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...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.展开更多
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 b...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.展开更多
Principal Component Analysis (PCA) is a widely used technique for data analysis and dimensionality reduction, but its sensitivity to feature scale and outliers limits its applicability. Robust Principal Component Anal...Principal Component Analysis (PCA) is a widely used technique for data analysis and dimensionality reduction, but its sensitivity to feature scale and outliers limits its applicability. Robust Principal Component Analysis (RPCA) addresses these limitations by decomposing data into a low-rank matrix capturing the underlying structure and a sparse matrix identifying outliers, enhancing robustness against noise and outliers. This paper introduces a novel RPCA variant, Robust PCA Integrating Sparse and Low-rank Priors (RPCA-SL). Each prior targets a specific aspect of the data’s underlying structure and their combination allows for a more nuanced and accurate separation of the main data components from outliers and noise. Then RPCA-SL is solved by employing a proximal gradient algorithm for improved anomaly detection and data decomposition. Experimental results on simulation and real data demonstrate significant advancements.展开更多
As the development of smart grid and energy internet, this leads to a significantincrease in the amount of data transmitted in real time. Due to the mismatch withcommunication networks that were not designed to carry ...As the development of smart grid and energy internet, this leads to a significantincrease in the amount of data transmitted in real time. Due to the mismatch withcommunication networks that were not designed to carry high-speed and real time data,data losses and data quality degradation may happen constantly. For this problem,according to the strong spatial and temporal correlation of electricity data which isgenerated by human’s actions and feelings, we build a low-rank electricity data matrixwhere the row is time and the column is user. Inspired by matrix decomposition, we dividethe low-rank electricity data matrix into the multiply of two small matrices and use theknown data to approximate the low-rank electricity data matrix and recover the missedelectrical data. Based on the real electricity data, we analyze the low-rankness of theelectricity data matrix and perform the Matrix Decomposition-based method on the realdata. The experimental results verify the efficiency and efficiency of the proposed scheme.展开更多
A parallel hybrid linear solver based on the Schur complement method has the potential to balance the robustness of direct solvers with the efficiency of preconditioned iterative solvers.However,when solving large-sca...A parallel hybrid linear solver based on the Schur complement method has the potential to balance the robustness of direct solvers with the efficiency of preconditioned iterative solvers.However,when solving large-scale highly-indefinite linear systems,this hybrid solver often suffers from either slow convergence or large memory requirements to solve the Schur complement systems.To overcome this challenge,we in this paper discuss techniques to preprocess the Schur complement systems in parallel. Numerical results of solving large-scale highly-indefinite linear systems from various applications demonstrate that these techniques improve the reliability and performance of the hybrid solver and enable efficient solutions of these linear systems on hundreds of processors,which was previously infeasible using existing state-of-the-art solvers.展开更多
Given a symmetric matrix X, we consider the problem of finding a low-rank positive approximant of X. That is, a symmetric positive semidefinite matrix, S, whose rank is smaller than a given positive integer, , which i...Given a symmetric matrix X, we consider the problem of finding a low-rank positive approximant of X. That is, a symmetric positive semidefinite matrix, S, whose rank is smaller than a given positive integer, , which is nearest to X in a certain matrix norm. The problem is first solved with regard to four common norms: The Frobenius norm, the Schatten p-norm, the trace norm, and the spectral norm. Then the solution is extended to any unitarily invariant matrix norm. The proof is based on a subtle combination of Ky Fan dominance theorem, a modified pinching principle, and Mirsky minimum-norm theorem.展开更多
Due to the fine-grained communication scenarios characterization and stability,Wi-Fi channel state information(CSI)has been increasingly applied to indoor sensing tasks recently.Although spatial variations are explici...Due to the fine-grained communication scenarios characterization and stability,Wi-Fi channel state information(CSI)has been increasingly applied to indoor sensing tasks recently.Although spatial variations are explicitlyreflected in CSI measurements,the representation differences caused by small contextual changes are easilysubmerged in the fluctuations of multipath effects,especially in device-free Wi-Fi sensing.Most existing datasolutions cannot fully exploit the temporal,spatial,and frequency information carried by CSI,which results ininsufficient sensing resolution for indoor scenario changes.As a result,the well-liked machine learning(ML)-based CSI sensing models still struggling with stable performance.This paper formulates a time-frequency matrixon the premise of demonstrating that the CSI has low-rank potential and then proposes a distributed factorizationalgorithm to effectively separate the stable structured information and context fluctuations in the CSI matrix.Finally,a multidimensional tensor is generated by combining the time-frequency gradients of CSI,which containsrich and fine-grained real-time contextual information.Extensive evaluations and case studies highlight thesuperiority of the proposal.展开更多
