Signal recovery from random measurements via extended orthogonal matching pursuit

Abstract

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 (mln d) number of measurements, whereas BP needs only O mln d m 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 OMPα, which runs OMP for (m + αm)-iterations instead of m-iterations, by choosing a value of α ??? [0, 1]. It is shown that OMPα guarantees a high probability signal recovery with O mln d αm+1 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 OMPα to illustrate another recovery scheme called OMP∞, which runs OMP until the signal residue vanishes. It is shown that OMP∞ can achieve a close to 0-norm recovery without any knowledge of m like BP.