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On a general iteratively reweighted algorithm for solving force reconstruction problems

JOURNAL OF SOUND AND VIBRATION 2019年 458卷 376-388页

作者： Aucejo, M. De Smet, O. Conservatoire Natl Arts & Metiers Struct Mech & Coupled Syst Lab 2 Rue Conte F-75003 Paris France

The multiplicative l(q)-regularization has been recently introduced in structural dynamics for solving force reconstruction problems. Practically, the resolution of this regularization strategy requires the implementation of an iterative procedure. To this end, an iteratively reweighted Least-Squares algorithm has been originally implemented. The core idea of this algorithm is to replace the direct resolution of the inverse problem by an equivalent iterative procedure having an explicit and unique solution at each iteration. However, the exploitation of this very general idea allows defining other iteratively reweighted schemes. The present paper aims at comparing the overall performances of three particular iteratively reweighted algorithms for solving force reconstruction problems derived from a more general and original iterative procedure. The numerical applications proposed in this contribution highlight the ability of the considered algorithms in providing consistent regularized solutions with respect to various parameters such as the measurement noise level or the tolerance chosen to stop the iterative process. (C) 2019 Elsevier Ltd. All rights reserved.

关键词： Linear inverse problem Force reconstruction Multiplicative regularization iteratively reweighted algorithm

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Global convergence of block Bregman proximal iteratively reweighted algorithm with extrapolation

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JOURNAL OF GLOBAL OPTIMIZATION 2024年 1-30页

作者： Zhang, Jie Yang, Xinmin Chongqing Univ Posts & Telecommun Sch Sci Chonngqing 400065 Peoples R China Chongqing Normal Univ Sch Math Sci Chongqing Peoples R China

In this paper, we propose a Bregman proximal iteratively reweighted algorithm with extrapolation based on block coordinate update aimed at solving a class of optimization problems which is the sum of a smooth possibly nonconvex loss function and a general nonconvex regularizer with a separable structure. The proposed algorithm can be used to solve the & ell;(p )(0 < p < 1) regularization problem by employing an update strategy of the smoothing parameter in its smooth approximation model. When the extrapolation parameter satisfies certain conditions, the global convergence and local convergence rate are obtained by using the Kurdyka-& Lstrok;ojasiewicz (KL) property on the objective function. Numerical experiments are given to indicate the superiority of the proposed algorithm.

关键词： iteratively reweighted algorithm Bregman distance Block coordinate update Extrapolation Kurdyka-& Lstrok ojasiewicz property

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On iteratively reweighted algorithms for Nonsmooth Nonconvex Optimization in Computer Vision

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SIAM JOURNAL ON IMAGING SCIENCES 2015年第1期8卷 331-372页

作者： Ochs, Peter Dosovitskiy, Alexey Brox, Thomas Pock, Thomas Univ Freiburg Dept Comp Sci D-79110 Freiburg Germany Univ Freiburg BIOSS Ctr Biol Signalling Studies D-79110 Freiburg Germany Graz Univ Technol Inst Comp Graph & Vis A-8010 Graz Austria AIT Austrian Inst Technol GmbH Digital Safety & Secur Dept A-1220 Vienna Austria

Natural image statistics indicate that we should use nonconvex norms for most regularization tasks in image processing and computer vision. Still, they are rarely used in practice due to the challenge of optimization. Recently, iteratively reweighed l(1) minimization (IRL1) has been proposed as a way to tackle a class of nonconvex functions by solving a sequence of convex l(2)-l(1) problems. We extend the problem class to the sum of a convex function and a (nonconvex) nondecreasing function applied to another convex function. The proposed algorithm sequentially optimizes suitably constructed convex majorizers. Convergence to a critical point is proved when the Kurdyka-Lojasiewicz property and additional mild restrictions hold for the objective function. The efficiency and practical importance of the algorithm are demonstrated in computer vision tasks such as image denoising and optical flow. Most applications seek smooth results with sharp discontinuities. These are achieved by combining nonconvexity with higher order regularization.

