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iterative mix thresholding algorithm with continuation technique for mix sparse optimization and application

JOURNAL OF GLOBAL OPTIMIZATION 2025年第3期91卷 511-534页

作者： Hu, Yaohua Lu, Jian Yang, Xiaoqi Zhang, Kai Shenzhen Univ Sch Math Sci Shenzhen 518060 Peoples R China Hong Kong Polytech Univ Dept Appl Math Kowloon Hong Kong Peoples R China Shenzhen Univ Shenzhen Audencia Financial Technol Inst Shenzhen 518060 Guangdong Peoples R China Guangdong Univ Finance & Econ Sch Finance Guangzhou 510320 Guangdong Peoples R China

Mix sparse structure is inherited in a wide class of practical applications, namely, the sparse structure appears as the inter-group and intra-group manners simultaneously. In this paper, we propose an iterative mix thresholding algorithm with continuation technique (IMTC) to solve the & ell;(0) regularized mix sparse optimization. The significant advantage of the IMTC is that it has a closed-form expression and low storage requirement, and it is able to promote the mix sparse structure of the solution. We prove the convergence property and the linear convergence rate of the ITMC to a local minimum;moreover, we show that the ITMC approaches an approximate true mix sparse solution within a tolerance relevant to the noise level under an assumption of restricted isometry property. We also apply the mix sparse optimization to model the differential optical absorption spectroscopy analysis with the wavelength misalignment, and numerical results indicate that the IMTC can exactly and quantitatively predict the existing materials and the factual wavelength misalignment simultaneously within 0.1 s, which meets the demand of improvement of the automatic analysis software.

关键词： Mix sparse optimization & ell (0) Regularization iterative thresholding algorithm Continuation technique Convergence theory

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GAITA: A Gauss-Seidel iterative thresholding algorithm for l_q regularized least squares regression

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2017年 319卷 220-235页

作者： Zeng, Jinshan Peng, Zhiming Lin, Shaobo Jiangxi Normal Univ Coll Comp Informat Engn Nanchang 330022 Jiangxi Peoples R China Univ Calif Los Angeles Dept Math Los Angeles CA 90095 USA Wenzhou Univ Coll Math & Informat Sci Wezhou 325035 Peoples R China

This paper studies the l(q)(0 < q < 1) regularized least squares regression (l(q)LS) problem, which arises in many applications of signal processing and machine learning. The iterative thresholding algorithm is an important algorithm for solving the l(q)LS problem, and can be viewed as a Jacobi-type iterative method. This paper proposes a Gauss-Seidel version of iterative thresholding algorithm called GAITA for solving the l(q)LS problem. Under certain conditions, we establish its global convergence(1), eventual linear rate, and the convergence to a local minimizer. Compared to the Jacobi counterpart, the proposed algorithm can allow larger step sizes and converge much faster. The effectiveness of the proposed algorithm is justified with numerical experiments on both synthetic data and real data. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Gauss-Seidel Jacobi iterative thresholding algorithm l(q) regularized least squares

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L_2/3 regularization: Convergence of iterative thresholding algorithm

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JOURNAL OF VISUAL COMMUNICATION AND IMAGE REPRESENTATION 2015年第Nov.期33卷 350-357页

作者： Zhang, Yong Ye, Wanzhou Shanghai Univ Coll Sci Dept Math Shanghai 200444 Peoples R China

The L-2/3 regularization is a nonconvex and nonsmooth optimization problem. Cao et al. (2013) investigated that the L-2/3 regularization is more effective in imaging deconvolution. The convergence issue of the iterative thresholding algorithm of L-2/3 regularization problem (the L-2/3 algorithm) hasn't been addressed in Cao et al. (2013). In this paper, we study the convergence of the L-2/3 algorithm. As the main result, we show that under certain conditions, the sequence {X-(n)} generated by the L-2/3 algorithm converges to a local minimizer of L-2/3 regularization, and its asymptotical convergence rate is linear. We provide a set of experiments to verify our theoretical assertions and show the performance of the algorithm on sparse signal recovery. The established results provide a theoretical guarantee for a wide range of applications of the algorithm. (C) 2015 Elsevier Inc. All rights reserved.

关键词： L-1/2 regularization L-2/3 regularization iterative thresholding algorithm thresholding formula Convergence Sparse signal recovery Asymptotical convergence rate Local minimizer

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L_q regularisation-based synthetic aperture radar image feature enhancement via iterative thresholding algorithm

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ELECTRONICS LETTERS 2016年第15期52卷 1336-1337页

作者： Bi, Hui Zhang, Bingchen Wang, Zhengdao Hong, Wen Chinese Acad Sci Inst Elect Sci & Technol Microwave Imaging Lab Beijing Peoples R China Univ Chinese Acad Sci Beijing Peoples R China Iowa State Univ Dept Elect & Comp Engn Ames IA USA

A novel L-q(0 < q <= 1) regularisation-based synthetic aperture radar (SAR) image feature enhancement method via SAR complex image data is presented. Compared with conventional L-q regularisation-based compressive sensing imaging technique with full-sampling raw data, the proposed method can achieve the same effect of image feature enhancement while enjoying much less computational cost. Through simulations and experiments, the superiority of the proposed method is demonstrated in terms of sidelobe, azimuth ambiguity, clutter suppression and super-resolution.

