A Perturbation Analysis of Low-Rank Matrix Recovery by Schatten p-Minimization

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.

RGBD Salient Object Detection by Structured Low-Rank Matrix Recovery and Laplacian Constraint

A structured low-rank matrix recovery model for RGBD salient object detection is proposed. Firstly, the problem is described by a low-rank matrix recovery, and the hierarchical structure of RGB image is added to the sparsity term. Secondly, the depth information is fused into the model by a Laplacian regularization term to ensure that the image regions which share similar depth value will be allocated to similar saliency value. Thirdly, a variation of alternating direction method is proposed to solve the proposed model. Finally, both quantitative and qualitative experimental results on NLPR1000 and NJU400 show the advantage of the proposed RGBD salient object detection model. © 2017, Tianjin University and Springer-Verlag Berlin Heidelberg.

Extremal ranks of the solution to a system of real quaternion matrix equations

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.

The Rank of Rank-2 Modified Matrix

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.

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

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.

Discussions on the Relation between Rank and the Number of Non-zero Eigenvalue of Matrix

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.

Constrained Low Rank Approximation of the Hermitian Nonnegative-Definite Matrix

In this paper, we consider a constrained low rank approximation problem: , where E is a given complex matrix, p is a positive integer, and is the set of the Hermitian nonnegative-definite least squares solution to the matrix equation . We discuss the range of p 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.

Robust Principal Component Analysis Integrating Sparse and Low-Rank Priors

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.

中文搜索引擎中的PageRank算法及实现

由于网页质量千差万别,对网页进行基于网络链接图的质量排序变成了现代搜索引擎的一个重要部件。分析了对网络排序模块的实现进行优化时,造成大规模稀疏矩阵-向量乘法运算低效的原因,并结合网络链接图的实际情况提出了几种不同的优化策略。然后,对几种优化策略做了实验性能比较,并综合考虑各种优化策略的运算效率和存储量需求,选择了适合实际系统的优化策略。同时,提出PageRank算法在实现时的一个变通处理——除汇。

基于Ranking的泊松矩阵分解兴趣点推荐算法

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.

RANKL/RANK通路介导神经母细胞瘤细胞迁移的机制探讨

目的:探讨RANKL/RANK通路对神经母细胞瘤(neuroblastoma,NB)SH-SY5Y细胞系细胞侵袭和转移的作用机制。方法:通过pcDNA3.1^+-RANKL和siRNA-RANKL转染NB SH-SY5Y细胞系过表达或沉默外源基因RANKL;细胞增殖实验(cell counting kit-8,CCK-8)检测外源基因RANKL转染NB SH-SY5Y细胞系对细胞增殖的影响;划痕试验检测外源基因RANKL转染NB SH-SY5Y细胞系对细胞迁移的影响;Western blot实验检测外源性基因RANKL转染NB SH-SY5Y细胞系后细胞内基质金属蛋白酶9(matrix metalloproteinase 9,MMP9)、基质金属蛋白酶2(matrix metalloproteinase 2,MMP2)表达变化。结果:通过pcDNA3.1^+-RANKL和siRNA-RANKL对NB SH-SY5Y细胞系转染外源性RANKL基因后三组NB细胞增殖能力无统计学差异;RANKL基因过表达后NB SH-SY5Y细胞迁移能力增强(P<0.01),RANKL沉默后NB SH-SY5Y细胞迁移能力减弱(P<0.01);RANKL基因过表达后NB SH-SY5Y细胞内MMP9、MMP2表达增强(P<0.05、P<0.001),RANKL沉默后NB SH-SY5Y细胞内MMP9、MMP2表达减弱(P<0.05、P<0.01)。结论:RANKL/RANK通路介导NB细胞迁移,并可能通过抑制细胞内MMP2、MMP9表达实现。

A Characterization of Graphs with Rank No More Than 5

The rank of a graph is defined to be the rank of its adjacency matrix. In this paper, the Matlab was used to explore the graphs with rank no more than 5;the performance of the proposed method was compared with former methods, which is simpler and clearer;and the results show that all graphs with rank no more than 5 are characterized.

