A Bound Of Restricted Isometry Constant For Recovery Of Sparse Signals And Low-rank Matrices

Compressed sensing is studied as a new theory about signal transmission in recent years.Sparse representation of signal,encoding measuring and reconstruction algorithm form three aspects of the compressed sensing theory.Sparse representation of signal is a priori condition of compressed sensing,when projecting signals to the orthogonal transformation base,the resulting transform vector is sparse or approximate sparse.In order to keep the original structure of the signal,in the measurement of coding,projection matrix must satisfy restricted isometry conditions,and then obtain linear projection measurement of the original signal through the product of original signal and measure matrix.Finally,reconstruct the original signal by the measured value and the projection matrix using the reconstruction algorithm.In this paper,a new bound on the restricted isometry conditions for sparse signal-s and low-rank matrices recovery is established.For the recovery of high-dimensional sparse signals,this paper considers constraint l₁ minimization methods in three set-tings:noiseless,bounded error,and Gaussian noise.It is shown that if the sensing matrix A satisfies the corresponding RIP condition,then all k-sparse signals ? can be recovered exactly via the constrained l₁ minimization based on y =A?,which has improved the bound that was established by T.Cai and A.Zhang?IEEE Trans.Inf.Theory,2014?.Similar results are established for low-rank matrix recovery.Finally,in the noisy case,sufficient conditions for the stable recovery of sparse signals and low-rank matrices are also given.