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
In this paper, the maximal and minimal ranks of the solution to a system of matrix equations over H, the real quaternion algebra, were derived. A previous known result could be regarded as a special case of the new re...In this paper, the maximal and minimal ranks of the solution to a system of matrix equations over H, the real quaternion algebra, were derived. A previous known result could be regarded as a special case of the new result.展开更多
Let Q be the quaternion division algebra over real field F, Denote by Hn（Q） the set of all n x n hermitian matrices over Q. We characterize the additive maps from Hn（Q） into Hm（Q） that preserve rank-1 matrices w...Let Q be the quaternion division algebra over real field F, Denote by Hn（Q） the set of all n x n hermitian matrices over Q. We characterize the additive maps from Hn（Q） into Hm（Q） that preserve rank-1 matrices when the rank of the image of In is equal to n. Let QR be the quaternion division algebra over the field of real number R. The additive maps from Hn （QR） into Hm （QR） that preserve rank-1 matrices are also given.展开更多
In this paper, the authors discuss the relationship in detail between the rank of M in the modified matrix M = A ＋ BC^＊ and the rank of matrix A. The authors do believe the results are useful tools in the modified m...In this paper, the authors discuss the relationship in detail between the rank of M in the modified matrix M = A ＋ BC^＊ and the rank of matrix A. The authors do believe the results are useful tools in the modified matrices.展开更多
<span style="line-height:1.5;"><span>In this paper, we consider a constrained low rank approximation problem: </span><img src="Edit_57d85c54-7822-4512-aafc-f0b0295a8f75.png" wi...<span style="line-height:1.5;"><span>In this paper, we consider a constrained low rank approximation problem: </span><img src="Edit_57d85c54-7822-4512-aafc-f0b0295a8f75.png" width="100" height="24" alt="" /></span><span style="line-height:1.5;"><span>, where </span><i><span>E</span></i><span> is a given complex matrix, </span><i><span>p</span></i><span> is a positive integer, and </span></span><span style="line-height:1.5;"></span><span style="line-height:1.5;"><span> is the set of the Hermitian nonnegative-definite least squares solution to the matrix equation </span><img src="Edit_ced08299-d2dc-4dbb-907a-4d8d36d2e87a.png" width="60" height="16" alt="" /></span><span style="line-height:1.5;"><span>. We discuss the range of </span><i><span>p</span></i><span> and derive the corresponding explicit solution expression of the constrained low rank approximation problem by matrix decompositions. And an algorithm for the problem is proposed and the numerical example is given to show its feasibility.展开更多
This paper gives the rank of matrix and equalities and inequalities of the difference number of non-zero eigenvalue, and discuss the equivalent description of multi angle of equalities for upper and lower bounds of th...This paper gives the rank of matrix and equalities and inequalities of the difference number of non-zero eigenvalue, and discuss the equivalent description of multi angle of equalities for upper and lower bounds of the inequality.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
随着基于位置社交网络(location-based social network,LBSN)的发展,兴趣点推荐成为满足用户个性化需求、减轻信息过载问题的重要手段.然而,已有的兴趣点推荐算法存在如下的问题:1)多数已有的兴趣点推荐算法简化用户签到频率数据,仅使...随着基于位置社交网络(location-based social network,LBSN)的发展,兴趣点推荐成为满足用户个性化需求、减轻信息过载问题的重要手段.然而,已有的兴趣点推荐算法存在如下的问题:1)多数已有的兴趣点推荐算法简化用户签到频率数据,仅使用二进制值来表示用户是否访问一个兴趣点;2)基于矩阵分解的兴趣点推荐算法把签到频率数据和传统推荐系统中的评分数据等同看待,使用高斯分布模型建模用户的签到行为;3)忽视用户签到数据的隐式反馈属性.为解决以上问题,提出一个基于Ranking的泊松矩阵分解兴趣点推荐算法.首先,根据LBSN中用户的签到行为特点,利用泊松分布模型替代高斯分布模型建模用户在兴趣点上签到行为;然后采用BPR(Bayesian personalized ranking)标准优化泊松矩阵分解的损失函数,拟合用户在兴趣点对上的偏序关系;最后,利用包含地域影响力的正则化因子约束泊松矩阵分解的过程.在真实数据集上的实验结果表明:基于Ranking的泊松矩阵分解兴趣点推荐算法的性能优于传统的兴趣点推荐算法.展开更多
为比选长输管道线路路由方案,定量分析各方案的适应性,提出了一种基于“层次分析法—逼近理想解排序法”(Analytic Hierarchy Process-Technique for Order Preference by Similarity to Ideal Solution,AHP-TOPSIS)的线路路由比选方法...为比选长输管道线路路由方案,定量分析各方案的适应性,提出了一种基于“层次分析法—逼近理想解排序法”(Analytic Hierarchy Process-Technique for Order Preference by Similarity to Ideal Solution,AHP-TOPSIS)的线路路由比选方法。以陕京二线输气管道大盂—石家庄分输站段路由方案比选为例,选取投资估算、隧道穿越情况和通过活动断裂带情况等10个指标,确定线路路由比选指标体系,构建目标层—准则层—方案层的递阶层次模型;通过构建AHP-TOPSIS评估模型,计算各指标权重并排序,构建加权规范化决策矩阵,计算各方案与理想方案的相对接近程度,从而确定最佳方案。分析结果表明:投资估算、隧道穿越情况和通过活动断裂带情况对线路路由方案的选择影响较大;通过定量计算,路由方案中南线方案更优,与工程实践也相符合。工程实例证实AHP-TOPSIS法能够用于长输管道线路路由比选,为最佳方案的选择提供依据。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Project supported by the National Natural Science Foundation of China （Grant No.60672160）
文摘In this paper, the maximal and minimal ranks of the solution to a system of matrix equations over H, the real quaternion algebra, were derived. A previous known result could be regarded as a special case of the new result.
