Primal and dual alternating direction algorithms for ℓ 1-ℓ 1-norm minimization problems in compressive sensing

Xiao, Yunhai; Zhu, Hong; Wu, Soon‐Yi

doi:10.1007/s10589-012-9475-x

Cited by 42 publications

(36 citation statements)

References 29 publications

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“…However, they can be solved using the alternating direction method of multipliers (ADMM) [79,80]. ADMM solves the extended 1 problem (19) by finding a saddle point of the augmented Lagrangian function…”

Section: -Based Alternating Direction Methodsmentioning

confidence: 99%

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Robust compressive sensing of sparse signals: a review

Carrillo

Ramirez

Arce

et al. 2016

EURASIP J. Adv. Signal Process.

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Compressive sensing generally relies on the 2 norm for data fidelity, whereas in many applications, robust estimators are needed. Among the scenarios in which robust performance is required, applications where the sampling process is performed in the presence of impulsive noise, i.e., measurements are corrupted by outliers, are of particular importance. This article overviews robust nonlinear reconstruction strategies for sparse signals based on replacing the commonly used 2 norm by M-estimators as data fidelity functions. The derived methods outperform existing compressed sensing techniques in impulsive environments, while achieving good performance in light-tailed environments, thus offering a robust framework for CS.

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Section: -Based Alternating Direction Methodsmentioning

confidence: 99%

“…The parameters μ and ν are step sizes. Convergence conditions for μ and ν and strategies to select β are detailed in [80].…”

Section: -Based Alternating Direction Methodsmentioning

confidence: 99%

Robust compressive sensing of sparse signals: a review

Carrillo

Ramirez

Arce

et al. 2016

EURASIP J. Adv. Signal Process.

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“…Since a standard CS method (e.g., BPDN or LASSO) may break down in highly impulsive noise, it is more appropriate to employ a robust method for initialization such as 1 -LA (6). The 1 -LA problem can be solved via the ADMM update steps (17), (18) and (16), which is guaranteed to converge to the global minimizer if τ 1 < 1/λ max (A T A) [45].…”

Section: Convergence Analysismentioning

confidence: 99%

Efficient and Robust Recovery of Sparse Signal and Image Using Generalized Nonconvex Regularization

Wen

Pei

Yang

et al. 2017

IEEE Trans. Comput. Imaging

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Abstract-This work addresses the robust reconstruction problem of a sparse signal from compressed measurements. We propose a robust formulation for sparse reconstruction which employs the 1-norm as the loss function for the residual error and utilizes a generalized nonconvex penalty for sparsity inducing. The 1-loss is less sensitive to outliers in the measurements than the popular 2-loss, while the nonconvex penalty has the capability of ameliorating the bias problem of the popular convex LASSO penalty and thus can yield more accurate recovery. To solve this nonconvex and nonsmooth minimization formulation efficiently, we propose a first-order algorithm based on alternating direction method of multipliers (ADMM). A smoothing strategy on the 1-loss function has been used in deriving the new algorithm to make it convergent. Further, a sufficient condition for the convergence of the new algorithm has been provided for generalized nonconvex regularization. In comparison with several state-of-the-art algorithms, the new algorithm showed better performance in numerical experiments in recovering sparse signals and compressible images. The new algorithm scales well for large-scale problems, as often encountered in image processing.

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“…It is a competitive method compared with other algorithms for compressive sensing. In [34], Xiao et al applied ADM method to ℓ 1 + ℓ 1 problem directly. Numerical results illustrated that the proposed algorithm performs better than Yall1 [18].…”

Section: Mathematical Problems In Engineeringmentioning

confidence: 99%

Sparse Signal Inversion with Impulsive Noise by Dual Spectral Projected Gradient Method

Ding

Zhao

Dou

2017

Mathematical Problems in Engineering

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We consider sparse signal inversion with impulsive noise. There are three major ingredients. The first is regularizing properties; we discuss convergence rate of regularized solutions. The second is devoted to the numerical solutions. It is challenging due to the fact that both fidelity and regularization term lack differentiability. Moreover, for ill-conditioned problems, sparsity regularization is often unstable. We propose a novel dual spectral projected gradient (DSPG) method which combines the dual problem of multiparameter regularization with spectral projection gradient method to solve the nonsmooth l1+l1 optimization functional. We show that one can overcome the nondifferentiability and instability by adding a smooth l2 regularization term to the original optimization functional. The advantage of the proposed functional is that its convex duality reduced to a constraint smooth functional. Moreover, it is stable even for ill-conditioned problems. Spectral projected gradient algorithm is used to compute the minimizers and we prove the convergence. The third is numerical simulation. Some experiments are performed, using compressed sensing and image inpainting, to demonstrate the efficiency of the proposed approach.

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Primal and dual alternating direction algorithms for ℓ 1-ℓ 1-norm minimization problems in compressive sensing

Cited by 42 publications

References 29 publications

Robust compressive sensing of sparse signals: a review

Robust compressive sensing of sparse signals: a review

Efficient and Robust Recovery of Sparse Signal and Image Using Generalized Nonconvex Regularization

Sparse Signal Inversion with Impulsive Noise by Dual Spectral Projected Gradient Method

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