In this paper,we study the low-rank matrix completion problem with Poisson observations,where only partial entries are available and the observations are in the presence of Poisson noise.We propose a novel model compo...In this paper,we study the low-rank matrix completion problem with Poisson observations,where only partial entries are available and the observations are in the presence of Poisson noise.We propose a novel model composed of the Kullback-Leibler(KL)divergence by using the maximum likelihood estimation of Poisson noise,and total variation(TV)and nuclear norm constraints.Here the nuclear norm and TV constraints are utilized to explore the approximate low-rankness and piecewise smoothness of the underlying matrix,respectively.The advantage of these two constraints in the proposed model is that the low-rankness and piecewise smoothness of the underlying matrix can be exploited simultaneously,and they can be regularized for many real-world image data.An upper error bound of the estimator of the proposed model is established with high probability,which is not larger than that of only TV or nuclear norm constraint.To the best of our knowledge,this is the first work to utilize both low-rank and TV constraints with theoretical error bounds for matrix completion under Poisson observations.Extensive numerical examples on both synthetic data and real-world images are reported to corroborate the superiority of the proposed approach.展开更多
Low-rank matrix decomposition with first-order total variation(TV)regularization exhibits excellent performance in exploration of image structure.Taking advantage of its excellent performance in image denoising,we app...Low-rank matrix decomposition with first-order total variation(TV)regularization exhibits excellent performance in exploration of image structure.Taking advantage of its excellent performance in image denoising,we apply it to improve the robustness of deep neural networks.However,although TV regularization can improve the robustness of the model,it reduces the accuracy of normal samples due to its over-smoothing.In our work,we develop a new low-rank matrix recovery model,called LRTGV,which incorporates total generalized variation(TGV)regularization into the reweighted low-rank matrix recovery model.In the proposed model,TGV is used to better reconstruct texture information without over-smoothing.The reweighted nuclear norm and Li-norm can enhance the global structure information.Thus,the proposed LRTGV can destroy the structure of adversarial noise while re-enhancing the global structure and local texture of the image.To solve the challenging optimal model issue,we propose an algorithm based on the alternating direction method of multipliers.Experimental results show that the proposed algorithm has a certain defense capability against black-box attacks,and outperforms state-of-the-art low-rank matrix recovery methods in image restoration.展开更多
文摘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.
基金supported by the National Natural Science Foundation of China(No.61271014)the Specialized Research Fund for the Doctoral Program of Higher Education(No.20124301110003)the Graduated Students Innovation Fund of Hunan Province(No.CX2012B238)
文摘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.
文摘Principal Component Analysis (PCA) is a widely used technique for data analysis and dimensionality reduction, but its sensitivity to feature scale and outliers limits its applicability. Robust Principal Component Analysis (RPCA) addresses these limitations by decomposing data into a low-rank matrix capturing the underlying structure and a sparse matrix identifying outliers, enhancing robustness against noise and outliers. This paper introduces a novel RPCA variant, Robust PCA Integrating Sparse and Low-rank Priors (RPCA-SL). Each prior targets a specific aspect of the data’s underlying structure and their combination allows for a more nuanced and accurate separation of the main data components from outliers and noise. Then RPCA-SL is solved by employing a proximal gradient algorithm for improved anomaly detection and data decomposition. Experimental results on simulation and real data demonstrate significant advancements.
文摘As the development of smart grid and energy internet, this leads to a significantincrease in the amount of data transmitted in real time. Due to the mismatch withcommunication networks that were not designed to carry high-speed and real time data,data losses and data quality degradation may happen constantly. For this problem,according to the strong spatial and temporal correlation of electricity data which isgenerated by human’s actions and feelings, we build a low-rank electricity data matrixwhere the row is time and the column is user. Inspired by matrix decomposition, we dividethe low-rank electricity data matrix into the multiply of two small matrices and use theknown data to approximate the low-rank electricity data matrix and recover the missedelectrical data. Based on the real electricity data, we analyze the low-rankness of theelectricity data matrix and perform the Matrix Decomposition-based method on the realdata. The experimental results verify the efficiency and efficiency of the proposed scheme.
基金supported in part by the Director,Office of Science,Office of Advanced Scientific Computing Research,of the U.S.Department of Energy under Contract No.DE-AC02-05CH11231.
文摘A parallel hybrid linear solver based on the Schur complement method has the potential to balance the robustness of direct solvers with the efficiency of preconditioned iterative solvers.However,when solving large-scale highly-indefinite linear systems,this hybrid solver often suffers from either slow convergence or large memory requirements to solve the Schur complement systems.To overcome this challenge,we in this paper discuss techniques to preprocess the Schur complement systems in parallel. Numerical results of solving large-scale highly-indefinite linear systems from various applications demonstrate that these techniques improve the reliability and performance of the hybrid solver and enable efficient solutions of these linear systems on hundreds of processors,which was previously infeasible using existing state-of-the-art solvers.
