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Linearized ADMM for Nonconvex Nonsmooth Optimization With Convergence Analysis

IEEE ACCESS 2019年 7卷 76131-76144页

作者： Liu, Qinghua Shen, Xinyue Gu, Yuantao Tsinghua Univ Bejing Natl Res Ctr Informat Sci & Technol BNRist Beijing 100084 Peoples R China Tsinghua Univ Dept Elect Engn Beijing 100084 Peoples R China

Linearized alternating direction method of multipliers (ADMM) as an extension of ADMM has been widely used to solve linearly constrained problems in signal processing, machine learning, communications, and many other fields. Despite its broad applications in nonconvex optimization, for a great number of nonconvex and nonsmooth objective functions, its theoretical convergence guarantee is still an open problem. In this paper, we propose a two-block linearized ADMM and a multi-block parallel linearized ADMM for problems with nonconvex and nonsmooth objectives. Mathematically, we present that the algorithms can converge for a broader class of objective functions under less strict assumptions compared with previous works. Furthermore, our proposed algorithm can update coupled variables in parallel and work for less restrictive nonconvex problems, where the traditional ADMM may have difficulties in solving subproblems.

关键词： Linearized ADMM multi-block ADMM nonconvex optimization parallel computation proximal algorithm

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A Short Note on the Twice Differentiability of the Marginal Function of a Convex Function

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2023年第2期198卷 857-867页

作者： Crouzeix, Jean-Pierre Hassouni, Abdelhak Ocana, Eladio Univ Clermont Auvergne LIMOS 1 Rue Chebarde F-63178 Aubiere France Univ Mohammed 5 Fac Sci BP 1014 Rabat Morocco Univ Nacl Ingn IMCA Inst Matemat & Ciencias Afines Calle Biologos 245 Urb San Cesar Lima Peru

The main result concerns the second-order differentiability of the marginal function of a convex twice differentiable function. The inf-convolution and the proximal algorithm are revised according to the result as exa... 详细信息

关键词： Convex analysis Convex duality Conjugacy Marginal function Inf-convolution proximal algorithm

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Point Process Estimation with Mirror Prox algorithms

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APPLIED MATHEMATICS AND OPTIMIZATION 2020年第3期82卷 919-947页

作者： He, Niao Harchaoui, Zaid Wang, Yichen Song, Le Univ Illinois Dept Ind Enterprise Syst Engn Urbana IL 61801 USA Univ Washington Dept Stat Seattle WA 98195 USA Georgia Inst Technol Dept Computat Sci & Engn Atlanta GA 30332 USA

Point process models have been extensively used in many areas of science and engineering, from quantitative sociology to medical imaging. Computing the maximum likelihood estimator of a point process model often leads to a convex optimization problem displaying a challenging feature, namely the lack of Lipschitz-continuity of the objective function. This feature can be a barrier to the application of common first order convex optimization methods. We present an approach where the estimation of a point process model is framed as a saddle point problem instead. This formulation allows us to develop Mirror Prox algorithms to efficiently solve the saddle point problem. We introduce a general Mirror Prox algorithm, as well as a variant appropriate for large-scale problems, and establish worst-case complexity guarantees for both algorithms. We illustrate the performance of the proposed algorithms for point process estimation on real datasets from medical imaging, social networks, and recommender systems.

关键词： Mirror Prox proximal algorithm Point process Saddle point problem

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A novel variational approach for multiphoton microscopy image restoration: from PSF estimation to 3D deconvolution

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INVERSE PROBLEMS 2024年第6期40卷 065003-065003页

作者： Ajdenbaum, Julien Chouzenoux, Emilie Lefort, Claire Martin, Segolene Pesquet, Jean-Christophe Univ Paris Saclay Ctr Vis Numer Inria Cent Supelec Gif Sur Yvette France Univ Limoges XLIM Res Inst CNRS UMR 7252 Limoges France

