New conditions for uniformly recovering sparse signals via orthogonal matching pursuit

Recently, lots of work has been done on conditions of guaranteeing sparse signal recovery using orthogonal matching pursuit (OMP). However, none of the existing conditions is both necessary and sufficient in terms of the so-called restricted isometric property, coherence, cumulative coherence (Babel function), or other verifiable quantities in the literature. Motivated by this observation, we propose a new measure of a matrix, named as union cumulative coherence, and present both sufficient and necessary conditions under which the OMP algorithm can uniformly recover sparse signals for all sensing matrices. The proposed condition guarantees a uniform recovery of sparse signals using OMP, and reveals the capability of OMP in sparse recovery. We demonstrate by examples that the proposed condition can be used to more effectively determine the recoverable sparse signals via OMP than the conditions existing in the literature. Furthermore, sparse recovery from noisy measurements is also considered in terms of the proposed union cumulative coherence.