Orthogonal Matching Pursuit from Noisy Measurements: A New Analysis∗

A well-known analysis of Tropp and Gilbert shows that orthogonal matching pursuit (OMP) can recover a k-sparse n-dimensional real vector from m = 4k log(n) noise-free linear measurements obtained through a random Gaussian measurement matrix with a probability that approaches one as n → ∞. This work strengthens this result by showing that a lower number of measurements, m = 2k log(n − k), is in fact sufficient for asymptotic recovery. More generally, when the sparsity level satisfies kmin ≤ k ≤ kmax but is unknown, m = 2kmax log(n − kmin) measurements is sufficient. Furthermore, this number of measurements is also sufficient for detection of the sparsity pattern (support) of the vector with measurement errors provided the signal-to-noise ratio (SNR) scales to infinity. The scaling m = 2k log(n− k) exactly matches the number of measurements required by the more complex lasso method for signal recovery in a similar SNR scaling.