Orthogonal matching pursuit: A Brownian motion 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 k _min≤ k ≤ k _max but is unknown, m=2k _max log(n-k _min) 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 with a similar SNR scaling.