关键词： image enhancement image resolution iterative methods radar clutter radar imaging synthetic aperture radar Lq regularisation SAR complex image data SAR image feature enhancement azimuth ambiguity clutter suppression compressive sensing imaging technique full-sampling raw data iterative thresholding algorithm superresolution synthetic aperture radar image

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Optimality condition and iterative thresholding algorithm for l_p-regularization problems

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SPRINGERPLUS 2016年第1期5卷 1873页

作者： Jiao, Hongwei Chen, Yongqiang Yin, Jingben Henan Inst Sci & Technol Sch Math Sci Xinxiang 453003 Peoples R China Henan Normal Univ Coll Math & Informat Sci Xinxiang 453007 Peoples R China

This paper investigates the l(p)-regularization problems, which has a broad applications in compressive sensing, variable selection problems and sparse least squares fitting for high dimensional data. We derive the exact lower bounds for the absolute value of nonzero entries in each global optimal solution of the model, which clearly demonstrates the relation between the sparsity of the optimum solution and the choice of the regularization parameter and norm. We also establish the necessary condition for global optimum solutions of lp-regularization problems, i.e., the global optimum solutions are fixed points of a vector thresholding operator. In addition, by selecting parameters carefully, a global minimizer which will have certain desired sparsity can be obtained. Finally, an iterative thresholding algorithm is designed for solving the lp-regularization problems, and any accumulation point of the sequence generated by the designed algorithm is convergent to a fixed point of the vector thresholding operator.

关键词： Optimality condition lp-regularization problems iterative thresholding algorithm Global optimum solution Fixed point

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Suppression of Ghost Targets in Focusing Azimuth Periodically Gapped SAR Raw Data with Complex iterative thresholding algorithm 11

Suppression of Ghost Targets in Focusing Azimuth Periodicall...

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IEEE 11th Sensor Array and Multichannel Signal Processing Workshop (SAM)

作者： Qian, Yulei Zhu, Daiyin Nanjing Univ Aeronaut & Astronaut Key Lab Radar Imaging & Microwave Photon Minist Educ Coll Elect Informat Engn Nanjing Peoples R China

ISBN: (纸本)9781728119465

An algorithm is presented in this paper to focus the azimuth periodically gapped SAR (Synthetic Aperture Radar) raw data. The proposed algorithm mainly contains phase multiplication in range frequency domain and sparse reconstruction in range Doppler domain. The phase multiplication in range frequency domain aims to acquire sparser data in range Doppler domain. Then, the complex iterative thresholding algorithm is utilized to reconstruct the complete SAR data in range Doppler domain. The iterative thresholding algorithm is extended to cope with the complex SAR data. Afterwards, the traditional SAR focusing methods are capable of obtaining image from the recovered data. The proposed method performs well on suppressing ghost targets induced by gapping. Point target simulation is implemented to assess the validity of the proposed method. In addition, real SAR data experiment is also utilized to demonstrate the effectiveness of the proposed method.

关键词： Synthetic Aperture Radar Gapping Sparse Reconstruction iterative thresholding algorithm

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A NOVEL iterative thresholding algorithm FOR ARCTANGENT REGULARIZATION PROBLEM 49

A NOVEL ITERATIVE THRESHOLDING ALGORITHM FOR ARCTANGENT REGU...

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49th IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP)

作者： He, Zihao Shu, Qianyu Wen, Jinming So, Hing Cheung Jinan Univ Dept Informat Sci & Technol Guangzhou Peoples R China Sichuan Normal Univ Sch Math Sci Chengdu Peoples R China Pazhou Lab Guangzhou Peoples R China City Univ Hong Kong Dept Elect Engn Hong Kong Peoples R China

ISBN: (纸本)9798350344868;9798350344851

In this work, we derive the proximity operator of an arctangent penalty, which is expressed using hyperbolic functions of sine and cosine. This penalty is then applied to sparse signal recovery, and an efficient arctangent regularization iterative thresholding (ARIT) algorithm is proposed, offering closed-form solutions for the subproblems associated with the arctangent penalty. Extensive experiments are conducted to compare the performance of ARIT with several existing iterative thresholding algorithms, and the results demonstrate that our algorithm achieves the best overall performance in terms of the probability of successful recovery, phase transition and running time.

