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(m ln 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 OMP_α, which runs OMP for (m+⌊αm⌋)-iterations instead of m-iterations, by choosing a value of α ∈ [0,1]. It is shown that OMP_α guarantees a high probability signal recovery with O(m ln 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 OMP_α 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 ℓ₀-norm recovery without any knowledge of m like BP.