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Differentially Private Precision Matrix Estimation

Acta Mathematica Sinica,English Series 2020年第10期36卷 1107-1124页

作者： Wen Qing SU Xiao GUO Hai ZHANG School of Mathematics Northwest UniversityXi'an 710127P.R.China Faculty of Information Technology Macao University of Science and TechnologyMacao 999078P.R.China Key Laboratory of Advanced Theory and Application in Statistics and Data Science(East China Normal University) Ministry of EducationShanghai 200062P.R.China

In this paper, we study the problem of precision matrix estimation when the dataset contains sensitive information. In the differential privacy framework, we develop a differentially private ridge estimator by perturbing the sample covariance matrix. Then we develop a differentially private graphical lasso estimator by using the alternating direction method of multipliers(admm) ***, we prove theoretical results showing that the differentially private ridge estimator for the precision matrix is consistent under fixed-dimension asymptotic, and establish a convergence rate of differentially private graphical lasso estimator in the Frobenius norm as both data dimension p and sample size n are allowed to grow. The empirical results that show the utility of the proposed methods are also provided.

关键词： Differential privacy graphical model admm algorithm

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Robust compressed sensing with bounded and structured uncertainties

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IET SIGNAL PROCESSING 2014年第7期8卷 783-791页

作者： Qing, Xiangyun Hu, Guosheng Wang, Xingyu Minist Educ Key Lab Adv Control & Optimizat Chem Proc Shanghai Peoples R China E China Univ Sci & Technol Dept Automat Shanghai 200237 Peoples R China Univ Surrey Ctr Vis Speech & Signal Proc Guildford GU2 5XH Surrey England

The robust compressed sensing problem subject to a bounded and structured perturbation in the sensing matrix is solved in two steps. The alternating direction method of multipliers (admm) is first applied to obtain a robust support set. Unlike the existing robust signal recovery solutions, the proposed optimisation problem is convex. The admm algorithm that every subproblem has a global minimum is employed to solve the optimisation problem. Then, the standard robust regularised least-squares problem restrained to the support is solved to reduce the recovery error. The numerical tests show that the proposed approach provides a robust estimation of support set, although it is conservative to recover signal magnitudes as a result of minimising the worst-cast data error across all bounded perturbations.

关键词： compressed sensing convex programming perturbation techniques matrix multiplication least mean squares methods estimation theory minimisation bounded uncertainty structured uncertainty robust compressed sensing problem structured perturbation bounded perturbation sensing matrix alternating direction method of multipliers admm algorithm robust signal recovery convex optimisation problem standard robust regularised least square problem recovery error reduction robust support set estimation signal magnitude recovery worst cast data error minimisation

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Alternating direction method of multipliers for a class of nonconvex bilinear optimization: convergence analysis and applications

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JOURNAL OF GLOBAL OPTIMIZATION 2018年第1期70卷 261-288页

作者： Hajinezhad, Davood Shi, Qingjiang Iowa State Univ Dept Ind & Mfg Syst Engn Ames IA 50011 USA Nanjing Univ Aeronaut & Astronaut Nanjing Jiangsu Peoples R China

In this paper, we study a class of nonconvex nonsmooth optimization problems with bilinear constraints, which have wide applications in machine learning and signal processing. We propose an algorithm based on the alternating direction method of multipliers, and rigorously analyze its convergence properties (to the set of stationary solutions). To test the performance of the proposed method, we specialize it to the nonnegative matrix factorization problem and certain sparse principal component analysis problem. Extensive experiments on real and synthetic data sets have demonstrated the effectiveness and broad applicability of the proposed methods.

关键词： Nonconvex optimization admm algorithm Bilinear constraint Nonsmooth regularization

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AN ALTERNATING DIRECTION METHOD OF MULTIPLIERS FOR THE NUMERICAL SOLUTION OF A FULLY NONLINEAR PARTIAL DIFFERENTIAL EQUATION INVOLVING THE JACOBIAN DETERMINANT

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2018年第1期40卷 A52-+页

作者： Caboussat, Alexandre Glowinski, Roland Univ Appl Sci Western Switzerland HES SO Haute Ecole Gest Geneve Geneva Sch Business Adm CH-1227 Carouge Switzerland Univ Houston Dept Math Houston TX 77204 USA

We consider the Dirichlet problem for a partial differential equation involving the Jacobian determinant in two dimensions of space. The problem consists in finding a vector-valued function such that the determinant of its gradient is given pointwise in a bounded domain, together with essential boundary conditions. This problem was initially considered in Dacorogna and Moser [Ann. Inst. H. Poincare Anal. Non Lineaire, 7 (1990), pp. 1{26], and several theoretical generalizations have been derived since. In this work, we design a numerical algorithm for the approximation of the solution of such a problem for various kinds of boundary data. The proposed method relies on an augmented Lagrangian algorithm with biharmonic regularization, and low order mixed finite element approximations. An iterative method allows us to decouple the nonlinearity and the differential operators. Numerical experiments show the capabilities of the method for benchmarks and then for more demanding test problems.

