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A novel dictionary learning method for sparse representation with nonconvex regularizations

NEUROCOMPUTING 2020年 417卷 128-141页

作者： Tan, Benying Li, Yujie Zhao, Haoli Li, Xiang Ding, Shuxue Guilin Univ Elect Technol Sch Artificial Intelligence Guilin Peoples R China Univ Aizu Grad Sch Comp Sci & Engn Aizu Wakamatsu Fukushima Japan Guilin Univ Elect Technol Sch Artificial Intelligence Guangxi Key Lab Wireless Broadband Commun & Signa Guilin 541004 Peoples R China Natl Inst Adv Ind Sci & Technol Tsukuba Ibaraki 3058560 Japan Sun Yat Sen Univ Sch Data & Comp Sci Guangzhou Guangdong Peoples R China

In dictionary learning, sparse regularization is used to promote sparsity and has played a major role in the developing of dictionary learning algorithms. l(1)-norm is of the most popular sparse regularization due to its convexity and the related tractable convex optimization problems. However, l(1)-norm leads to biased solutions and provides inferior performance on certain applications compared with nonconvex sparse regularizations. In this work, we propose a generalized minimax-concave (GMC) sparse regularization, which is nonconvex, to promote sparsity to design dictionary learning model. Applying the alternate optimization scheme, we use the forward-backward splitting (FBS) algorithm to solve the sparse coding problem. As the improvement, we incorporate Nesterov's acceleration technique and adaptive threshold scheme into the FBS algorithm to improve the convergence efficiency and performance. In the dictionary update step, we apply the difference of convex functions (DC) programming and the DC algorithm (DCA) to address the dictionary update. Two dictionary update algorithms are designed;one updates the dictionary atoms one by one, and the other one updates the dictionary atoms simultaneously. The presented dictionary learning algorithms perform robustly in dictionary recovery. Numerical experiments are designed to verify the performance of proposed algorithms and to compare with the state-of-the-art algorithms. (C) 2020 Published by Elsevier B.V.

关键词： Dictionary learning Nonconvex GMC regularization DC programming and DCA forward-backward splitting algorithm

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PROXIMAL THRESHOLDING algorithm FOR MINIMIZATION OVER ORTHONORMAL BASES

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SIAM JOURNAL ON OPTIMIZATION 2008年第4期18卷 1351-1376页

作者： Combettes, Patrick L. Pesquet, Jean-Christophe Univ Paris 06 Fac Math Lab Jacques Louis Lions UMR CNRS 7598 F-75005 Paris France Univ Marne la Vallee Inst Gaspard Monge F-77454 Marne La Vallee 2 France Univ Marne la Vallee UMR CNRS 8049 F-77454 Marne La Vallee 2 France

The notion of soft thresholding plays a central role in problems from various areas of applied mathematics, in which the ideal solution is known to possess a sparse decomposition in some orthonormal basis. Using convex-analytical tools, we extend this notion to that of proximal thresholding and investigate its properties, providing, in particular, several characterizations of such thresholders. We then propose a versatile convex variational formulation for optimization over orthonormal bases that covers a wide range of problems, and we establish the strong convergence of a proximal thresholding algorithm to solve it. Numerical applications to signal recovery are demonstrated.

关键词： convex programming deconvolution denoising forward-backward splitting algorithm Hilbert space orthonormal basis proximal algorithm proximal thresholding proximity operator signal recovery soft thresholding strong convergence

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New analysis of linear convergence of gradient-type methods via unifying error bound conditions

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MATHEMATICAL PROGRAMMING 2020年第1-2期180卷 371-416页

作者： Zhang, Hui Natl Univ Def Technol Dept Math Changsha 410073 Hunan Peoples R China

This paper reveals that a common and central role, played in many error bound (EB) conditions and a variety of gradient-type methods, is a residual measure operator. On one hand, by linking this operator with other optimality measures, we define a group of abstract EB conditions, and then analyze the interplay between them;on the other hand, by using this operator as an ascent direction, we propose an abstract gradient-type method, and then derive EB conditions that are necessary and sufficient for its linear convergence. The former provides a unified framework that not only allows us to find new connections between many existing EB conditions, but also paves a way to construct new ones. The latter allows us to claim the weakest conditions guaranteeing linear convergence for a number of fundamental algorithms, including the gradient method, the proximal point algorithm, and the forward-backward splitting algorithm. In addition, we show linear convergence for the proximal alternating linearized minimization algorithm under a group of equivalent EB conditions, which are strictly weaker than the traditional strongly convex condition. Moreover, by defining a new EB condition, we show Q-linear convergence of Nesterov's accelerated forward-backward algorithm without strong convexity. Finally, we verify EB conditions for a class of dual objective functions.

关键词： Residual measure operator Error bound Gradient descent Linear convergence Proximal point algorithm forward-backward splitting algorithm Proximal alternating linearized minimization Nesterov's acceleration Dual objective function

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Elastic-Net Regularization: Iterative algorithms and Asymptotic Behavior of Solutions

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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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Convergence of Stochastic Proximal Gradient algorithm

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APPLIED MATHEMATICS AND OPTIMIZATION 2020年第3期82卷 891-917页

作者： Rosasco, Lorenzo Villa, Silvia Vu, Bing Cong Univ Genoa DIBRIS Via Dodecaneso 35 I-16146 Genoa Italy Ist Italian Tecnol LCSL Bldg 46-515577 Massachusetts Ave Cambridge MA 02139 USA Univ Genoa Dipartimento Matemat Via Dodecaneso 35 I-16146 Genoa Italy EPFL STI IEL LIONS ELD 243 Batiment EL CH-1015 Lausanne Switzerland

We study the extension of the proximal gradient algorithm where only a stochastic gradient estimate is available and a relaxation step is allowed. We establish convergence rates for function values in the convex case, as well as almost sure convergence and convergence rates for the iterates under further convexity assumptions. Our analysis avoid averaging the iterates and error summability assumptions which might not be satisfied in applications, e.g. in machine learning. Our proofing technique extends classical ideas from the analysis of deterministic proximal gradient algorithms.