基金supported by the National Natural Science Foundation of China (No. 91320201 and No. 61471262)the International (Regional) Collaborative Key Research Projects (No. 61520106002)
文摘Let Q be the quaternion division algebra over real field F, Denote by Hn（Q） the set of all n x n hermitian matrices over Q. We characterize the additive maps from Hn（Q） into Hm（Q） that preserve rank-1 matrices when the rank of the image of In is equal to n. Let QR be the quaternion division algebra over the field of real number R. The additive maps from Hn （QR） into Hm （QR） that preserve rank-1 matrices are also given.
基金the National Natural Sciences Foundation of China(10371044)the Science and Technology Commission of Shanghai Municipality through Grant(04JC14031)+1 种基金the University Young Teacher Sciences Foundation of Anhui Province(2006jq1220zd)Supported by the Ph.D.,Program Scholarship Fund of ECNU(2007)
文摘In this paper, the authors discuss the relationship in detail between the rank of M in the modified matrix M = A ＋ BC^＊ and the rank of matrix A. The authors do believe the results are useful tools in the modified matrices.
文摘<span style="line-height:1.5;"><span>In this paper, we consider a constrained low rank approximation problem: </span><img src="Edit_57d85c54-7822-4512-aafc-f0b0295a8f75.png" width="100" height="24" alt="" /></span><span style="line-height:1.5;"><span>, where </span><i><span>E</span></i><span> is a given complex matrix, </span><i><span>p</span></i><span> is a positive integer, and </span></span><span style="line-height:1.5;"></span><span style="line-height:1.5;"><span> is the set of the Hermitian nonnegative-definite least squares solution to the matrix equation </span><img src="Edit_ced08299-d2dc-4dbb-907a-4d8d36d2e87a.png" width="60" height="16" alt="" /></span><span style="line-height:1.5;"><span>. We discuss the range of </span><i><span>p</span></i><span> and derive the corresponding explicit solution expression of the constrained low rank approximation problem by matrix decompositions. And an algorithm for the problem is proposed and the numerical example is given to show its feasibility.
文摘This paper gives the rank of matrix and equalities and inequalities of the difference number of non-zero eigenvalue, and discuss the equivalent description of multi angle of equalities for upper and lower bounds of the inequality.
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.
文摘随着基于位置社交网络(location-based social network,LBSN)的发展,兴趣点推荐成为满足用户个性化需求、减轻信息过载问题的重要手段.然而,已有的兴趣点推荐算法存在如下的问题:1)多数已有的兴趣点推荐算法简化用户签到频率数据,仅使用二进制值来表示用户是否访问一个兴趣点;2)基于矩阵分解的兴趣点推荐算法把签到频率数据和传统推荐系统中的评分数据等同看待,使用高斯分布模型建模用户的签到行为;3)忽视用户签到数据的隐式反馈属性.为解决以上问题,提出一个基于Ranking的泊松矩阵分解兴趣点推荐算法.首先,根据LBSN中用户的签到行为特点,利用泊松分布模型替代高斯分布模型建模用户在兴趣点上签到行为;然后采用BPR(Bayesian personalized ranking)标准优化泊松矩阵分解的损失函数,拟合用户在兴趣点对上的偏序关系;最后,利用包含地域影响力的正则化因子约束泊松矩阵分解的过程.在真实数据集上的实验结果表明:基于Ranking的泊松矩阵分解兴趣点推荐算法的性能优于传统的兴趣点推荐算法.
文摘为比选长输管道线路路由方案,定量分析各方案的适应性,提出了一种基于“层次分析法—逼近理想解排序法”(Analytic Hierarchy Process-Technique for Order Preference by Similarity to Ideal Solution,AHP-TOPSIS)的线路路由比选方法。以陕京二线输气管道大盂—石家庄分输站段路由方案比选为例,选取投资估算、隧道穿越情况和通过活动断裂带情况等10个指标,确定线路路由比选指标体系,构建目标层—准则层—方案层的递阶层次模型;通过构建AHP-TOPSIS评估模型,计算各指标权重并排序,构建加权规范化决策矩阵,计算各方案与理想方案的相对接近程度,从而确定最佳方案。分析结果表明:投资估算、隧道穿越情况和通过活动断裂带情况对线路路由方案的选择影响较大;通过定量计算,路由方案中南线方案更优,与工程实践也相符合。工程实例证实AHP-TOPSIS法能够用于长输管道线路路由比选,为最佳方案的选择提供依据。