关键词： Jacobian determinant augmented Lagrangian methods admm algorithm biharmonic regularization finite element method quadratically constrained minimization

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Trace regression model with simultaneously low rank and row(column) sparse parameter

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COMPUTATIONAL STATISTICS & DATA ANALYSIS 2017年 116卷 1-18页

作者： Zhao, Junlong Niu, Lu Zhan, Shushi Beijing Normal Univ Sch Stat Beijing Peoples R China Beihang Univ Sch Math & Syst Sci Beijing Peoples R China

In this paper, we consider the trace regression model with matrix covariates, where the parameter is a matrix of simultaneously low rank and row(column) sparse. To estimate the parameter, we formulate a convex optimization problem with the nuclear norm and group Lasso penalties, and propose an alternating direction method of multipliers (admm) algorithm. The asymptotic properties of the estimator are established. Simulation results confirm the effectiveness of our method. (C) 2017 Elsevier B.V. All rights reserved.

关键词： Trace regression model Matrix covariates Low rank Row/column sparse admm algorithm

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Profile extraction and defect detection for stereolithography curing process based on multi-regularized tensor decomposition

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JOURNAL OF MANUFACTURING SYSTEMS 2024年 74卷 100-111页

作者： Zhang, Yinwei Zhang, Tao Liu, Jian Kang, Wenjun Liang, Rongguang Potter, Barrett G. Univ Arizona Dept Syst & Ind Engn Tucson AZ 85721 USA Univ Sci & Technol Beijing Natl Ctr Mat Serv Safety Beijing 100083 Peoples R China Univ Arizona James C Wyant Coll Opt Sci Tucson AZ 85721 USA

Optical lenses cured from the stereolithography process are still at their primitive stage, where the detection of process faults and product defects is of great importance. Such needed capability is enabled by an in situ process monitoring system with advanced camera sensors that collect high -dimensional images of the outer geometric profile and/or inner defects of the cured lenses. The state -of -the -art methods fall short in distinguishing the profile of the cured lens and the sparsely distributed spots from the curable resin shown in noisy images, leaving profile assessment and defect detection ineffective and unreliable. In this paper, a Multi-Regularized Tensor Decomposition (MRTD) Method is proposed to improve the performance of profile extraction and defect detection. It decomposes an in situ image of the curing process into four components: the smooth background, the smooth cured lens profile, the sparse spots, and the white noises. A modified admm algorithm is developed to solve this decomposition as a large-scale optimization problem. A real -world case study is conducted to demonstrate that the proposed method outperforms existing state -of -the -art methods with respect to profile extraction accuracy and defect detection sensitivity.

关键词： Optimization Tensor decomposition Image processing admm algorithm Additive manufacturing

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Peer-to-peer decentralized energy trading in industrial town considering central shared energy storage using alternating direction method of multipliers algorithm

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IET RENEWABLE POWER GENERATION 2022年第12期16卷 2579-2589页

作者： Aminlou, Ali Mohammadi-Ivatloo, Behnam Zare, Kazem Razzaghi, Reza Anvari-Moghaddam, Amjad Univ Tabriz Fac Elect & Comp Engn Tabriz *** Iran Monash Univ Dept Elect & Computet Syst Engn Melbourne Vic Australia Aalborg Univ Dept Energy AAU Energy Aalborg Denmark

Distributed energy resources are being progressively deployed by industry. The penetration of distributed generation in low voltage (LV) networks can position traditional consumers as market participants. With the improvements in communication networks and the introduction of new energy markets, these prosumers are incentivized to sell their excess production to other industries by participating in peer-to-peer (P2P) energy markets. This market paves the way for developing new technologies such as a shared battery energy storage system (SBESS). In this paper, a central storage unit rents its capacity for the prosumers to reduce the overall peak-load of the microgrid. Each user requests the required SBESS capacity and calculates the best charging and discharging times to reduce their cost. To this end, this paper organized a P2P energy trading paradigm with the presence of SBESS. The optimization problem was simulated and solved using the alternating direction method of multipliers (admm) algorithm. Results demonstrate how combining features of P2P energy trading and SBESS can save up to 29% for the industrial town.

