Sparse Optimization Theory and Methods

Zhao, Yun-Bin

doi:10.1201/9781315113142

Cited by 43 publications

(84 citation statements)

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“…Motivated by the new analysis tool introduced in [29], we develop the stability result for the model (2) in this paper under the assumption of restricted weak range space property (RSP) of order k (which will be introduced in next section). Our result extends the stability theorem for ℓ 1 -minimization established by Zhao et al [28][29][30].…”

Section: Introductionsupporting

confidence: 88%

“…The problem (C1) is often called the standard ℓ 0 -minimization problem [8,17,28]. Two structured sparsity models, called the nonnegative sparsity model [7,8,17,28] and the monotonic sparsity model (isotonic regression) [23,24], are also the special cases of the model (1). It is well known that ℓ 1 -minimization is a useful method to solve the ℓ 0 -minimization problem.…”

Section: Introductionmentioning

confidence: 99%

“…(D1) min The problem (D2) is often called quadratically constrained basis pursuit [10,17,28], and it reduces to (D1) if ε = 0, which is called standard ℓ 1 -minimization or the basis pursuit [8,12,17,19,26]. The problem (D4) is the type of Dantzig Selectors [9,17].…”

Section: Introductionmentioning

confidence: 99%

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Stability analysis of a class of sparse optimization problems

Zhao

2020

Optimization Methods and Software

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The sparse optimization problems arise in many areas of science and engineering, such as compressed sensing, image processing, statistical and machine learning. The ℓ0-minimization problem is one of such optimization problems, which is typically used to deal with signal recovery. The ℓ1-minimization method is one of the plausible approaches for solving the ℓ0-minimization problems, and thus the stability of such a numerical method is vital for signal recovery. In this paper, we establish a stability result for the ℓ1-minimization problems associated with a general class of ℓ0-minimization problems. To this goal, we introduce the concept of restricted weak range space property (RSP) of a transposed sensing matrix, which is a generalized version of the weak RSP of the transposed sensing matrix introduced in [Zhao et al., Math. Oper. Res., 44(2019), 175-193]. The stability result established in this paper includes several existing ones as special cases.

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Section: Introductionsupporting

confidence: 88%

Section: Introductionmentioning

confidence: 99%

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Stability analysis of a class of sparse optimization problems

Zhao

2020

Optimization Methods and Software

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“…Both (1) and (2) are the central models for sparse signal recovery and sparse representation of data on their redundant bases. These models provide an essential basis for the development of the theory and algorithms for compressed sensing (see, e.g., [12,25,26,32,53]). The problem (1) has also been widely used in the fields of statistical regressions and wireless communications (see, e.g., [45,4,41]).…”

Section: Introductionmentioning

confidence: 99%

“…The problems (1) and (2) are NP-hard in general [46]. The plausible algorithms for such problems can be briefly categorized into the following classes: (i) Convex optimization methods (e.g., 1 -minimization [19], reweighed 1 -minimization [16,31,56], and dual-density-based reweighted 1 -minimization [53,54,55]); (ii) heuristic methods (such as matching pursuit [43], orthogonal matching pursuit [44,52], compressive sampling matching pursuit [47], and subspace pursuit [20]); (iii) thresholding methods (e.g., soft thresholding [21,22,24], hard thresholding [6,7,8,30], graded hard thresholding pursuits [10,11], and the 'firm' thresholding [51]); (iv) integer programming methods [4].…”

Section: Introductionmentioning

confidence: 99%

Optimal $k$-Thresholding Algorithms for Sparse Optimization Problems

Zhao¹

2020

SIAM J. Optim.

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117

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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.

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