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Multi-step fixed-point proximity algorithms for solving a class of optimization problems arising from image processing

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2015年第2期41卷 387-422页

作者： Li, Qia Shen, Lixin Xu, Yuesheng Zhang, Na Sun Yat Sen Univ Sch Math & Computat Sci Guangdong Prov Key Lab Computat Sci Guangzhou 510275 Guangdong Peoples R China Syracuse Univ Dept Math Syracuse NY 13244 USA Sun Yat Sen Univ Sch Math & Computat Sci Guangzhou 510275 Guangdong Peoples R China

We introduce in this paper a class of multi-step fixed-point proximity algorithms for solving optimization problems in the context of image processing. The objective functions of such optimization problems are the sum of two convex functions having one composed with an affine transformation which is often the regularization term. We are particularly interested in the scenario when the convex functions involved in the objective function have low regularity (not differentiable) since many practical problems encountered in image processing have this nature. We characterize the solutions of the optimization problem as fixed-points of a mapping defined in terms of the proximity operators of the two convex functions. The algorithmic and mathematical challenges come from the fact that the mapping is a composition of a firmly non-expansive operator with an expansive affine transform. A class of multi-step iterative schemes are developed based on the fixed-point equations that characterize the solutions. For the purpose of studying convergence of the proposed algorithms, we introduce a notion of weakly firmly non-expansive mappings and establish under certain conditions that the sequence generated from a weakly firmly non-expansive mapping is convergent. We use this general convergence result to conclude that the proposed multi-step algorithms converge. We in particular design a class of two-step algorithms for solving the optimization problem which include several existing algorithms as special examples and at the same time offer novel algorithms. Moreover, we identify the well-known alternating split Bregman iteration method as a special case of the proposed algorithm and modify it to improve its convergence result. A class of two-step algorithms for the total variation based image restoration models are presented.

关键词： fixed-point algorithm Proximity operator Image processing Convex optimization

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LINEAR CONVERGENCE OF STOCHASTIC BLOCK-COORDINATE fixed point algorithmS 26

LINEAR CONVERGENCE OF STOCHASTIC BLOCK-COORDINATE FIXED POIN...

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26th European Signal Processing Conference (EUSIPCO)

作者： Combettes, Patrick L. Pesquet, Jean-Christophe North Carolina State Univ Dept Math Box 8205 Raleigh NC 27695 USA Univ Paris Saclay Cent Supelec INRIA Ctr Visual Comp Gif Sur Yvette France

ISBN: (纸本)9789082797015

Recent random block-coordinate fixed point algorithms are particularly well suited to large-scale optimization in signal and image processing. These algorithms feature random sweeping rules to select arbitrarily the blocks of variables that are activated over the course of the iterations and they allow for stochastic errors in the evaluation of the operators. The present paper provides new linear convergence results. These convergence rates are compared to those of standard deterministic algorithms both theoretically and experimentally in an image recovery problem.

关键词： Block-coordinate algorithm fixed-point algorithm linear convergence stochastic algorithm

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Applications of accelerated computational methods for quasi-nonexpansive operators to optimization problems

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SOFT COMPUTING 2020年第23期24卷 17887-17911页

作者： Sahu, D. R. Banaras Hindu Univ Dept Math Varanasi 221005 Uttar Pradesh India

This paper studies the convergence rates of two accelerated computational methods without assuming nonexpansivity of the underlying operators with convex and affine domains in infinite-dimensional Hilbert spaces. One method is a noninertial method, and its convergence rate is estimated as R-T,R-{xn}(n) = o(1/root n) in worst case. The other is an inertial method, and its convergence rate is estimated as R-T,R-{yn}(n) = o(1/root n) under practical conditions. Then, we apply our results to give newresults on convergence rates for solving generalized split common fixed-point problems for the class of demimetric operators. We also apply our results to variational inclusion problems and convex optimization problems. Our results significantly improve and/or develop previously discussed fixed-point problems and splitting problems and related algorithms. To demonstrate the applicability of our methods, we provide numerical examples for comparisons and numerical experiments on regression problems for publicly available high-dimensional real datasets taken from different application domains.