关键词： power engineering computing central shared energy storage distributed power generation low voltage networks admm algorithm central storage unit power markets shared battery energy storage system power generation economics convex programming SBESS industrial town peer-to-peer computing distributed energy resources P2P energy trading paradigm SBESS capacity peer-to-peer energy markets communication networks peer-to-peer decentralized energy trading alternating direction method of multipliers algorithm distributed generation battery storage plants market participants

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Bayesian inference for adaptive low rank and sparse matrix estimation

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NEUROCOMPUTING 2018年 291卷 71-83页

作者： Jia, Xixi Feng, Xiangchu Wang, Weiwei Xu, Chen Zhang, Lei Xidian Univ Sch Math & Stat Xian 710126 Shaanxi Peoples R China Shenzhen Univ Shenzhen 518060 Guangdong Peoples R China Hong Kong Polytech Univ Hong Kong Hong Kong Peoples R China

Low rank and sparse matrix estimation has been attracting significant interest in recent years. Generally, such a problem is modeled by imposing the l(1)-norm to pursuit a sparse and low rank matrix decomposition. However, the l(1)-norm is only a conservative sparse regularizer which leads to over-penalty. To remedy this issue, this paper presents an adaptive regularizer learning strategy to provide advanced low rank solution and avoid over-penalty. The new method is termed ARLLR. In the Bayesian inference, the prior distribution of the singular values is assumed to be Laplacian with hyper scale parameters. With the help of full Maximize A Posterior (MAP), we learn the optimal scale parameters by revealing its correlation to the inherent variables. We indicate that the adaptively estimated regularizer corresponds to the log function and the global minimum is given for the proposed non-convex problem. Furthermore, by employing the adaptive regularizer on the sparse part, a double log regularized low rank and sparse matrix decomposition model which is denoted by ARLLRE, is proposed. The admm algorithm is utilized to solve the ARLLRE problem, and the convergence of the algorithm is proved. In experiment, we use ARLLR for image denoising and ARLLRE for foreground and background extraction, respectively. Experimental results show that ARLLR enhances image denoising performance compared with the state-of-the-art image denoising algorithms in both quantity value and visual quality. Meanwhile, ARLLRE delivers excellent results in foreground and background extraction. (c) 2018 Elsevier B.V. All rights reserved.

关键词： MAP estimation admm algorithm Image denoising Low rank and sparse decomposition

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Solution methodologies for minimizing a sum of pointwise minima of two functions

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OPTIMIZATION LETTERS 2023年第1期17卷 75-87页

作者： Zuo, Xinyi Jiang, Yi Sichuan Normal Univ Sch Math Sci Visual Comp & Virtual Real Key Lab Chengdu 610066 Sichuan Peoples R China

In this paper, an NP-hard problem of minimizing a sum of pointwise minima of two functions is considered. Using a new equivalent formula, we propose a smooth approximation and an admm algorithm for solving the problem. In numerical experiments, we survey four methods including the algorithms proposed in this paper and the known methods. The results of numerical experiments indicate that the performance of each of algorithms could highly depend on the problem and simulation settings.

关键词： Smooth approximation admm algorithm DC algorithm

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A relative error-based approach for variable selection

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COMPUTATIONAL STATISTICS & DATA ANALYSIS 2016年第0期103卷 250-262页

作者： Hao, Meiling Lin, Yunyuan Zhao, Xingqiu Hong Kong Polytech Univ Dept Appl Math Hong Kong Hong Kong Peoples R China Chinese Univ Hong Kong Dept Stat Hong Kong Hong Kong Peoples R China

The accelerated failure time model or the multiplicative regression model is well-suited to analyze data with positive responses. For the multiplicative regression model, the authors investigate an adaptive variable selection method via a relative error-based criterion and Lasso-type penalty with desired theoretical properties and computational convenience. With fixed or diverging number of variables in regression model, the resultant estimator achieves the oracle property. An alternating direction method of multipliers algorithm is proposed for computing the regularization paths effectively. A data-driven procedure based on the Bayesian information criterion is used to choose the tuning parameter. The finite-sample performance of the proposed method is examined via simulation studies. An application is illustrated with an analysis of one period of stock returns in Hong Kong Stock Exchange. (C) 2016 Elsevier B.V. All rights reserved.

关键词： Adaptive lasso admm algorithm Diverging number Oracle property Relative error

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