In multi-photon microscopy (MPM), a recent in-vivo fluorescence microscopy system, the task of image restoration can be decomposed into two interlinked inverse problems: firstly, the characterization of the point spread function (PSF) and subsequently, the deconvolution (i.e. deblurring) to remove the PSF effect, and reduce noise. The acquired MPM image quality is critically affected by PSF blurring and intense noise. The PSF in MPM is highly spread in 3D and is not well characterized, presenting high variability with respect to the observed objects. This makes the restoration of MPM images challenging. Common PSF estimation methods in fluorescence microscopy, including MPM, involve capturing images of sub-resolution beads, followed by quantifying the resulting ellipsoidal 3D spot. In this work, we revisit this approach, coping with its inherent limitations in terms of accuracy and practicality. We estimate the PSF from the observation of relatively large beads (approximately 1 mu m in diameter). This goes through the formulation and resolution of an original non-convex minimization problem, for which we propose a proximal alternating method along with convergence guarantees. Following the PSF estimation step, we then introduce an innovative strategy to deal with the high level multiplicative noise degrading the acquisitions. We rely on a heteroscedastic noise model for which we estimate the parameters. We then solve a constrained optimization problem to restore the image, accounting for the estimated PSF and noise, while allowing a minimal hyper-parameter tuning. Theoretical guarantees are given for the restoration algorithm. These algorithmic contributions lead to an end-to-end pipeline for 3D image restoration in MPM, that we share as a publicly available Python software. We demonstrate its effectiveness through several experiments on both simulated and real data.

关键词： non-convex optimisation multiphoton microscopy 3D deconvolution PSF calibration proximal algorithm

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ADAPTIVE FISTA FOR NONCONVEX OPTIMIZATION

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SIAM JOURNAL ON OPTIMIZATION 2019年第4期29卷 2482-2503页

作者： Ochs, Peter Pock, Thomas Saarland Univ Fac Math & Comp Sci Campus E1-7 D-66123 Saarbrucken Germany Graz Univ Technol Inst Comp Graph & Vis Inffeldgasse 16-2 A-8010 Graz Austria

In this paper we propose an adaptively extrapolated proximal gradient method, which is based on the accelerated proximal gradient method (also known as FISTA);however, we locally optimize the extrapolation parameter by carrying out an exact (or inexact) line search. It turns out that in some situations, the proposed algorithm is equivalent to a class of SR1 (identity minus rank 1) proximal quasi-Newton methods. Convergence is proved in a general nonconvex setting, and hence, as a byproduct, we also obtain new convergence guarantees for proximal quasi-Newton methods. The efficiency of the new method is shown in numerical experiments on a sparsity regularized nonlinear inverse problem.

关键词： FISTA proximal algorithm proximal quasi-Newton SR1 quasi-Newton

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A NOTE ON THE PAPER BY ECKSTEIN AND SVAITER ON "GENERAL PROJECTIVE SPLITTING METHODS FOR SUMS OF MAXIMAL MONOTONE OPERATORS"

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2009年第4期48卷 2513-2515页

作者： Bauschke, Heinz H. Univ British Columbia Okanagan Irving K Barber Sch Kelowna BC V1V 1V7 Canada

In their recent paper from 2009 [SIAM J. Control Optim., 48 (2009), pp. 787-811], J. Eckstein and B. F. Svaiter proposed a very general and flexible splitting framework for finding a zero of the sum of finitely many maximal monotone operators. In this short note, we provide a technical result that allows for the removal of Eckstein and Svaiter's assumption that the sum of the operators be maximal monotone or that the underlying Hilbert space be finite-dimensional.

关键词： firmly nonexpansive mapping maximal monotone operator nonexpansive mapping proximal algorithm splitting algorithm

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Alternative Design of DeepPDNet in the Context of Image Restoration

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IEEE SIGNAL PROCESSING LETTERS 2022年 29卷 932-936页

作者： Jiu, Mingyuan Pustelnik, Nelly Zhengzhou Univ Sch Comp & Artificial Intelligence Zhengzhou 450001 Peoples R China Univ Lyon 1 Univ Lyon ENS Lyon CNRSLab Phys F-69342 Lyon France UCLouvain ISPGrp ICTEAM B-1348 Louvain Belgium

This work designs an image restoration deep network relying on unfolded Chambolle-Pock primal-dual iterations. Each layer of our network is built from Chambolle-Pock iterations when specified for minimizing a sum of a l(2)-norm data-term and an analysis sparse prior. The parameters of our network are the step-sizes of the Chambolle-Pock scheme and the linear operator involved in sparsity-based penalization, including implicitly the regularization parameter. A backpropagation procedure is fully described. Preliminary experiments illustrate the good behavior of such a deep primal-dual network in the context of image restoration on BSD68 database.

