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Nearly optimal number of iterations for sparse signal recovery with orthogonal multi-matching pursuit

A signal x is called K-sparse if it has at most K nonzero entries. Recovering a K-sparse signal x from linear measurements y = Ax + w, where A is a sensing matrix and w is a noise vector, arises from numerous applications. Orthogonal multi-matching pursuit (OMMP), which is an extension of the orthogonal matching pursuit (OMP) algorithm and has better recovery performance than OMP, is a popular sparse recovery algorithm. One of the main challenges to study the recovery performance of OMMP is to investigate the optimal required number of iterations for ensuring stable reconstruction of x. This paper provides a nearly optimal number of iterations. Specifically, based on the restricted isometry property of the sensing matrix, we present a sufficient condition that can guarantee stable reconstruction of x in nearly optimal number of iterations by OMMP. Furthermore, we build an upper bound on the recovery error with fewer required iterations than existing results. Our results show that the required number of iterations to ensure stable recovery of any K-sparse signals is fewer than those required by the state-of-the-art results.