An algorithm for the minimization of mixed l/sub 1/ and l/sub 2/ norms with application to Bayesian estimation

Alliney, S.; Ruzinsky, S.A.

doi:10.1109/78.277854

Cited by 161 publications

(98 citation statements)

References 15 publications

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“…where n is white Gaussian noise of variance σ 2 , and the prior on x is Laplacian (that is, log p(x) = −λ x 1 + K) [1], [25], [54]. Problem (1) technique to overcome the ill-conditioned, or even singular, nature of matrix A, when trying to infer x from noiseless observations y = Ax or from noisy observations as in (2).…”

Section: A Backgroundmentioning

confidence: 99%

Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems

Figueiredo

Nowak

Wright

2007

IEEE J. Sel. Top. Signal Process.

3,059

1,847

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Abstract-Many problems in signal processing and statistical inference involve finding sparse solutions to under-determined, or ill-conditioned, linear systems of equations. A standard approach consists in minimizing an objective function which includes a quadratic (squared ℓ2) error term combined with a sparseness-inducing (ℓ1) regularization term.Basis pursuit, the least absolute shrinkage and selection operator (LASSO), waveletbased deconvolution, and compressed sensing are a few wellknown examples of this approach. This paper proposes gradient projection (GP) algorithms for the bound-constrained quadratic programming (BCQP) formulation of these problems. We test variants of this approach that select the line search parameters in different ways, including techniques based on the BarzilaiBorwein method. Computational experiments show that these GP approaches perform well in a wide range of applications, often being significantly faster (in terms of computation time) than competing methods. Although the performance of GP methods tends to degrade as the regularization term is de-emphasized, we show how they can be embedded in a continuation scheme to recover their efficient practical performance.

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Section: A Backgroundmentioning

confidence: 99%

Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems

Figueiredo

Nowak

Wright

2007

IEEE J. Sel. Top. Signal Process.

3,059

1,847

View full text Add to dashboard Cite

show abstract

“…One is the CRNN-LAD algorithm defined in (41), which is based on a least absolute deviation (LAD) method (Alliney and Ruzinsky 1994). Another is the CRNN-GLAD algorithm by using a generalized least absolute deviation (GLAD) method (Xia and Kamel 2008).…”

Section: Crnns For Parameter Estimation Of Ar Signalsmentioning

confidence: 99%

Cooperative recurrent modular neural networks for constrained optimization: a survey of models and applications

Kamel

Xia

2008

Cogn Neurodyn

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Constrained optimization problems arise in a wide variety of scientific and engineering applications. Since several single recurrent neural networks when applied to solve constrained optimization problems for real-time engineering applications have shown some limitations, cooperative recurrent neural network approaches have been developed to overcome drawbacks of these single recurrent neural networks. This paper surveys in details work on cooperative recurrent neural networks for solving constrained optimization problems and their engineering applications, and points out their standing models from viewpoint of both convergence to the optimal solution and model complexity. We provide examples and comparisons to shown advantages of these models in the given applications.

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“…As in many recent publications [15,16,17,18,19,20], we adopt the TV regularizer to handle the ill-posed nature of the problem of inferring x. This amounts to computing the herein termed TV estimate, which is given by…”

Section: Problem Formulationmentioning

confidence: 99%

“…Total variation (TV) regularization was introduced by Rudin, Osher, and Fatemi in [15] and has become popular in recent years [15,16,17,18,19,20]. Recently, the range of application of TV-based methods has been successfully extended to inpainting, blind deconvolution, and processing of vector-valued images (e.g., color).…”

Section: Introductionmentioning

confidence: 99%

Total Variation-Based Image Deconvolution: a Majorization-Minimization Approach

Bioucas-Dias

Figueiredo

Oliveira

2006 IEEE International Conference on Acoustics Speed and Signal Processing Proceedings

164

190

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The total variation regularizer is well suited to piecewise smooth images. If we add the fact that these regularizers are convex, we have, perhaps, the reason for the resurgence of interest on TV-based approaches to inverse problems. This paper proposes a new TV-based algorithm for image deconvolution, under the assumptions of linear observations and additive white Gaussian noise. To compute the TV estimate, we propose a majorization-minimization approach, which consists in replacing a difficult optimization problem by a sequence of simpler ones, by relying on convexity arguments. The resulting algorithm has O(N ) computational complexity, for finite support convolutional kernels. In a comparison with state-of-the-art methods, the proposed algorithm either outperforms or equals them, with similar computational complexity.

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An algorithm for the minimization of mixed l/sub 1/ and l/sub 2/ norms with application to Bayesian estimation

Cited by 161 publications

References 15 publications

Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems

Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems

Cooperative recurrent modular neural networks for constrained optimization: a survey of models and applications

Total Variation-Based Image Deconvolution: a Majorization-Minimization Approach

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