关键词： Image restoration deep learning proximal algorithm Chambolle-Pock algorithm sparsity

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A Survey of Sparse Representation: algorithms and Applications

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IEEE ACCESS 2015年 3卷 490-530页

作者： Zhang, Zheng Xu, Yong Yang, Jian Li, Xuelong Zhang, David Harbin Inst Technol Shenzhen Grad Sch Biocomp Res Ctr Shenzhen 518055 Peoples R China Key Lab Network Oriented Intelligent Computat Shenzhen 518055 Peoples R China Nanjing Univ Sci & Technol Coll Comp Sci & Technol Nanjing 210094 Jiangsu Peoples R China Chinese Acad Sci Xian Inst Opt & Precis Mech Ctr Opt Imagery Anal & Learning State Key Lab Transient Opt & Photon Xian 710119 Shaanxi Peoples R China Hong Kong Polytech Univ Biometr Res Ctr Hong Kong Hong Kong Peoples R China

Sparse representation has attracted much attention from researchers in fields of signal processing, image processing, computer vision, and pattern recognition. Sparse representation also has a good reputation in both theoretical research and practical applications. Many different algorithms have been proposed for sparse representation. The main purpose of this paper is to provide a comprehensive study and an updated review on sparse representation and to supply guidance for researchers. The taxonomy of sparse representation methods can be studied from various viewpoints. For example, in terms of different norm minimizations used in sparsity constraints, the methods can be roughly categorized into five groups: 1) sparse representation with L-0-norm minimization;2) sparse representation with L-p-norm (0 < p < 1) minimization;3) sparse representation with L-1-norm minimization;4) sparse representation with 12,1-norm minimization;and 5) sparse representation with 12-norm minimization. In this paper, a comprehensive overview of sparse representation is provided. The available sparse representation algorithms can also be empirically categorized into four groups: 1) greedy strategy approximation;2) constrained optimization;3) proximity algorithm-based optimization;and 4) homotopy algorithm-based sparse representation. The rationales of different algorithms in each category are analyzed and a wide range of sparse representation applications are summarized, which could sufficiently reveal the potential nature of the sparse representation theory. In particular, an experimentally comparative study of these sparse representation algorithms was presented.

关键词： Sparse representation compressive sensing greedy algorithm constrained optimization proximal algorithm homotopy algorithm dictionary learning

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Online sparse class imbalance learning on big data

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NEUROCOMPUTING 2016年 216卷 250-260页

作者： Maurya, Chandresh Kumar Toshniwal, Durga Venkoparao, Gopalan Vijendran Indian Inst Technol Dept Comp Sci & Engn Haridwar 247667 UK India RBEI Res & Technol Ctr Bangalore Karnataka India

Class imbalance learning is the study of problems in which some classes appear more frequently than the others. Most existing works that study this problem assume data set to be dense and do not exploit the rich structure of the data. One such structure is the sparsity. In the present work, we focus on solving the class imbalance problem under the sparsity assumption. More specifically, a well-known Gmean metric for class imbalance learning problem in binary classification setting has been maximized, which results in a non-convex loss function. Convex relaxation techniques are used to convert the non-convex problem to the convex problem. The problem formulation in the present work uses L-1 regularized proximal learning framework and is solved via accelerated-stochastic-proximal gradient descent algorithm. Our aim in the paper is to show: (i) the application of proximal algorithms to solve real world problems (class imbalance);(ii) how it scales to Big data;and (iii) how it outperforms some recently proposed algorithms in terms of Gmean, F-measure and Mistake rate on several benchmark data sets. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Class imbalance learning Online learning proximal algorithm Big data

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QUASI-NONEXPANSIVE ITERATIONS ON THE AFFINE HULL OF ORBITS: FROM MANN'S MEAN VALUE algorithm TO INERTIAL METHODS

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SIAM JOURNAL ON OPTIMIZATION 2017年第4期27卷 2356-2380页

作者： Combettes, Patrick L. Glaudin, Lilian E. North Carolina State Univ Dept Math Raleigh NC 27695 USA UPMC Univ Paris 06 Sorbonne Univ UMR 7598 Lab Jacques Louis Lions F-75005 Paris France

Fixed point iterations play a central role in the design and the analysis of a large number of optimization algorithms. We study a new iterative scheme in which the update is obtained by applying a composition of quasi-nonexpansive operators to a point in the affine hull of the orbit generated up to the current iterate. This investigation unifies several algorithmic constructs, including Mann's mean value method, inertial methods, and multilayer memoryless methods. It also provides a framework for the development of new algorithms, such as those we propose for solving monotone inclusion and minimization problems.

关键词： averaged operator fixed point iteration forward-backward algorithm inertial algorithm mean value iterations monotone operator splitting nonsmooth minimization Peaceman-Rachford algorithm proximal algorithm

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