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A Perturbation Analysis of low-rank matrix recovery by Schatten p-Minimization

Journal of Applied Mathematics and Physics 2024年第2期12卷 475-487页

作者： Zhaoying Sun Huimin Wang Zhihui Zhu Department of Mathematics Shaoxing University Shaoxing China

A number of previous papers have studied the problem of recovering low-rank matrices with noise, further combining the noisy and perturbed cases, we propose a nonconvex Schatten p-norm minimization method to deal with the recovery of fully perturbed low-rank matrices. By utilizing the p-null space property (p-NSP) and the p-restricted isometry property (p-RIP) of the matrix, sufficient conditions to ensure that the stable and accurate reconstruction for low-rank matrix in the case of full perturbation are derived, and two upper bound recovery error estimation ns are given. These estimations are characterized by two vital aspects, one involving the best r-approximation error and the other concerning the overall noise. Specifically, this paper obtains two new error upper bounds based on the fact that p-RIP and p-NSP are able to recover accurately and stably low-rank matrix, and to some extent improve the conditions corresponding to RIP.

关键词： Nonconvex Schatten p-Norm low-rank matrix recovery p-Null Space Property the Restricted Isometry Property

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Proximity point algorithm for low-rank matrix recovery from sparse noise corrupted data

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Applied Mathematics and Mechanics(English Edition) 2014年第2期35卷 259-268页

作者：朱玮舒适成礼智 School of Mathematics and Computational Science Xiangtan University Hunan Key Laboratory for Computation and Simulation in Science and Engineering Xiangtan University Department of Mathematics and Computational Science College of ScienceNational University of Defense Technology

The method of recovering a low-rank matrix with an unknown fraction whose entries are arbitrarily corrupted is known as the robust principal component analysis （RPCA）. This RPCA problem, under some conditions, can be exactly solved via convex optimization by minimizing a combination of the nuclear norm and the 11 norm. In this paper, an algorithm based on the Douglas-Rachford splitting method is proposed for solving the RPCA problem. First, the convex optimization problem is solved by canceling the constraint of the variables, and ~hen the proximity operators of the objective function are computed alternately. The new algorithm can exactly recover the low-rank and sparse components simultaneously, and it is proved to be convergent. Numerical simulations demonstrate the practical utility of the proposed algorithm.

关键词： low-rank matrix recovery sparse noise Douglas-Rachford splitting method,proximity operator

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Sparse recovery: from vectors to tensors

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National Science Review 2018年第5期5卷 756-767页

作者： Yao Wang Deyu Meng Ming Yuan School of Mathematics and Statistics Xi'an Jiaotong University Shenyang Institute of Automation Chinese Academy of Sciences Ministry of Education Key Lab of Intelligent Networks and Network Security Xi'an Jiaotong University Department of Statistics Columbia University

Recent advances in various fields such as telecommunications, biomedicine and economics, among others, have created enormous amount of data that are often characterized by their huge size and high dimensionality. It has become evident, from research in the past couple of decades, that sparsity is a flexible and powerful notion when dealing with these data, both from empirical and theoretical viewpoints. In this survey, we review some of the most popular techniques to exploit sparsity, for analyzing high-dimensional vectors, matrices and higher-order tensors.

关键词： high-dimensional data sparsity compressive sensing low-rank matrix recovery tensors

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SOLVING SYSTEMS OF QUADRATIC EQUATIONS VIA EXPONENTIAL-TYPE GRADIENT DESCENT ALGORITHM

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Journal of Computational Mathematics 2020年第4期38卷 638-660页

作者： Meng Huang Zhiqiang Xu LSEC Inst.Comp.Math.Academy of Mathematics and System ScienceChinese Academy of SciencesBeijing 100190China School of Mathematical Sciences University of Chinese Academy of SciencesBeijing 100049China

We consider the rank minimization problem from quadratic measurements,i.e.,recovering a rank r matrix X 2 Rn×r from m scalar measurements yi=ai XX⊤ai,ai 2 Rn,i=1,...,m.Such problem arises in a variety of applications such as quadratic regression and quantum state tomography.We present a novel algorithm,which is termed exponential-type gradient descent algorithm,to minimize a non-convex objective function f(U)=14m Pm i=1(yi−a⊤i UU⊤ai)2.This algorithm starts with a careful initialization,and then refines this initial guess by iteratively applying exponential-type gradient descent.Particularly,we can obtain a good initial guess of X as long as the number of Gaussian random measurements is O(nr),and our iteration algorithm can converge linearly to the true X(up to an orthogonal matrix)with m=O(nr log(cr))Gaussian random measurements。

关键词： low-rank matrix recovery Non-convex optimization Phase retrieval

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