关键词： iteratively reweighted algorithm majorization-minimization IRL1 IRLS nonsmooth nonconvex optimization Kurdyka-Lojasiewicz inequality computer vision nonconvex total generalized variation

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An accelerated IRNN-iteratively reweighted Nuclear Norm algorithm for nonconvex nonsmooth low-rank minimization problems

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2021年 396卷 113602-113602页

作者： Phan, Duy Nhat Nguyen, Thuy Ngoc Ho Chi Minh City Univ Educ Dept Math & Informat Ho Chi Minh City Vietnam Carnegie Mellon Univ Dynam Decis Making Lab Pittsburgh PA 15213 USA

Low-rank minimization problems arise in numerous important applications such as recommendation systems, machine learning, network analysis, and so on. The problems however typically consist of minimizing a sum of a smooth function and nonconvex nonsmooth composite functions, which solving them remains a big challenge. In this paper, we take inspiration from the Nesterov's acceleration technique to accelerate an iteratively reweighted nuclear norm algorithm for the considered problems ensuring that every limit point is a critical point. Our algorithm iteratively computes the proximal operator of a reweighted nuclear norm which has a closed-form solution by performing the SVD of a smaller matrix instead of the full SVD. This distinguishes our work from recent accelerated proximal gradient algorithms that require an expensive computation of the proximal operator of nonconvex nonsmooth composite functions. We also investigate the convergence rate with the Kurdyka-Lojasiewicz assumption. Numerical experiments are performed to demonstrate the efficiency of our algorithm and its superiority over well-known methods. (C) 2021 Elsevier B.V. All rights reserved.

关键词： Low-rank minimization iteratively reweighted algorithm Nesterov's acceleration Proximal gradient algorithm Matrix completion Subspace clustering

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Approximate versions of proximal iteratively reweighted algorithms including an extended IP-ICMM for signal and image processing problems

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2020年 376卷 112837-000页

作者： Kang, Myeongmin Chungnam Natl Univ Dept Math Daejeon South Korea

iteratively reweighted algorithms are popular methods for solving nonconvex unconstrained minimization problems. Applications are notably mathematical models in image processing or signal processing. They often have a convex subproblem and do not have closed form solutions in general. In this paper, we propose approximate versions of proximal iteratively reweighted algorithms for nonconvex and nonsmooth unconstrained minimization problems. Specifically, we can achieve an approximate solution for the subproblem by applying a computable inexact stopping rule. The convergence of our method is proved based on an inexact unified framework. Numerical applications for image deblurring or denoising problems validate the effectiveness of the proposed algorithms. (C) 2020 Elsevier B.V. All rights reserved.

关键词： iteratively reweighted algorithm Inexact stopping condition Nonconvex optimization Nonsmooth objective function Kurdyka-Lojasiewicz property Nonconvex regularization

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iteratively Linearized reweighted Alternating Direction Method of Multipliers for a Class of Nonconvex Problems

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2018年第20期66卷 5380-5391页

作者： Sun, Tao Jiang, Hao Cheng, Lizhi Zhu, Wei Natl Univ Def Technol Dept Math Changsha 410073 Hunan Peoples R China Natl Univ Def Technol Coll Comp Changsha 410073 Hunan Peoples R China Natl Univ Def Technol State Key Lab High Performance Computat Changsha 410073 Hunan Peoples R China Xiangtan Univ Sch Math & Computat Sci Hunan Key Lab Computat & Simulat Sci & Engn Xiangtan 411105 Peoples R China

In this paper, we consider solving a class of nonconvex and nonsmooth problems frequently appearing in signal processing andmachine learning research. The traditional alternating directionmethod of multipliers encounters troubles in both mathematics and computations in solving the nonconvex and nonsmooth subproblem. In view of this, we propose a reweighted alternating direction method of multipliers. In this algorithm, all subproblems are convex and easy to solve. We also provide several guarantees for the convergence and prove that the algorithm globally converges to a critical point of an auxiliary function with the help of the Kurdyka-Lojasiewicz property. Several numerical results are presented to demonstrate the efficiency of the proposed algorithm.

关键词： Alternating direction method of multipliers iteratively reweighted algorithm Nonconvex and nonsmooth minimization Kurdyka-Lojasiewicz property Semi-algebraic functions

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iteratively reweighted GROUP LASSO BASED ON LOG-COMPOSITE REGULARIZATION

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2021年第5期43卷 S655-S678页

作者： Ke, Chengyu Ahn, Miju Shin, Sunyoung Lou, Yifei Southern Methodist Univ Dept Engn Management Informat & Syst Dallas TX 75205 USA Univ Texas Dallas Dept Math Sci Richardson TX 75080 USA

The paper considers supervised learning problems of labeled data with grouped input features. The groups are nonoverlapped such that the model coefficients corresponding to the input features form disjoint groups. The coefficients have group sparsity structure in the sense that coefficients corresponding to each group shall be simultaneously either zero or nonzero. To make effective use of such group sparsity structure given a priori, we introduce a novel log-composite regularizer, which can be minimized by an iterative algorithm. In particular, our algorithm iteratively solves for a traditional group least absolute shrinkage and selection operator (LASSO) problem that involves summing up the l(2) norm of each group until convergent. By updating group weights, our approach enforces a group of smaller coefficients from the previous iterate to be more likely to set to zero compared to the group LASSO. Theoretical results include a minimizing property of the proposed model as well as the convergence of the iterative algorithm to a stationary solution under mild conditions. We conduct extensive experiments on synthetic and real datasets, indicating that our method yields a performance that is superior to that of the state-of-the-art methods in linear regression and binary classification.

