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elastic-net regularization: Iterative Algorithms and Asymptotic Behavior of Solutions

NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 2010年第12期31卷 1406-1432页

作者： Umanita, Veronica Villa, Silvia Univ Genoa DIMA I-16146 Genoa Italy Univ Genoa DISI I-16146 Genoa Italy

We derive strongly convergent algorithms to solve inverse problems involving elastic-net regularization. Moreover, using functional analysis techniques, we provide a rigorous study of the asymptotic properties of the regularized solutions that allows to cast in a unified framework 1, elastic-net and classical Tikhonov regularization.

关键词： elastic-net regularization Forward-backward splitting algorithm Linear inverse problems

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elastic-net regularization for nonlinear electrical impedance tomography with a splitting approach

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APPLICABLE ANALYSIS 2019年第12期98卷 2201-2217页

作者： Wang, Jing Han, Bo Wan, Wei Heilongjiang Univ Sch Math Sci Harbin Heilongjiang Peoples R China Harbin Inst Technol Dept Math Harbin Heilongjiang Peoples R China Jiaxing Univ Coll Math Phys & Informat Engn Jiaxing Zhejiang Peoples R China

Image reconstruction of EIT mathematically is a typical nonlinear and severely ill-posed inverse problem. Appropriate priors or penalties are required to enable the reconstruction. The commonly used -norm can enforce the stability to preserve local smoothness, and the current -norm can enforce the sparsity to preserve sharp edges. Considering the fact that -norm penalty always makes the solution overly smooth and -norm penalty always makes the solution too sparse, elastic-net regularization approach with a convex combination term of -norm and -norm emerges for fully nonlinear EIT inverse problems. Our aim is to combine the strength of both terms: sparsity in the transform domain and smoothness in the physical domain, in an attempt to improve the reconstruction resolution and robustness to noise. Nonlinearity and non-smoothness of the generated composite minimization problem make it challenging to find an efficient numerical solution. Then we develop one simple and fast numerical optimization scheme based on the split Bregman technique for the joint penalties regularization. The method is validated using simulated data for some typical conductivity distributions. Results indicate that the proposed inversion model with an appropriate parameter choice achieves an efficient regularization solution and enhances the quality of the reconstructed image.

关键词： Electrical impedance tomography image reconstruction regularization method elastic-net regularization split Bregman iteration

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elastic-net regularization based multi-point vibration response prediction in situation of unknown uncorrelated multiple sources load

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INTERNATIONAL JOURNAL OF APPLIED ELECTROMAGnetICS AND MECHANICS 2020年第1-4期64卷 649-657页

作者： Chen, Delei Wang, Cheng Lai, Xiongming Zhang, Huizhen Li, Haibo Chen, Jianwei Huaqiao Univ Coll Comp Sci & Technol 668 Jimei Rd Xiamen 361021 Fujian Peoples R China Huaqiao Univ Coll Mech Engn & Automat Xiamen Fujian Peoples R China San Diego State Univ Dept Math & Stat San Diego CA 92182 USA

In order to reduce the influence of ill-posed inverse on response prediction in the situation of unknown uncorrelated multiple sources load, a response prediction method based on elastic-net regularization in the frequency domain was proposed. This method utilized the linear relationship between known responses and the unknown responses instead of the transfer function to predict the response. Moreover, the elastic-net regularization model has two regularization parameters combining l1, l2 regularization to reduce the influence of ill-posed inverse. The experiment results on the data of acoustic and vibration sources on cylindrical shells showed that the elastic-net regularization in predicting response could obtain higher accurate results compared with the method of transfer function and the method of ordinary least squares, and predict vibration response effectively and satisfy industrial requirements.

关键词： Multi-point vibration response prediction elastic-net regularization ill-posed ordinary least squares

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elastic-net regularization based multi-point vibration response prediction in situation of unknown uncorrelated multiple sources load

Elastic-net regularization based multi-point vibration respo...

