Signal And Low Rank Matrix Recovery Based On Restricted Equidistant Properties

Compressed sensing(CS)is a new type of sampling theory which guides informa-tion acquisition.CS achieves data acquisition with sampling rate significantly lower than the Nyquist-Shannon rate,and reduces the amount of data needed to signal re-covery.In this dissertation,we consider the problem of reconstruction of signals which have different characteristics and low rank matrices.We mainly investigate the re-covery condition that different reconstruction methods can successfully recover signals and low rank matrices.The recovery guarantees being considered are all based on RIP.The existing recovery conditions are improved and extended of the weighted l1 minimization for for signal recovery with prior support information,the mixed l2/l1 minimization for signal reconstruction with block structure,the l1-analysis for signal recovery with sparse expansion in redundant dictionaries and the Schatten-p minimiza-tion for low rank matrix reconstruction.We obtain the general recovery conditions.Comparing with the existing conclusions,we discuss the optimality of recovery condi-tions and the superiority of reconstruction performance.The main research contents and dissertation structure is as follows.In Chapter 1,we introduce briefly the research background and research develop-ment of compressed sensing,the notations and concepts appearing in the dissertation as well as the main results of the dissertation.In Chapter 2,we study the reconstruction condition of signals whose partial prior support information is available.Firstly,the recovery conditions based on the high order RIP and based on RIC and ROC are established for the stable and exact recov-ery of sparse signals in the noisy and noiseless cases via the weighted l1 minimization,respectively.When the accuracy of prior support information is at least 50%,the re-covery conditions we establish are weaker than the existing optimal recovery conditions by the l1 minimization method.Meanwhile,the better reconstruction performances are provided.Furthermore,we prove that the recovery conditions is sharp by means of the weighted l1 minimization method.In Chapter 3,we consider the recovery condition of signals which have a block structure.Firstly,the block sparse representation of a polytope is established.A recovery condition in terms of the high order block RIP is obtained by using this representation.The recovery condition guarantees the stable recovery of all block s-parse signals in the presence of noise,and exact recovery of block sparse signals in the noiseless case via the mixed l2/l1 minimization.Moreover,a concrete example is constructed to prove that the recovery condition is sharp.The significance of this chapter is that signals can be recovered under more general conditions by exploit-ing the block sparse structure instead of the conventional sparsity pattern.Finally,the number of measurements which make the Gaussian random matrices satisfy the recovery condition we establish with overwhelming probability are obtained and less than that the Gaussian random matrices meeting standard RIP with overwhelming probability needed.In Chapter 4,we investigate the reconstruction condition of signals which have sparse representations in redundant dictionaries.In view of the high order D-RIP condition,all signals with sparse representation can be recovered stably and exactly via the l1-analysis method in the noisy setting and in the noiseless setting,respectively.In Chapter 5,we consider the recovery condition of low rank matrices.According to the singular value decomposition of matrices,a recovery condition is established to ensure the stable and exact recovery of all low rank matrices via the Schatten-p minimization method.This recovery condition breaks through the range of the recovery condition RIP-based which the nuclear norm minimization method need.