关键词： Convex optimization fixed-point set Nonexpansive Proximal gradient method fixed-point algorithm S-iteration method Maximal monotone operator Lasso problem

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Facial expression recognition by supervised independent component analysis using MAP estimation

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IEICE TRANSACTIONS ON INFORMATION AND SYSTEMS 2008年第2期E91D卷 341-350页

作者： Chen, Fan Kotani, Kazunori Japan Adv Inst Sci & Technol Sch Informat Sci Ishikawa 9231292 Japan

Permutation ambiguity of the classical Independent Component Analysis (ICA) may cause problems in feature extraction for pattern classification. Especially when only a small subset of components is derived from data, these components may not be most distinctive for classification, because ICA is an unsupervised method. We include a selective prior for de-mixing coefficients into the classical ICA to alleviate the problem. Since the prior is constructed upon the classification information from the training data, we refer to the proposed ICA model with a selective prior as a supervised ICA (sICA). We formulated the learning rule for sICA by taking a Maximum a Posteriori (MAP) scheme and further derived a fixed point algorithm for learning the de-mixing matrix. We investigate the performance of sICA in facial expression recognition from the aspects of both correct rate of recognition and robustness even with few independent components.

关键词： facial expression recognition supervised independent component analysis fixed-point algorithm

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A fast blind source separation algorithm based on the temporal structure of signals

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NEUROCOMPUTING 2014年 139卷 261-271页

作者： Zhang, Hongjuan Wang, Guinan Cai, Pingmei Wu, Zikai Ding, Shuxue Shanghai Univ Dept Math Shanghai 200444 Peoples R China Shanghai Univ Sci & Technol Sch Business Shanghai 200093 Peoples R China Univ Aizu Sch Comp Sci & Engn Aizu Wakamatsu Fukushima 9658580 Japan

Classical independent component analysis (ICA) has been reasonably successful;however, the performance and the convergence of the conventional ICA algorithms have reached limitations of further improvement since they utilize only the statistical independency among the sources. For circumventing this situation, in this paper, we incorporate some other kinds of temporal priori information, i.e., the generalized autocorrelation and the nonlinear predictability of each source, and make a convex combination of them to formulate a novel cost function for blind source separation (BSS). With this cost function, a fixed-point BSS algorithm is developed. This algorithm inherits the advantages of the well-known FastICA algorithm of ICA, which converges fast and does not need to choose any learning step sizes. Its higher separation accuracy is verified by numerical experiments. Meanwhile, we also give the consistency analysis and prove convergence properties of the algorithm, which has a (locally) consistent estimator and at least quadratic convergence. (C) 2014 Elsevier B.V. All rights reserved.

关键词： Blind sources separation Nonlinear autocorrelation Nonlinear predictability fixed-point algorithm

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Spatio-temporal FastICA algorithms for the blind separation of convolutive mixtures

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IEEE TRANSACTIONS ON AUDIO SPEECH AND LANGUAGE PROCESSING 2007年第5期15卷 1511-1520页

作者： Douglas, Scott C. Gupta, Malay Sawada, Hiroshi Makino, Shoji So Methodist Univ Dept Elect Engn Dallas TX 75275 USA NTT Corp NTT Commun Sci Labs Kyoto 6190237 Japan

This paper derives two spatio-temporal extensions of the well-known FastICA algorithm of Hyvarinen and Oja that are applicable to the convolutive blind source separation task. Our time-domain algorithms combine multichannel spatio-temporal prewhitening via multistage least-squares linear prediction with novel adaptive procedures that impose paraunitary, constraints on the multichannel separation filter. The techniques converge quickly to a separation solution without any step size selection or divergence difficulties, and unlike other methods, ours do not require special coefficient initialization procedures to obtain good separation performance. They also allow for the efficient reconstruction of individual signals as observed in the sensor measurements directly from the system parameters for single-input multiple-output blind source separation tasks. An analysis of one of the adaptive constraint procedures shows its fast convergence to a paraunitary filter bank solution. Numerical evaluations of the proposed algorithms and comparisons with several existing convolutive blind source separation techniques indicate the excellent relative performance of the proposed methods.

关键词： blind source separation (BSS) fixed-point algorithm independent component analysis speech enhancement

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A numerical analysis of the Nash strategy for weakly coupled large-scale systems

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL 2006年第8期51卷 1371-1377页

作者： Mukaidani, Hiroaki Hiroshima Univ Grad Sch Educ Higashihiroshima 7398524 Japan

This note discusses the feedback Nash equilibrium of linear quadratic N-player Nash games for infinite-horizon large-scale interconnected systems. The asymptotic structure along with the uniqueness and positive semidefiniteness of the solutions of the cross-coupled algebraic Riccati equations (CAREs) is newly established via the Newton-Kantorovich theorem. The main contribution of this study is the proposal of a new algorithm for solving the CAREs. In order to improve the convergence rate of the algorithm, Newton's method is combined with a new decoupling algorithm;it is shown that the proposed algorithm attains quadratic convergence. Moreover, it is shown for the first time that solutions to the CAREs can be obtained by solving the independent algebraic Lyapunov equation (ALE) by using the reduced-order calculation.

