Improved RIP-Based Bounds for Guaranteed Performance of Two Compressed Sensing Algorithms

Iterative hard thresholding (IHT) and compressive sampling matching pursuit (CoSaMP) are two mainstream compressed sensing algorithms using the hard thresholding operator. The guaranteed performance of the two algorithms for signal recovery was mainly analyzed in terms of the restricted isometry property (RIP) of sensing matrices. At present, the best known bound using RIP of order … Read more

Dual-density-based reweighted $\ell_{1}hBcalgorithms for a class of $\ell_{0}hBcminimization problems

The optimization problem with sparsity arises in many areas of science and engineering such as compressed sensing, image processing, statistical learning and data sparse approximation. In this paper, we study the dual-density-based reweighted $\ell_{1}$-algorithms for a class of $\ell_{0}$-minimization models which can be used to model a wide range of practical problems. This class of … Read more

Optimal K-Thresholding Algorithms for Sparse Optimization Problems

The simulations indicate that the existing hard thresholding technique independent of the residual function may cause a dramatic increase or numerical oscillation of the residual. This inherit drawback of the hard thresholding renders the traditional thresholding algorithms unstable and thus generally inefficient for solving practical sparse optimization problems. How to overcome this weakness and develop … Read more

Stability Analysis for a Class of Sparse Optimization Problems

The sparse optimization problems arise in many areas of science and engineering, such as compressed sensing, image processing, statistical and machine learning. The $\ell_{0}$-minimization problem is one of such optimization problems, which is typically used to deal with signal recovery. The $\ell_{1}$-minimization method is one of the plausible approaches for solving the $\ell_{0}$-minimization problems, and … Read more

Weak Stability of $\ell_1hBcminimization Methods in Sparse Data Reconstruction

As one of the most plausible convex optimization methods for sparse data reconstruction, $\ell_1$-minimization plays a fundamental role in the development of sparse optimization theory. The stability of this method has been addressed in the literature under various assumptions such as restricted isometry property (RIP), null space property (NSP), and mutual coherence. In this paper, … Read more

Constructing New Weighted l1-Algorithms for the Sparsest Points of Polyhedral Sets

The l0-minimization problem that seeks the sparsest point of a polyhedral set is a longstanding challenging problem in the fields of signal and image processing, numerical linear algebra and mathematical optimization. The weighted l1-method is one of the most plausible methods for solving this problem. In this paper, we develop a new weighted l1-method through … Read more

1-Bit Compressive Sensing: Reformulation and RRSP-Based Sign Recovery Theory

Recently, the 1-bit compressive sensing (1-bit CS) has been studied in the field of sparse signal recovery. Since the amplitude information of sparse signals in 1-bit CS is not available, it is often the support or the sign of a signal that can be exactly recovered with a decoding method. In this paper, we first … Read more

Uniqueness Conditions for A Class of $\ell_0hBcMinimization Problems

We consider a class of $\ell_0$-minimization problems, which is to search for the partial sparsest solution to an underdetermined linear system with additional constraints. We introduce several concepts, including $l_p$-induced quasi-norm ($0

Equivalence and Strong Equivalence between Sparsest and Least $\ell_1hBcNorm Nonnegative Solutions of Linear Systems and Their Application

Many practical problems can be formulated as $\ell_0$-minimization problems with nonnegativity constraints, which seek the sparsest nonnegative solutions to underdetermined linear systems. Recent study indicates that $\ell_1$-minimization is efficient for solving some classes of $\ell_0$-minimization problems. From a mathematical point of view, however, the understanding of the relationship between $\ell_0$- and $\ell_1$-minimization remains incomplete. In … Read more

Equivalence and Strong Equivalence between Sparsest and Least l1-Norm Nonnegative Solutions of Linear Systems and Their Application

Many practical problems can be formulated as $\ell_0$-minimization problems with nonnegativity constraints, which seek the sparsest nonnegative solutions to underdetermined linear systems. Recent study indicates that $\ell_1$-minimization is efficient for solving some classes of $\ell_0$-minimization problems. From a mathematical point of view, however, the understanding of the relationship between $\ell_0$- and $\ell_1$-minimization remains incomplete. In … Read more