关键词： Compressed sensing iterative thresholding algorithm arctangent regularization

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Sparse Regularization: Convergence Of iterative Jumping thresholding algorithm

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2016年第19期64卷 5106-5118页

作者： Zeng, Jinshan Lin, Shaobo Xu, Zongben Jiangxi Normal Univ Coll Comp Informat Engn Nanchang 330022 Peoples R China Wenzhou Univ Coll Math & Informat Sci Wenzhou 325035 Peoples R China Xi An Jiao Tong Univ Sch Math & Stat Xian 710049 Peoples R China

In recent studies on sparse modeling, nonconvex penalties have received considerable attentions due to their superiorities on sparsity-inducing over the convex counterparts. In this paper, we study the convergence of a nonconvex iterative thresholding algorithm for solving a class of sparse regularized optimization problems, where the corresponding thresholding functions of the penalties are discontinuous with jump discontinuities. Therefore, we call the algorithm the iterative jumping thresholding (IJT) algorithm. The finite support and sign convergence of IJT algorithm is first verified via taking advantage of such jump discontinuity. Together with the introduced restricted Kurdyka-Lojasiewicz property, then the global convergence(1) of the entire sequence can be further proved. Furthermore, we can show that the IJT algorithm converges to a strictly local minimizer at an eventual linear rate(2) under some additional conditions. Moreover, we derive a posteriori computable error estimate, which can be used to design an efficient terminate rule. It should be pointed out that the l(q) quasinorm (0 < q < 1) is an important subclass of the nonconvex penalties studied in this paper. In particular, when applied to the l(q) regularization, IJT algorithm can converge to a local minimizer with an eventual linear rate under certain concentration conditions. We also apply the proposed algorithm to sparse signal recovery and synthetic aperture radar imaging problems. The experiment results show the effectiveness of the proposed algorithm.

关键词： Sparse regularization non-convex optimization iterative thresholding algorithm l(q) regularization (0 < q < 1) Kurdyka-Lojasiewicz inequality

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On convergence of iterative thresholding algorithms to approximate sparse solution for composite nonconvex optimization

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MATHEMATICAL PROGRAMMING 2025年第1-2期211卷 181-206页

作者： Hu, Yaohua Hu, Xinlin Yang, Xiaoqi Shenzhen Univ Sch Math Sci Shenzhen 518060 Peoples R China Shenzhen Univ WeBank Inst Fintech Shenzhen Audencia Business Sch Shenzhen 518060 Peoples R China Hong Kong Polytech Univ Dept Appl Math Kowloon Hong Kong Peoples R China

This paper aims to find an approximate true sparse solution of an underdetermined linear system. For this purpose, we propose two types of iterative thresholding algorithms with the continuation technique and the truncation technique respectively. We introduce a notion of limited shrinkage thresholding operator and apply it, together with the restricted isometry property, to show that the proposed algorithms converge to an approximate true sparse solution within a tolerance relevant to the noise level and the limited shrinkage magnitude. Applying the obtained results to nonconvex regularization problems with SCAD, MCP and lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _p$$\end{document} penalty (0 <= p <= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le p \le 1$$\end{document}) and utilizing the recovery bound theory, we establish the convergence of their proximal gradient algorithms to an approximate global solution of nonconvex regularization problems. The established results include the existing convergence theory for l1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _1$$\end{document} or l0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _0$$\end{document} regularization problems for finding a true sparse solution as special c

关键词： Nonconvex sparse optimization iterative thresholding algorithm Proximal gradient algorithm Sparse solution Global solution

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A lifting-wavelet-based iterative thresholding correction for atomic force microscopy images with vertical distortion

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NANOTECHNOLOGY AND PRECISION ENGINEERING 2025年第3期8卷

作者： Bai, Yifan Wu, Yinan Fang, Yongchun Nankai Univ Inst Robot & Automat Informat Syst Coll Artificial Intelligence Tianjin 300350 Peoples R China Tianjin Key Lab Intelligent Robot Tianjin 300350 Peoples R China

To eliminate distortion caused by vertical drift and illusory slopes in atomic force microscopy (AFM) imaging, a lifting-wavelet-based iterative thresholding correction method is proposed in this paper. This method achieves high-quality AFM imaging via line-by-line corrections for each distorted profile along the fast axis. The key to this line-by-line correction is to accurately simulate the profile distortion of each scanning row. Therefore, a data preprocessing approach is first developed to roughly filter out most of the height data that impairs the accuracy of distortion modeling. This process is implemented through an internal double-screening mechanism. A line-fitting method is adopted to preliminarily screen out the obvious specimens. Lifting wavelet analysis is then carried out to identify the base parts that are mistakenly filtered out as specimens so as to preserve most of the base profiles and provide a good basis for further distortion modeling. Next, an iterative thresholding algorithm is developed to precisely simulate the profile distortion. By utilizing the roughly screened base profile, the optimal threshold, which is used to screen out the pure bases suitable for distortion modeling, is determined through iteration with a specified error rule. On this basis, the profile distortion is accurately modeled through line fitting on the finely screened base data, and the correction is implemented by subtracting the modeling result from the distorted profile. Finally, the effectiveness of the proposed method is verified through experiments and applications.

关键词： Atomic force microscopy lifting wavelet analysis iterative thresholding algorithm vertical distortion

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