关键词： cross-coupled algebraic Riccati equations (CARE) fixed-point algorithm Nash games Newton's method weakly coupled large-scale systems

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CONVERGENCE RATE ANALYSIS OF THE FORWARD-DOUGLAS-RACHFORD SPLITTING SCHEME

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SIAM JOURNAL ON OPTIMIZATION 2015年第3期25卷 1760-1786页

作者： Davis, Damek Univ Calif Los Angeles Dept Math Los Angeles CA 90025 USA

Operator splitting schemes are a class of powerful algorithms that solve complicated monotone inclusion and convex optimization problems that are built from many simpler pieces. They give rise to algorithms in which all simple pieces of the decomposition are processed individually. This leads to easily implementable and highly parallelizable or distributed algorithms, which often obtain nearly state-of-the-art performance. In this paper, we analyze the convergence rate of the forward-Douglas-Rachford splitting (FDRS) algorithm, which is a generalization of the forward-backward splitting (FBS) and Douglas-Rachford splitting algorithms. Under general convexity assumptions, we derive the ergodic and nonergodic convergence rates of the FDRS algorithm, and show that these rates are the best possible. Under Lipschitz differentiability assumptions, we show that the best iterate of FDRS converges as quickly as the last iterate of the FBS algorithm. Under strong convexity assumptions, we derive convergence rates for a sequence that strongly converges to a minimizer. Under strong convexity and Lipschitz differentiability assumptions, we show that FDRS converges linearly. We also provide examples where the objective is strongly convex, yet FDRS converges arbitrarily slowly. Finally, we relate the FDRS algorithm to a primal-dual FBS scheme and clarify its place among existing splitting methods. Our results show that the FDRS algorithm automatically adapts to the regularity of the objective functions and achieves rates that improve upon the sharp worst case rates that hold in the absence of smoothness and strong convexity.

关键词： forward-Douglas-Rachford splitting Douglas-Rachford splitting forward-backward splitting generalized forward-backward splitting fixed-point algorithm primal-dual algorithm

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Complexity Pursuit for Unifying Model

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NEURAL PROCESSING LETTERS 2010年第1期31卷 17-24页

作者： Yang, Yumin Anshan Normal Univ Dept Math Anshan 114001 Peoples R China

Complexity pursuit is an extension of projection pursuit to time series data and the method is closely related to blind separation of time-dependent source signals and independent component analysis. The goal is to find projections of time series that have interesting structure, defined using criteria related to Kolmogoroff complexity or coding length. In this paper, we first derive a simple approximation of coding length for unifying model that takes into account nongaussianity of sources, their autocorrelations and their smoothly changing nonstationary variances. Next, a fixed-point algorithm is proposed by using approximate Newton method. Finally, simulations verify the fixed-point algorithm converges faster than the existing gradient algorithm and it is more simple to implement due to it does not need any learning rate.

关键词： Independent component analysis Blind source separation Complexity pursuit Kolmogoroff complexity fixed-point algorithm

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Semi-blind source extraction algorithm for fetal electrocardiogram based on generalized autocorrelations and reference signals

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2009年第1期223卷 409-420页

作者： Zhang, Hongjuan Shi, Zhenwei Guo, Chonghui Feng, Enmin Dalian Univ Technol Inst Syst Engn Dalian 116024 Peoples R China Dalian Univ Technol Dept Appl Math Dalian 116024 Peoples R China Beijing Univ Aeronaut & Astronaut Sch Astronaut Image Proc Ctr Beijing 100083 Peoples R China

Blind source extraction (BSE) has become one of the processing methods in the field of signal processing and analysis, which only desires to extract "interesting" source signals with specific stochastic property or features so as to save lots of computing time and resources. This paper addresses BSE problem, in which desired source signals have some available reference signals. Based oil this prior information, we develop an objective function for extraction of temporally correlated sources. Maximizing this objective function, a semi-blind source extraction fixed-point algorithm is proposed. Simulations on artificial electrocardiograph (ECG) signals and the real-world ECG data demonstrate the better performance of the new algorithm. Moreover, comparisons with existing algorithms further indicate the validity of our new algorithm, and also show its robustness to the estimated error of time delay. (c) 2008 Elsevier B.V All rights reserved.

关键词： Independent component analysis Semi-blind source extraction Constrained optimization Generalized autocorrelation fixed-point algorithm

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