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19th International Symposium on Applied Electromagnetics and Mechanics (ISEM)

关键词： Multi-point vibration response prediction elastic-net regularization ill-posed ordinary least squares

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Single Image Blind Deblurring Based on Salient Edge-Structures and elastic-net regularization

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JOURNAL OF MATHEMATICAL IMAGING AND VISION 2020年第8期62卷 1049-1061页

作者： Yu, XiaoYuan Xie, Wei South China Univ Technol Coll Automat Sci & Technol Guangzhou 510640 Peoples R China

In single image blind deblurring, the blur kernel and latent image are estimated from a single observed blurry image. The associated mathematical problem is ill-posed, and an acceptable solution is difficult to obtain without additional priors or heuristics. Inspired by the nonlocal self-similarity in image denoising problem, we introduce elastic-net regularization as a rank prior to improve the estimation of the intermediate image. Furthermore, it is well known that salient edge-structures can provide reliable information for kernel estimation. Therefore, we propose a new blind image deblurring method by combining the salient edge-structures and the elastic-net regularization. The salient edge-structures are selected from the intermediate image and used to guide the estimation of the blur kernel. Then, we employ the elastic-net regularization and edge-structures to further estimate intermediate latent image, by retaining the dominant edge and removing slight texture, for a better kernel estimation. Finally, quantitative and qualitative evaluations are conducted by comparing the results with those obtained by state-of-the-art methods. We conclude that the proposed method performs favorably when considering both synthetic and real blurry images.

关键词： Single image blind deblurring Salient edge-structures elastic-net regularization Kernel and latent image estimation

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Learning performance of elastic-net regularization

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MATHEMATICAL AND COMPUTER MODELLING 2013年第5-6期57卷 1395-1407页

作者： Zhao, Yu-long Feng, Yun-long City Univ Hong Kong Dept Math Kowloon Hong Kong Peoples R China

In this paper, within the framework of statistical learning theory we address the elastic-net regularization problem. Based on the capacity assumption of hypothesis space composed by infinite features, significant contributions are made in several aspects. First, concentration estimates for sample error are presented by introducing l(2)-empirical covering number and utilizing an iteration process. Second, a constructive approximation approach for estimating approximation error is presented. Third, the elastic-net learning with infinite features is studied and the role that the tuning parameter. plays is also discussed. Finally, our learning rate is shown to be faster compared with existing results. (C) 2012 Elsevier Ltd. All rights reserved.

关键词： Learning theory elastic-net regularization l(2)-empirical covering number Learning rate

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Error Analysis for Matrix elastic-net regularization Algorithms

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IEEE TRANSACTIONS ON NEURAL netWORKS AND LEARNING SYSTEMS 2012年第5期23卷 737-748页

作者： Li, Hong Chen, Na Li, Luoqing Huazhong Univ Sci & Technol Sch Math & Stat Wuhan 430074 Peoples R China Hubei Univ Fac Math & Comp Sci Wuhan 430062 Peoples R China

elastic-net regularization is a successful approach in statistical modeling. It can avoid large variations which occur in estimating complex models. In this paper, elastic-net regularization is extended to a more general setting, the matrix recovery (matrix completion) setting. Based on a combination of the nuclear-norm minimization and the Frobenius-norm minimization, we consider the matrix elastic-net (MEN) regularization algorithm, which is an analog to the elastic-net regularization scheme from compressive sensing. Some properties of the estimator are characterized by the singular value shrinkage operator. We estimate the error bounds of the MEN regularization algorithm in the framework of statistical learning theory. We compute the learning rate by estimates of the Hilbert-Schmidt operators. In addition, an adaptive scheme for selecting the regularization parameter is presented. Numerical experiments demonstrate the superiority of the MEN regularization algorithm.

关键词： Approximation error elastic-net regularization matrix recovery sample error singular value shrinkage operator

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elastic-net regularization versus l¹-regularization for linear inverse problems with quasi-sparse solutions

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INVERSE PROBLEMS 2017年第1期33卷 015004-015004页

