Recovery conditions for generalized orthogonal matching pursuit based coherence

Abstract

In sparse approximation, a key theoretical issue is the guarantee conditions for the exact recovery of $s$-sparse signals. The Orthogonal Matching Pursuit (OMP) and the Generalized Orthogonal Matching Pursuit (GOMP) are two important algorithms commonly used in sparse approximation. The main difference is that the OMP algorithm selects one atom in each iteration, while the GOMP algorithm selects multiple atoms. In the current theoretical analysis, the GOMP algorithm can only guarantee the selection of at least one correct atom in each iteration. However, in practical applications, the GOMP algorithm has been shown to select multiple correct atoms in each iteration but lacks theoretical guarantee conditions. In this paper, we discuss the extended coherence-based conditions for exact support recovery of the $s$-sparse signals using the GOMP algorithm. We propose several sufficient conditions for the GOMP algorithm to select $M$ ($1\le M\le s$) correct atoms in each iteration in noiseless and bounded-noise cases respectively. Some of the conditions involve the decay of nonzero entries in sparse signals. Numerical experiments demonstrate the effectiveness of the proposed sufficient conditions.