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