Signal Recovery from Random Measurements via Extended Orthogonal Matching Pursuit

Orthogonal Matching Pursuit (OMP) and Basis Pursuit (BP) are two well-known recovery algorithms in compressed sensing. To recover a
$d$
-dimensional
$m$
-sparse signal with high probability, OMP needs
$O(m\ln{d})$
number of measurements, whereas BP needs only
$O\left(m\ln{{d\over m}}\right)$
number of measurements. In contrary, OMP is a practically more appealing algorithm due to its superior execution speed. In this piece of work, we have proposed a scheme that brings the required number of measurements for OMP closer to BP. We have termed this scheme as
${\rm OMP}_{\alpha}$
, which runs OMP for
$(m+\lfloor\alpha{m}\rfloor)$
-iterations instead of
$m$
-iterations, by choosing a value of
$\alpha\in[0,1]$
. It is shown that
${\rm OMP}_{\alpha}$
guarantees a high probability signal recovery with
$O\left(m\ln{{d\over\lfloor\alpha{m}\rfloor+1}}\right)$
number of measurements. Another limitation of OMP unlike BP is that it requires the knowledge of
$m$
. In order to overcome this limitation, we have extended the idea of
${\rm OMP}_{\alpha}$
to illustrate another recovery scheme called
${\rm OMP}_{\infty}$
, which runs OMP until the signal residue vanishes. It is shown that
${\rm OMP}_{\infty}$
can achieve a close to
$\ell_{0}$
-norm recovery without any knowledge of
$m$
like BP.