SIAM Journal on Control and Optimization
Nonlinear total variation based noise removal algorithms
Proceedings of the eleventh annual international conference of the Center for Nonlinear Studies on Experimental mathematics : computational issues in nonlinear science: computational issues in nonlinear science
Mathematical Programming: Series A and B
Atomic Decomposition by Basis Pursuit
SIAM Journal on Scientific Computing
Wavelets and curvelets for image deconvolution: a combined approach
Signal Processing - Special section: Security of data hiding technologies
Smooth minimization of non-smooth functions
Mathematical Programming: Series A and B
Sparse reconstruction by separable approximation
IEEE Transactions on Signal Processing
Fixed-Point Continuation for $\ell_1$-Minimization: Methodology and Convergence
SIAM Journal on Optimization
Probing the Pareto Frontier for Basis Pursuit Solutions
SIAM Journal on Scientific Computing
Bregman Iterative Algorithms for $\ell_1$-Minimization with Applications to Compressed Sensing
SIAM Journal on Imaging Sciences
A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems
SIAM Journal on Imaging Sciences
A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems
SIAM Journal on Imaging Sciences
The Split Bregman Method for L1-Regularized Problems
SIAM Journal on Imaging Sciences
A New Alternating Minimization Algorithm for Total Variation Image Reconstruction
SIAM Journal on Imaging Sciences
Exact Matrix Completion via Convex Optimization
Foundations of Computational Mathematics
The power of convex relaxation: near-optimal matrix completion
IEEE Transactions on Information Theory
Dense error correction via l1-minimization
IEEE Transactions on Information Theory
Fast image recovery using variable splitting and constrained optimization
IEEE Transactions on Image Processing
A coordinate gradient descent method for l1-regularized convex minimization
Computational Optimization and Applications
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Why Simple Shrinkage Is Still Relevant for Redundant Representations?
IEEE Transactions on Information Theory
Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit
IEEE Transactions on Information Theory
An EM algorithm for wavelet-based image restoration
IEEE Transactions on Image Processing
Inexact Alternating Direction Methods for Image Recovery
SIAM Journal on Scientific Computing
Alternating Direction Method for Image Inpainting in Wavelet Domains
SIAM Journal on Imaging Sciences
Alternating Direction Method for Covariance Selection Models
Journal of Scientific Computing
An alternating direction method for finding Dantzig selectors
Computational Statistics & Data Analysis
Joint sparsity-based robust multimodal biometrics recognition
ECCV'12 Proceedings of the 12th international conference on Computer Vision - Volume Part III
An ADM-based splitting method for separable convex programming
Computational Optimization and Applications
Computational Optimization and Applications
Advances in Computational Mathematics
Advances in Computational Mathematics
Compressed sensing photoacoustic imaging based on fast alternating direction algorithm
Journal of Biomedical Imaging
Proceedings of the 2nd International Workshop on Big Data, Streams and Heterogeneous Source Mining: Algorithms, Systems, Programming Models and Applications
International Journal of Computer Vision
A comparison of typical ℓp minimization algorithms
Neurocomputing
Letters: Two-dimensional relaxed representation
Neurocomputing
Recovering low-rank matrices from corrupted observations via the linear conjugate gradient algorithm
Journal of Computational and Applied Mathematics
A proximal parallel splitting method for minimizing sum of convex functions with linear constraints
Journal of Computational and Applied Mathematics
A simple and efficient algorithm for fused lasso signal approximator with convex loss function
Computational Statistics
Computers & Mathematics with Applications
Bilinear discriminative dictionary learning for face recognition
Pattern Recognition
Sparse semi-supervised learning on low-rank kernel
Neurocomputing
Computational Optimization and Applications
Novel document detection for massive data streams using distributed dictionary learning
IBM Journal of Research and Development
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In this paper, we propose and study the use of alternating direction algorithms for several $\ell_1$-norm minimization problems arising from sparse solution recovery in compressive sensing, including the basis pursuit problem, the basis pursuit denoising problems of both unconstrained and constrained forms, and others. We present and investigate two classes of algorithms derived from either the primal or the dual form of $\ell_1$-problems. The construction of the algorithms consists of two main steps: (1) to reformulate an $\ell_1$-problem into one having blockwise separable objective functions by adding new variables and constraints; and (2) to apply an exact or inexact alternating direction method to the augmented Lagrangian function of the resulting problem. The derived alternating direction algorithms can be regarded as first-order primal-dual algorithms because both primal and dual variables are updated at every iteration. Convergence properties of these algorithms are established or restated when they already exist. Extensive numerical experiments are performed, using randomized partial Walsh-Hadamard sensing matrices, to demonstrate the versatility and effectiveness of the proposed approach. Moreover, we present numerical results to emphasize two practically important but perhaps overlooked points: (i) that algorithm speed should be evaluated relative to appropriate solution accuracy; and (ii) that when erroneous measurements possibly exist, the $\ell_1$-fidelity should generally be preferable to the $\ell_2$-fidelity.