文摘Given a symmetric matrix X, we consider the problem of finding a low-rank positive approximant of X. That is, a symmetric positive semidefinite matrix, S, whose rank is smaller than a given positive integer, , which is nearest to X in a certain matrix norm. The problem is first solved with regard to four common norms: The Frobenius norm, the Schatten p-norm, the trace norm, and the spectral norm. Then the solution is extended to any unitarily invariant matrix norm. The proof is based on a subtle combination of Ky Fan dominance theorem, a modified pinching principle, and Mirsky minimum-norm theorem.
文摘We exploit the theory of reproducing kernels to deduce a matrix inequality for the inverse of the restriction of a positive definite Hermitian matrix.
基金the National Natural Science Foundation of China under Grant 61771258 and Grant U1804142the Key Science and Technology Project of Henan Province under Grants 202102210280,212102210159,222102210192,232102210051the Key Scientific Research Projects of Colleges and Universities in Henan Province under Grant 20B460008.
文摘Due to the fine-grained communication scenarios characterization and stability,Wi-Fi channel state information(CSI)has been increasingly applied to indoor sensing tasks recently.Although spatial variations are explicitlyreflected in CSI measurements,the representation differences caused by small contextual changes are easilysubmerged in the fluctuations of multipath effects,especially in device-free Wi-Fi sensing.Most existing datasolutions cannot fully exploit the temporal,spatial,and frequency information carried by CSI,which results ininsufficient sensing resolution for indoor scenario changes.As a result,the well-liked machine learning(ML)-based CSI sensing models still struggling with stable performance.This paper formulates a time-frequency matrixon the premise of demonstrating that the CSI has low-rank potential and then proposes a distributed factorizationalgorithm to effectively separate the stable structured information and context fluctuations in the CSI matrix.Finally,a multidimensional tensor is generated by combining the time-frequency gradients of CSI,which containsrich and fine-grained real-time contextual information.Extensive evaluations and case studies highlight thesuperiority of the proposal.
基金supported in part by the National Natural Science Foundation of China(Grant No.12201473)by the Science Foundation of Wuhan Institute of Technology(Grant No.K202256)+3 种基金The research of M.K.Ng was supported in part by the HKRGC GRF(Grant Nos.12300218,12300519,17201020,17300021)The research of X.Zhang was supported in part by the National Natural Science Foundation of China(Grant No.12171189)by the Knowledge Innovation Project of Wuhan(Grant No.2022010801020279)by the Fundamental Research Funds for the Central Universities(Grant No.CCNU22JC023).
文摘In this paper,we study the low-rank matrix completion problem with Poisson observations,where only partial entries are available and the observations are in the presence of Poisson noise.We propose a novel model composed of the Kullback-Leibler(KL)divergence by using the maximum likelihood estimation of Poisson noise,and total variation(TV)and nuclear norm constraints.Here the nuclear norm and TV constraints are utilized to explore the approximate low-rankness and piecewise smoothness of the underlying matrix,respectively.The advantage of these two constraints in the proposed model is that the low-rankness and piecewise smoothness of the underlying matrix can be exploited simultaneously,and they can be regularized for many real-world image data.An upper error bound of the estimator of the proposed model is established with high probability,which is not larger than that of only TV or nuclear norm constraint.To the best of our knowledge,this is the first work to utilize both low-rank and TV constraints with theoretical error bounds for matrix completion under Poisson observations.Extensive numerical examples on both synthetic data and real-world images are reported to corroborate the superiority of the proposed approach.
基金Project supported by the National Natural Science Foundation of China(No.62072024)the Outstanding Youth Program of Beijing University of Civil Engineering and Architecture,China(No.JDJQ20220805)the Shenzhen Stability Support General Project(Type A),China(No.20200826104014001)。
文摘Low-rank matrix decomposition with first-order total variation(TV)regularization exhibits excellent performance in exploration of image structure.Taking advantage of its excellent performance in image denoising,we apply it to improve the robustness of deep neural networks.However,although TV regularization can improve the robustness of the model,it reduces the accuracy of normal samples due to its over-smoothing.In our work,we develop a new low-rank matrix recovery model,called LRTGV,which incorporates total generalized variation(TGV)regularization into the reweighted low-rank matrix recovery model.In the proposed model,TGV is used to better reconstruct texture information without over-smoothing.The reweighted nuclear norm and Li-norm can enhance the global structure information.Thus,the proposed LRTGV can destroy the structure of adversarial noise while re-enhancing the global structure and local texture of the image.To solve the challenging optimal model issue,we propose an algorithm based on the alternating direction method of multipliers.Experimental results show that the proposed algorithm has a certain defense capability against black-box attacks,and outperforms state-of-the-art low-rank matrix recovery methods in image restoration.
