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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 a truly efficient thresholding method is a fundamental question in this field. The aim of this paper is to address this question by proposing a new thresholding technique based on the notion of optimal k-thresholding. The central idea for this new development is to connect the k-thresholding directly to the residual reduction during the course of algorithms. This leads to a natural design principle for the efficient thresholding methods. Under the restricted isometry property (RIP), we prove that the optimal thresholding based algorithms are globally convergent to the solution of sparse optimization problems. The numerical experiments demonstrate that when solving sparse optimization problems, the traditional hard thresholding methods have been significantly transcended by the proposed algorithms which can even outperform the classic 1 -minimization method in many situations.
Abstract-Recently, the worse-case analysis, probabilistic analysis and empirical justification have been employed to address the fundamental question: When does ℓ1-minimization find the sparsest solution to an underdetermined linear system? In this paper, a deterministic analysis, rooted in the classic linear programming theory, is carried out to further address this question. We first identify a necessary and sufficient condition for the uniqueness of least ℓ1-norm solutions to linear systems. From this condition, we deduce that a sparsest solution coincides with the unique least ℓ1-norm solution to a linear system if and only if the so-called range space property (RSP) holds at this solution. This yields a broad understanding of the relationship between ℓ0-and ℓ1-minimization problems. Our analysis indicates that the RSP truly lies at the heart of the relationship between these two problems. Through RSP-based analysis, several important questions in this field can be largely addressed. For instance, how to efficiently interpret the gap between the current theory and the actual numerical performance of ℓ1-minimization by a deterministic analysis, and if a linear system has multiple sparsest solutions, when does ℓ1-minimization guarantee to find one of them? Moreover, new matrix properties (such as the RSP of order K and the Weak-RSP of order K) are introduced in this paper, and a new theory for sparse signal recovery based on the RSP of order K is established.Index Terms-Underdetermined linear system, sparsest solution, least ℓ1-norm solution, range space property, strict complementarity, sparse signal recovery, compressed sensing.
Many practical problems can be formulated as ℓ 0 -minimization problems with nonnegativity constraints, which seek the sparsest nonnegative solutions to underdetermined linear systems. Recent study indicates that ℓ 1 -minimization is efficient for solving some classes of ℓ 0 -minimization problems. From a mathematical point of view, however, the understanding of the relationship between ℓ 0 -and ℓ 1 -minimization remains incomplete. In this paper, we further discuss several theoretical questions associated with these two problems. For instance, how to completely characterize the uniqueness of least ℓ 1 -norm nonnegative solutions to a linear system, and is there any alternative matrix property that is different from existing ones, and can fully characterize the uniform recovery of K-sparse nonnegative vectors? We prove that the fundamental strict complementarity theorem of linear programming can yield a necessary and sufficient condition for a linear system to have a unique least ℓ 1 -norm nonnegative solution. This condition leads naturally to the so-called range space property (RSP) and the 'full-column-rank' property, which altogether provide a broad understanding of the relationship between ℓ 0 -and ℓ 1 -minimization. Motivated by these results, we introduce the concept of the 'RSP of order K' that turns out to be a full characterization of the uniform recovery of K-sparse nonnegative vectors. This concept also enables us to develop certain conditions for the non-uniform recovery of sparse nonnegative vectors via the so-called weak range space property.
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