Sharp sufficient conditions for stable recovery of block sparse signals by block orthogonal matching pursuit

In this paper, we use the block orthogonal matching pursuit (BOMP) algorithm to recover block sparse signals x from measurements y = Ax + v, where v is an ℓ2-bounded noise vector (i.e., kvk2 ≤ ǫ for some constant ǫ). We investigate some sufficient conditions based on the block restricted isometry property (block-RIP) for exact (when v = 0) and stable (when v , 0) recovery of block sparse signals x. First, on the one hand, we show that if A satisfies the block-RIP with δK+1 1 and √2/2 ≤ δ < 1, the recovery of x may fail in K iterations for a sensingmatrix A which satisfies the block-RIP with δK+1 = δ. Finally, we study some sufficient conditions for partial recovery of block sparse signals. Specifically, if A satisfies the block-RIP with δK+1 < √2/2, then BOMP is guaranteed to recover some blocks of x if these blocks satisfy a sufficient condition. We further show that this condition is also sharp

Greed is good: algorithmic results for sparse approximation

This article presents new results on using a greedy algorithm, orthogonal matching pursuit (OMP), to solve the sparse approximation problem over redundant dictionaries. It provides a sufficient condition under which both OMP and Donoho's basis pursuit (BP) paradigm can recover the optimal representation of an exactly sparse signal. It leverages this theory to show that both OMP and BP succeed for every sparse input signal from a wide class of dictionaries. These quasi-incoherent dictionaries offer a natural generalization of incoherent dictionaries, and the cumulative coherence function is introduced to quantify the level of incoherence. This analysis unifies all the recent results on BP and extends them to OMP. Furthermore, the paper develops a sufficient condition under which OMP can identify atoms from an optimal approximation of a nonsparse signal. From there, it argues that OMP is an approximation algorithm for the sparse problem over a quasi-incoherent dictionary. That is, for every input signal, OMP calculates a sparse approximant whose error is only a small factor worse than the minimal error that can be attained with the same number of terms

Joint Block-Sparse Recovery Using Simultaneous BOMP/BOLS

We consider the greedy algorithms for the joint recovery of high-dimensional sparse signals based on the block multiple measurement vector (BMMV) model in compressed sensing (CS). To this end, we first put forth two versions of simultaneous block orthogonal least squares (S-BOLS) as the baseline for the OLS framework. Their cornerstone is to sequentially check and select the support block to minimize the residual power. Then, parallel performance analysis for the existing simultaneous block orthogonal matching pursuit (S-BOMP) and the two proposed S-BOLS algorithms is developed. It indicates that under the conditions based on the mutual incoherence property (MIP) and the decaying magnitude structure of the nonzero blocks of the signal, the algorithms select all the significant blocks before possibly choosing incorrect ones. In addition, we further consider the problem of sufficient data volume for reliable recovery, and provide its MIP-based bounds in closed-form. These results together highlight the key role of the block characteristic in addressing the weak-sparse issue, i.e., the scenario where the overall sparsity is too large. The derived theoretical results are also universally valid for conventional block-greedy algorithms and non-block algorithms by setting the number of measurement vectors and the block length to 1, respectively.Comment: This work has been submitted to the IEEE for possible publicatio