Sparse Recovery With Orthogonal Matching Pursuit Under RIP

This paper presents a new analysis for the orthogonal matching pursuit (OMP) algorithm. It is shown that if the restricted isometry property (RIP) is satisfied at sparsity level
$O(\bar{k})$
, then OMP can stably recover a
$\bar{k}$
-sparse signal in 2-norm under measurement noise. For compressed sensing applications, this result implies that in order to uniformly recover a
$\bar{k}$
-sparse signal in
${\BBR}^d$
, only
$O(\bar{k} \ln d)$
random projections are needed. This analysis improves some earlier results on OMP depending on stronger conditions that can only be satisfied with
$\Omega(\bar{k}^2 \ln d)$
or
$\Omega(\bar{k}^{1.6} \ln d)$
random projections.