On the limited memory BFGS method for large scale optimization
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Projected Barzilai-Borwein methods for large-scale box-constrained quadratic programming
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A simple test to check the optimality of a sparse signal approximation
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Image Restoration with Discrete Constrained Total Variation Part I: Fast and Exact Optimization
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SIAM Journal on Optimization
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SIAM Journal on Optimization
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Sparse reconstruction by separable approximation
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Fixed-Point Continuation for $\ell_1$-Minimization: Methodology and Convergence
SIAM Journal on Optimization
An Efficient TVL1 Algorithm for Deblurring Multichannel Images Corrupted by Impulsive Noise
SIAM Journal on Scientific Computing
Bregman Iterative Algorithms for $\ell_1$-Minimization with Applications to Compressed Sensing
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Linearized Bregman Iterations for Frame-Based Image Deblurring
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A Fast Algorithm for Edge-Preserving Variational Multichannel Image Restoration
SIAM Journal on Imaging Sciences
A New Alternating Minimization Algorithm for Total Variation Image Reconstruction
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Parametric Maximum Flow Algorithms for Fast Total Variation Minimization
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Compressed sensing with cross validation
IEEE Transactions on Information Theory
A Singular Value Thresholding Algorithm for Matrix Completion
SIAM Journal on Optimization
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Bregmanized Nonlocal Regularization for Deconvolution and Sparse Reconstruction
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Fixed point and Bregman iterative methods for matrix rank minimization
Mathematical Programming: Series A and B
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A generalized uncertainty principle and sparse representation in pairs of bases
IEEE Transactions on Information Theory
On sparse representation in pairs of bases
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Why Simple Shrinkage Is Still Relevant for Redundant Representations?
IEEE Transactions on Information Theory
Iterative Regularization and Nonlinear Inverse Scale Space Applied to Wavelet-Based Denoising
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A New TwIST: Two-Step Iterative Shrinkage/Thresholding Algorithms for Image Restoration
IEEE Transactions on Image Processing
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SIAM Journal on Imaging Sciences
SIAM Journal on Imaging Sciences
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Error Forgetting of Bregman Iteration
Journal of Scientific Computing
Accelerated Linearized Bregman Method
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A fixed-point augmented Lagrangian method for total variation minimization problems
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A coupled variational model for image denoising using a duality strategy and split Bregman
Multidimensional Systems and Signal Processing
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This paper analyzes and improves the linearized Bregman method for solving the basis pursuit and related sparse optimization problems. The analysis shows that the linearized Bregman method has the exact regularization property; namely, it converges to an exact solution of the basis pursuit problem whenever its smooth parameter $\alpha$ is greater than a certain value. The analysis is based on showing that the linearized Bregman algorithm is equivalent to gradient descent applied to a certain dual formulation. This result motivates generalizations of the algorithm enabling the use of gradient-based optimization techniques such as line search, Barzilai-Borwein, limited memory BFGS (L-BFGS), nonlinear conjugate gradient, and Nesterov's methods. In the numerical simulations, the two proposed implementations, one using Barzilai-Borwein steps with nonmonotone line search and the other using L-BFGS, gave more accurate solutions in much shorter times than the basic implementation of the linearized Bregman method with a so-called kicking technique.