Improved Conditions Of Block-Sparse Signal Recovery Via Non-convex Optimization Model

The concept of compressed sensing(CS)initiated by Candes et al.is a new sampling theory which is different from Nyquist theorem.Recently,it is growing fast,and has received extensive attention of researchers in signal processing,applied mathematics,statistics and so on.The goal of compressed sensing is to reconstruct a unknown high dimensional signal accurately with fewer number of linear measurements.In this theoretical framework,the sampling rate no longer depends on the bandwidth of signals,but greatly depends on two basic norms : the sparsity and the incoherence.The standard framework of compressed sensing does not take the structural characteristics of signals into account,which leads that signals can not be reconstructed well in most cases.The typical one is block-sparse signals,which the nonzero coefficients occur in some blocks.In this study,based on the linear compressed sensing framework,we mainly use the prior knowledge of signals and the non-convex optimization algorithm to complete the research of reconstructing block-sparse signals.The main research contents are as follows:(1)For block-sparse signals,based on the b-RIP framework and the b-ROC lemma,the first-order b-RIP condition is obtained to ensure the accurate recovery of blocksparse signals in the noiseless case via the non-convex optimization model.(2)Based on the traditional research of solving the sparse signal reconstruction problems via the non-convex optimization model,we obtain the g(q,k)(t-1)(10)k-order bRIP condition in block sparse field,which can be considered as a improvement of the existing results.(3)The traditional sparse representation lemma of a polytope is generalized into block sparse field,and correspondingly,obtain the block-sparse representation lemma of a polytope.Finally,we obtain the second-order b-RIP condition of guaranteeing the accurate recovery of block-sparse signals in the noiseless case based on the non-convex optimization model.