Two-stage orthogonal Subspace Matching Pursuit for joint sparse recovery

Abstract

The joint sparse recovery problem addresses simultaneous recovery of jointly sparse signals (signal matrix) and their union support whose cardinality is k from their multiple measurement vectors (MMV) obtained through a common sensing matrix. k + 1 is the ideal lower bound on the minimum required number of measurements for perfect recovery for almost all signals, i.e., excluding a set of Lebesgue measure zero. To get close to the lower bound by taking advantage of the signal structure, Lee, et al. proposed the Subspace-Augmented MUltiple SIgnal Classification (SA-MUSIC) method which is guaranteed to achieve the lower bound when the rank of signal matrix is k and provided less restrictive conditions than existing methods in approaching k +1 in the practically important case when the rank of the signal matrix is smaller than k. The conditions, however, are still restrictive despite its empirically superior performance. We propose an efficient algorithm called the Two-stage orthogonal Subspace Matching Pursuit (TSMP) which has less theoretical restriction in approaching the lower bound than existing algorithms. Empirical results show that the TSMP method with low complexity outperforms most existing methods. The proposed scheme has better empirical performance than most existing methods even in the single measurement vectors (SMV) problem case. Variants of restricted isometry property or mutual coherence are used to improve the theoretical guarantees of TSMP and to cover the noisy case as well.