Analysis And Applications Of Sparse Signal Recovery Algorithm With High Probability

Compressive sensing(CS)theory has been proven to be a powerful tool for signal processing,which is an interdisciplinary field of probability and statistics,matrix analysis,geometric topology,functional analysis,etc.Iterative CS algorithms based on thresholding,feedback and null space tuning(NST+HT+FB)are exceedingly effective and fast,particularly for large scale problems.This paper turns out that further efforts at understanding the NST+HT+FB recovery algorithm with high probability by probability and statistics theory reveal that the finite number of convergence steps can be explicitly estimated.Since most natural signals are sparse or highly compressible under a basis,CS has a wide range of applications and extensions.In fact,CS is a single measurement vector(SMV)model.When a sequence of measurement vectors are available such as in source localization and DOA estimation,the CS model can be extended to the multiple measurement vector(MMV)model.If one aims at recovering a matrix rather than a sparse vector from incomplete measurements,sparsity can be replaced by the assumption that the original matrix is low-rank.As a popular special case of low-rank matrix recovery,robust principal component analysis(RPCA)based on statistics refers to the decomposition of an observed matrix into the low-rank component and the sparse component.In many CS problems,the key factor is a more realistic structured sparsity that goes beyond the simple sparsity by considering the interdependency structure among the sparse signal.Numerous works show that imposing structured sparsity on the support of the signal can also boost the recovery performance.Motivated by this trajectory,we introduce the sparse Bayesian learning algorithms to solve MMV and RPCA problems with considering structured sparsity by statistic theory.The application of CS and low-rank to the related problem of image super-resolution are contained in this paper.The major contributions of this paper are as follows:1.In this paper,we present a new perspective to analyze NST+HT+FB,which turns out that the efficiency of the algorithm can be further elaborated by an estimate of the number of iterations for the guaranteed convergence.The convergence condition of NST+HT+FB is also improved.Moreover,an adaptive scheme without the knowledge of the sparsity level is proposed with its convergence guarantee.The number of iterations for the finite step of convergence of the Adpt NST+HT+FB scheme is also derived.2.The suboptimal feedback scheme avoiding matrix inversion becomes exceedingly effective for large system recovery problems.An adaptive algorithm based on thresholding,suboptimal feedback and null space tuning without a prior knowledge of the sparsity level is proposed and analyzed with convergence guarantee.3.We apply inherent structures to MMV problem and introduce a Bayesian model with taking both spatial and temporal dependencies into account.Due to the property of the beta process that the sparse representation can be decomposed to values and sparsity indicators,the proposed algorithm ingeniously captures the temporal correlation structure by the learning of amplitudes and the spatial correlation structure by the estimation of clustered sparsity patterns.4.This paper proposes a Bayesian framework for RPCA with structured sparse component,where both amplitude and support correlation structure are considered simultaneously.For model learning,we resort to variational Bayesian inference,which can potentially be applied to estimate the posteriors of all latent variables.5.We develop a concept of cluster rather than using patch as the basic unit to solve single image super-resolution.For the proposed algorithm,all patches are splitted into numerous subspaces,and the optimal representation problem is solved with jointly low-rank and sparse regularization for each subspace.