions, a variant of IHT can recover the signal in cases where regular IHT fails.