An &amp;#x2113;&lt;inf&gt;1&lt;/inf&gt;-algorithm for underdetermined systems and applications

Argaez, Miguel; Ramírez, Carlos; Sánchez, Raquel Criado

doi:10.1109/nafips.2011.5752016

Cited by 9 publications

(12 citation statements)

References 7 publications

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“…To perform this comparison, we applied each of the two algorithms 30 times, and for each application, we computed the distance ∥x − x∥ 2 The average of these values is 1191.01, which is smaller than the average distance corresponding to the original algorithm.…”

Section: Testing the New Algorithm: Preliminary Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Rotation-invariance can further improve state-of-the-art blind deconvolution techniques

Cervantes

Usevitch

Kreinovich

2016

2016 IEEE International Conference on Systems, Man, and Cybernetics (SMC)

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Section: Testing the New Algorithm: Preliminary Resultsmentioning

confidence: 99%

“…|a i | instead of the discontinuous expression ∥a∥ 0 ; see, e.g., [2]. The ℓ 1 -norm is convex, so if J is also convex (and it often is), then we get an additional advantage of being able to use known algorithms for minimizing convex functions.…”

Section: Need To Use Sparsity-based Techniquesmentioning

confidence: 99%

Rotation-invariance can further improve state-of-the-art blind deconvolution techniques

Cervantes

Usevitch

Kreinovich

2016

2016 IEEE International Conference on Systems, Man, and Cybernetics (SMC)

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“…While numerous algorithms and methodologies have been proposed to solve this problem in the presence of Gaussian noise e.g. (Chen et al, 2001;Kim et al, 2007;Figueiredo et al, 2007;Wright et al, 2008;Argaez et al, 2011;Becker et al, 2010), strategies addressing the case of impulse or sparse noise are more limited. Recently, the works of Nikolova (2004), Fu et al (2006) and Bar et al (2006) studied the recovery problem in the presence of impulse noise but in an image processing setting and for complete datasets.…”

Section: Discussionmentioning

confidence: 99%

“…While several numerical methods have proliferated for solving (1) in which the fidelity term is measured with the 2 norm, e.g. (Chen et al, 2001;Kim et al, 2007;Figueiredo et al, 2007;Wright et al, 2008;Argaez et al, 2011;Becker et al, 2010), research efforts aimed at solving (1) as it is presented here have been more limited. Such works include Fu et al (2006), where problem (1) is posed as a linear programming problem and solved using interior point methods; Nikolova (2004), who considers the denoising case, i.e.…”

Section: Methodsmentioning

confidence: 99%

Interpolating Sparsely Corrupted Signals in Micrometeorology

Ramírez¹,

Argaez²,

Jaimes³

et al. 2014

JMR

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In real applications where data acquisition is carried out under extreme conditions, post-processing techniques for systematic corrections are of critical importance. In micrometeorological studies, it is often the case that acquired data contains both missing information and impulse noise due to instrumentation failure, data transmission and data rejection for quality assurance. In this work, we propose a simple algorithm based on an 1 − 1 variational formulation that simultaneously suppresses impulse noise and interpolates missing information. Our approach consists of relaxing the objective function in the variational formulation with a strictly convex and continuously differentiable function that depends on a regularization parameter. We solve a sequence of strictly convex optimization subproblems as the regularization parameter goes to zero, converging to the solution of the original problem. Numerical experiments on real micrometeorological datasets are conducted showing the effectiveness of the proposed approach. Furthermore, a convergence analysis is presented providing theoretical guaranties of our method.

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“…Motivated by the recent successful algorithms proposed for sparse signal recovery problems in [1], [7], [14], we apply the selective nature of sparse representation to solve classification problems. A test sample is represented in an overcomplete dictionary with the training samples as base elements.…”

Section: Introductionmentioning

confidence: 99%

Sparse representation via ℓ<inf>1</inf>-minimization for underdetermined systems in classification of tumors with gene expression data

Sánchez

Argaez

Guillén

2011

2011 Annual International Conference of the IEEE Engineering in Medicine and Biology Society

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The development of cancer diagnosis models and cancer discovery from DNA microarray data are of great interest in bioinformatics and medicine. In pattern recognition and machine learning, a classification problem refers to finding an algorithm for assigning a given input data into one of several categories. Many natural signals are sparse or compressible in the sense that they have short representations when expressed in a suitable basis. Motivated by the recent successful algorithm developments for sparse signal recovery, we apply the selective nature of sparse representation to perform the above mentioned classification. In order to find such sparse representation we implement an ℓ(1)-minimization algorithm. This methodology overcomes the lack of robustness with respect to outliers. In contrast to other classification algorithms, no model selection dependency is involved. The minimization algorithm is a convex relaxation-like that has been proven to efficiently recover sparse signals. To study its performance, the proposed method is applied to six tumor gene expression datasets and numerically compared with various support vector machine methods (SVM). The numerical results show that the ℓ(1)-minimization algorithm proposed performs at least comparably and often better than SVMs.

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An ℓ<inf>1</inf>-algorithm for underdetermined systems and applications

Cited by 9 publications

References 7 publications

Rotation-invariance can further improve state-of-the-art blind deconvolution techniques

Rotation-invariance can further improve state-of-the-art blind deconvolution techniques

Interpolating Sparsely Corrupted Signals in Micrometeorology

Sparse representation via ℓ<inf>1</inf>-minimization for underdetermined systems in classification of tumors with gene expression data

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An &#x2113;<inf>1</inf>-algorithm for underdetermined systems and applications

Cited by 9 publications

References 7 publications

Rotation-invariance can further improve state-of-the-art blind deconvolution techniques

Rotation-invariance can further improve state-of-the-art blind deconvolution techniques

Interpolating Sparsely Corrupted Signals in Micrometeorology

Sparse representation via &#x2113;<inf>1</inf>-minimization for underdetermined systems in classification of tumors with gene expression data

Contact Info

Product

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An ℓ<inf>1</inf>-algorithm for underdetermined systems and applications

Sparse representation via ℓ<inf>1</inf>-minimization for underdetermined systems in classification of tumors with gene expression data