关键词： group sparsity nonconvex regularization iteratively reweighted algorithm directional stationarity variable selection

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Convergence rate analysis of proximal iteratively reweighted l1 methods for lp regularization problems

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OPTIMIZATION LETTERS 2023年第2期17卷 413-435页

作者： Wang, Hao Zeng, Hao Wang, Jiashan ShanghaiTech Univ Sch Informat Sci & Technol Shanghai Peoples R China Univ Washington Dept Math Seattle WA 98195 USA

In this paper, we focus on the local convergence rate analysis of the proximal iteratively reweighted l(1) algorithms for solving l(p) regularization problems, which are widely applied for inducing sparse solutions. We show that if the KurdykaLojasiewicz property is satisfied, the algorithm converges to a unique first-order stationary point;furthermore, the algorithm has local linear convergence or local sublinear convergence. The theoretical results we derived are much stronger than the existing results for iteratively reweighted l(1) algorithms.

关键词： Kurdyka-Lojasiewicz property iteratively reweighted algorithm l(p) regularization Convergence rate

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Total Generalized Variation Based Denoising Models for Ultrasound Images

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JOURNAL OF SCIENTIFIC COMPUTING 2017年第1期72卷 172-197页

作者： Kang, Myeongmin Kang, Myungjoo Jung, Miyoun Seoul Natl Univ Dept Math Sci Seoul South Korea Hankuk Univ Foreign Studies Dept Math Yongin South Korea

In this paper, we introduce a class of variational models for the restoration of ultrasound images corrupted by noise. The proposed models involve the convex or nonconvex total generalized variation regularization. The total generalized variation regularization ameliorates the staircasing artifacts that appear in the restored images of total variation based models. Incorporating total generalized variation regularization with nonconvexity helps preserve edges in the restored images. To realize the proposed convex model, we adopt the alternating direction method of multipliers, and the iteratively reweighted algorithm is employed to handle the nonconvex model. These methods result in fast and efficient optimization algorithms for solving our models. Numerical experiments demonstrate that the proposed models are superior to other state-of-the-art models.

关键词： Ultrasound image denoising Total generalized variation Nonconvex regularization Alternating direction method of multipliers iteratively reweighted algorithm

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An enhanced sparse regularization method for impact force identification

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MECHANICAL SYSTEMS AND SIGNAL PROCESSING 2019年 126卷 341-367页

作者： Qiao, Baijie Liu, Junjiang Liu, Jinxin Yang, Zhibo Chen, Xuefeng State Key Lab Mfg Syst Engn Xian 710061 Shaanxi Peoples R China Xi An Jiao Tong Univ Sch Mech Engn Xian 710049 Shaanxi Peoples R China

The standard sparse regularization method based on l(1)-norm minimization for impact force identification has already proved to be an interesting alternative to the classical regularization method based on l(2)-norm minimization. However, choosing the l(1)-norm as a convex relaxation of the l(0)-norm, the corresponding sparse regularization model generally offers a sparse but underestimated solution. In this paper, considering the sparsity of impact force, an enhanced sparse regularization method based on reweighted l(1)-norm minimization is developed for reducing the peak force error and improving the identification accuracy of impact force. First, a weighted l(1)-norm convex optimization model is presented to overcome the ill-posed nature of the inverse problem of impact force identification. Second, to solve such a regularized model efficiently, an iteratively reweighted l(1)-norm minimization algorithm is introduced, where the weights are adaptively updated from the previous solution. The application of the iteratively reweighted scheme is to overcome the mismatch between l(1)-norm minimization and l(0)-norm minimization, while keeping the enhanced sparse regularization problem solvable and convex. Finally, numerical simulation and experimental verification including the single and double impact force identification on a plate structure are presented to illustrate the superior performance of the enhanced sparse regularization method compared to classical regularization approaches. Effects of reweighting iteration number, tuning parameters, initial conditions and response locations are successfully investigated in detail. Results demonstrate that compared with the standard l(1)-norm regularization method and the classical l(2)-norm regularization method, the enhanced sparse regularization method based on reweighted l(1)-norm minimization whose solution is much sparser, can greatly improve the identification accuracy of impact force. Moreover, the proposed method

关键词： Impact force identification Enhanced sparse regularization Weighted l(1)-norm minimization iteratively reweighted algorithm

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