Theory And Numerical Methods For Joint Sparse Signals And Low Rank Plus Sparse Recovery

Compressed sensing(CS)has been a very active field of recent research with a wide range of applications,including signal processing,medical imaging,radar system,and image compression,just to name a few.CS theory may be possible to surpass the traditional limits of sampling theory,and make the signal sampling and compression be performed simultaneously.Therefore,CS can reduce the cost of signal acquisition and storage in practice,and has received widespread attention in recent years.In CS com-munity,a central goal is to develop fast algorithms that can recover sparse signals from a relatively small number of linear measurements.The dissertation mainly focuses on the joint sparse recovery(also known as the multiple measurement vector,MMV)problem which is a natural extension of the single measurement vector(SMV)problem,and on low rank and sparse matrix decomposition(LRSD)problem from noisy observations.Meanwhile,the theory and algorithm for solving MMV and LRSD problem are studied in our analysis.The main work and innovation are described as follows.Firstly,due to lack of theoretical analysis of MMV problem,the sparse represen-tation of MMV has received wide attention.The Orthogonal Matching Pursuit(OMP)algorithm can find the sparsest representation of MMV,and OMP can exactly recover the joint sparse signals under a certain condition.We establish a new sufficient con-dition associated with the asymmetric restricted isometry property(ARIP)constants,under which all joint sparse signals can be recovered exactly via OMP in k iterations from the given linear measurement.In particular,if the symmetric RIP constant of the sensing matrix satisfies some condition for SMV,then all k sparse signals can also be recovered exactly via the OMP algorithm.Secondly,based the symmetric RIP constant,we improve the sufficient condition to recover exactly joint sparse signal via OMP.Moreover,a special sensing matrix is constructed such that the OMP algorithm fails to recover joint sparse signals in k itera-tions,Similar results also hold for k sparse signals recovery.In addition,our main result further improves the proposed bound by Mo et al.,which can not guarantee OMP to exactly recover some k sparse signals.Thirdly,since the MMV problem is a combinational optimization problem,its con-vex relaxation problem can be solved much efficiently.The classical alternating direction method of multipliers(ADMM)is an convex optimization algorithm that has recently become very popular due to its capabilities to solve large-scale or distributed problems.The MMV-ADMM algorithm to solve the MMV problem has been proposed by Lv et al.,but the theoretical result about the convergence of matrix iteration sequence generated by the algorithm is left as a future research topic.Based on the subdifferential property of the 2-norm for vector,a shrink operator associated with matrix is established.By using the operator,a convergence theorem is proved,which shows the MMV-ADMM algorithm can recover the joint sparse signals.Finally,we present 3-block ADMM to solve the LRSD problem.The 3-block AD-MM which is a natural extension of classical ADMM has been applied successfully to solve convex problem with 3-block variables,but the existing sufficient conditions to guarantee the convergence of the 3-block ADMM usually require the penalty parameter to satisfy a certain bound,which may affect the performance of solving the large scale problem in practice.However,we prove that the 3-block ADMM is convergent when the penalty parameters satisfy a weaker condition,and the objective function value se-quences generated by 3-block ADMM converge to the optimal value.Moreover,some preliminary numerical experiments verify that the 3-block ADMM for solving the LRSD problem from noisy observations can achieve higher precision and is more efficient than existing methods.