关键词： Proximal methods forward-backward splitting algorithm Stochastic optimization

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On the optimal relaxation parameters of Krasnosel'ski-Mann iteration

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OPTIMIZATION 2021年第9期70卷 1959-1986页

作者： He, Songnian Dong, Qiao-Li Tian, Hanlin Li, Xiao-Huan Civil Aviat Univ China Tianjin Key Lab Adv Signal Proc Tianjin Peoples R China Civil Aviat Univ China Coll Sci Tianjin Peoples R China

This paper is devoted to the optimal selection of the relaxation parameter sequence for Krasnosel'ski-Mann iteration. Firstly, we establish the optimal relaxation parameter sequence of the Krasnosel'ski-Mann iteration, with which the algorithm is proved to achieve the optimal convergence rate. Then we present an approximation to the optimal relaxation parameter sequence since the optimal relaxation parameter sequence involves fixed points of the operators and can't be used in actual computing. Thirdly, we apply our results to the operators splitting method, such as forward-backward splitting methods and Douglas-Rachford splitting method. Finally, numerical experiments are provided to illustrate that Krasnosel'ski-Mann iteration and the relaxed projected method with our proposed relaxation parameter sequence behave better than others.

关键词： Krasnosel'ski-Mann iteration nonexpansive mapping averaged mapping Douglas-Rachford splitting algorithm optimal relaxation parameter forward-backward splitting algorithm

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Convergence rate analysis of proximal gradient methods with applications to composite minimization problems

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OPTIMIZATION 2021年第1期70卷 75-100页

作者： Sahu, D. R. Yao, J. C. Verma, M. Shukla, K. K. Banaras Hindu Univ Dept Math Varanasi Uttar Pradesh India China Med Univ China Med Univ Hosp Res Ctr Interneural Comp Taichung Taiwan Inst Res & Dev Banking Technol Hyderabad India Indian Inst Technol BHU Dept Comp Sci & Engn Varanasi Uttar Pradesh India

First-order methods such as proximal gradient, which use forward-backward splitting techniques have proved to be very effective in solving nonsmooth convex minimization problem, which is useful in solving various practical problems in different fields such as machine learning and image processing. In this paper, we propose few new forward-backward splitting algorithms, which consume less number of iterations to converge to an optimum. In addition, we derive convergence rates for the proposed formulations and show that the speed of convergence of these algorithms is significantly better than the traditional forward-backward algorithm. To demonstrate the practical applicability, we apply them to two real-world problems of machine learning and image processing. The first issue deals with the regression on high-dimensional datasets, whereas the second one is the image deblurring problem. Numerical experiments have been conducted on several publicly available real datasets to verify the obtained theoretical results. Results demonstrate the superiority of our algorithms in terms of accuracy, the number of iterations required to converge and the rate of convergence against the classical first-order methods.

关键词： Accretive operator convex minimization problem forward-backward splitting algorithm Mann iteration method S-iteration method

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L1 Norm Minimization approach to MIMO detector 8

L1 Norm Minimization approach to MIMO detector

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8th International Conference on Signal Processing and Communication Systems, (ICSPCS)

作者： Hashimoto, Yo Konishi, Katsumi Takahashi, Tomohiro Uruma, Kazunori Furukawa, Toshihiro Tokyo Univ Sci Tokyo 162 Japan Kogakuin Univ Tokyo 160 Japan

ISBN: (纸本)9781479952557

This paper deals with the quadrature amplitude modulation (QAM) problem for the multiple-input multiple-output (MIMO) channel. Based on the maximum likelihood estimation, the QAM detection problem is formulated as an integer quadratic programming, which is a combinatorial problem and difficult to obtain exact solutions. In order to overcome combinatorial difficulties, this paper formulates the QAM detection problem as the l(o) norm minimization problem and relaxes it into a quadratic programming with the l(1) norm regularization. Utilizing and modifying the forward-backward splitting (FOBOS) algorithm, a new QAM detection algorithm is proposed. This algorithm has a trade-off between the computational cost and the detection accuracy, which depends on a parameter of the algorithm. Numerical simulations show that the proposed algorithm works well and achieves a good detection performance with less computational cost comparing with the semidefinite relaxation (SDR) based algorithm.

关键词： MIMO communication integer programming minimisation numerical analysis quadratic programming quadrature amplitude modulation FOBOS algorithm L1 norm minimization approach MIMO detector QAM detection algorithm SDR based algorithm combinatorial problem computational cost detection accuracy forward-backward splitting algorithm integer quadratic programming maximum likelihood estimation multiple-input multiple-output channel numerical simulations semidefinite relaxation based algorithm Approximation algorithms Detectors MIMO Quadrature amplitude modulation Signal processing algorithms Signal to noise ratio Vectors Quadrature amplitude modulation quadratic programming multiple input multiple output systems maximum likelihood estimation Signal processing algorithms signal to noise ratios MIMO communication Approximation algorithms Integer Programming Numerical Analysis

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