作者： Chen, De-Han Hofmann, Bernd Zou, Jun Chinese Univ Hong Kong Dept Math Shatin Hong Kong Peoples R China Tech Univ Chemnitz Fak Math D-09107 Chemnitz Germany

y We consider the ill-posed operator equation Ax = y with an injective and bounded linear operator A mapping between l(2) and a Hilbert space Y, possessing the unique solution x={x(k)}(k-1)(infinity). For the cases that sparsity x is an element of l(0) is expected but often slightly violated in practice, we investigate in comparison with the l(1)-regularization the elastic-net regularization, where the penalty is a weighted superposition of the l(1)-norm and the l(2)-norm square, under the assumption that x is an element of l(1). There occur two positive parameters in this approach, the weight parameter. and the regularization parameter as the multiplier of the whole penalty in the Tikhonov functional, whereas only one regularization parameter arises in l(1)-regularization. Based on the variational inequality approach for the description of the solution smoothness with respect to the forward operator A and exploiting the method of approximate source conditions, we present some results to estimate the rate of convergence for the elastic-net regularization. The occurring rate function contains the rate of the decay x(k) -> 0 for k -> infinity and the classical smoothness properties of x as an element in l(2).

关键词： linear ill-posed problems sparsity constraints elastic-net regularization l(1)-regularization convergence rates source conditions

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elastic-net regularization FOR LOW-RANK MATRIX RECOVERY

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INTERNATIONAL JOURNAL OF WAVELETS MULTIRESOLUTION AND INFORMATION PROCESSING 2012年第5期10卷 1250050-1250050页

作者： Li, Hong Chen, Na Li, Luoqing Huazhong Univ Sci & Technol Sch Math & Stat Wuhan 430074 Peoples R China Hubei Univ Fac Math & Comp Sci Wuhan 430062 Peoples R China

This paper considers the problem of recovering a low-rank matrix from a small number of measurements consisting of linear combinations of the matrix entries. We extend the elastic-net regularization in compressive sensing to a more general setting, the matrix recovery setting, and consider the elastic-net regularization scheme for matrix recovery. To investigate on the statistical properties of this scheme and in particular on its convergence properties, we set up a suitable mathematic framework. We characterize some properties of the estimator and construct a natural iterative procedure to compute it. The convergence analysis shows that the sequence of iterates converges, which then underlies successful applications of the matrix elastic-net regularization algorithm. In addition, the error bounds of the proposed algorithm for low-rank matrix and even for full-rank matrix are presented in this paper.

关键词： elastic-net regularization matrix recovery proximity operator singular value shrinkage operator

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Group sparse structure and elastic-net regularization for compressive sensing of pulse-type physiological signals

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BIOMEDICAL SIGNAL PROCESSING AND CONTROL 2020年第0期57卷 101685-101685页

作者： Pant, Jeevan K. Krishnan, Sridhar Ryerson Univ Dept Elect Comp & Biomed Engn Toronto ON Canada

Objective: Group Gini (Gr. Gini) and average-group Gini (AvgGr. Gini) indexes are introduced for the evaluation of group-sparsity of rhythmic pulse-type physiological signals, and a new method for the reconstruction of physiological pulse-type rhythmic signals for compressive sensing (CS) applications is proposed. Methods: Gr. Gini and AvgGr. Gini indexes are developed by generalizing the conventional Gini index, for the evaluation of sparsity of signals with grouped nonzero components. The signal reconstruction CS method is based on applying generalized elastic-net regularized least-squares. The elastic-net function is generalized by including nonconvex lp-pseudonorm, so that its minimization results in enhanced group-sparse structure on the first-order difference (FoD) and second-order difference (SoD) of the signal. It operates by taking advantage of the pre-existing group-sparse structure on the FoD and SoD components of the original signal. This method is effective for reconstructing signals with pulse-type structures by encouraging the grouping of significant signal components. Results: Simulation results demonstrate that the Gr. Gini and AvgGr. Gini indexes can effectively evaluate group sparse-structure. The proposed CS method can offer improvement in signal-to-noise ratio (SNR), structural similarity index measure, Gr. Gini index, and AvgGr. Gini index. It is found to offer improvement of *** index, *** index, and SNR by 0.4%, 0.4%, and 5.29 dB respectively, for electrocardiogram (ECG) signals and by 5.89%, 5.89%, 16 dB, respectively, for foot-gait signals. Conclusion: The proposed CS method is effective for enhancing reconstruction performance of pulse-type physiological signals. Significance: It can be useful for the development of healthcare system for remote and long-term telemonitoring of pulse-type physiological signals. (C) 2019 Published by Elsevier Ltd.

关键词： Pulse-type physiological signals Group-sparsity First-order difference Second-order difference Compressive sensing elastic-net regularization

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