Eigenvector Method for Ranking Alternatives with Vague Value Measurements

An eigenvector method for ranking alternatives whose measurements are given as vague values is provided. Firstly, a positive matrix is constructed which is defined as evaluation information matrix （EIM）. Based on four assumptions for evaluating alternatives, a ranking eigenvector is defined. And then it is proved, based on positive matrix theory, that the EIM＇s eigenvector corresponding to the maximal eigenvalue is the ranking vector. For alternatives whose characteristics are presented by vague sets, the proposed techniques can evaluate the degree of suitability to which an alternative satisfies the decision-maker＇ s requirement efficiently.

A PERTURBATION ANALYSIS FOR THE PROJECTION OF A STIFFLY SCALED MATRIX

In this paper we study the perturbation bound of the projection ( W A ) ( W A )+,where both the matrices A and W are given with W positive diagonal and severely stiff.When the perturbed matrix (A)= A + δA satisfy several row rank preserving conditions,we derive a new perturbation bound of the projection.

自适应Frank-Wolfe算法及其在矩阵填充上的应用

提出一种矩阵填充问题的自适应Frank-Wolfe算法。首先,采用Nesterov加速策略加速Frank-Wolfe算法,然后,在迭代过程中对矩阵降秩,提高标准Frank-Wolfe算法收敛速率的同时,降低了迭代成本;最后,通过数值实验验证所提算法的有效性。

Method for multiple attribute decision making based on incomplete linguistic judgment matrix

With respect to the multiple attribute decision making problems with linguistic preference relations on alternatives in the form of incomplete linguistic judgment matrix, a method is proposed to analyze the decision problem. The incomplete linguistic judgment matrix is transformed into incomplete fuzzy judgment matrix and an optimization model is developed on the basis of incomplete fuzzy judgment matrix provided by the decision maker and the decision matrix to determine attribute weights by Lagrange multiplier method. Then the overall values of all alternatives are calculated to rank them. A numerical example is given to illustrate the feasibility and practicality of the proposed method.

Correction of failure in antenna array using matrix pencil technique

In this paper a non-iterative technique is developed for the correction of faulty antenna array based on matrix pencil technique（MPT）. The failure of a sensor in antenna array can damage the radiation power pattern in terms of sidelobes level and nulls. In the developed technique, the radiation pattern of the array is sampled to form discrete power pattern information set. Then this information set can be arranged in the form of Hankel matrix（HM） and execute the singular value decomposition（SVD）. By removing nonprincipal values, we obtain an optimum lower rank estimation of HM. This lower rank matrix corresponds to the corrected pattern. Then the proposed technique is employed to recover the weight excitation and position allocations from the estimated matrix. Numerical simulations confirm the efficiency of the proposed technique, which is compared with the available techniques in terms of sidelobes level and nulls.

Unsupervised Multi-Level Non-Negative Matrix Factorization Model: Binary Data Case

Rank determination issue is one of the most significant issues in non-negative matrix factorization (NMF) research. However, rank determination problem has not received so much emphasis as sparseness regularization problem. Usually, the rank of base matrix needs to be assumed. In this paper, we propose an unsupervised multi-level non-negative matrix factorization model to extract the hidden data structure and seek the rank of base matrix. From machine learning point of view, the learning result depends on its prior knowledge. In our unsupervised multi-level model, we construct a three-level data structure for non-negative matrix factorization algorithm. Such a construction could apply more prior knowledge to the algorithm and obtain a better approximation of real data structure. The final bases selection is achieved through L2-norm optimization. We implement our experiment via binary datasets. The results demonstrate that our approach is able to retrieve the hidden structure of data, thus determine the correct rank of base matrix.

乘型模糊判断矩阵排序向量的递推方法

文章首先在模糊判断矩阵乘型一致性以及矩阵元素和权重之间关系的基础上,结合模糊判断矩阵的上三角矩阵元素,构建了一个关于权重和矩阵上三角元素的方程组,并证明了该方程组存在唯一的正解。随后指出方程组的证明过程就是模糊判断矩阵排序向量的求解过程,从而给出了乘型一致性模糊判断矩阵排序向量的一种递推方法。然后,在偏差函数基础上,通过构造并求解一个优化模型,求出了非乘型一致性模糊判断矩阵的排序向量,结果显示,其解的形式与采用乘型一致性模糊判断矩阵递推方法得到的排序向量完全一样。最后,通过实例以及相关方法对比说明排序向量递推方法是可行有效的。

应用线性方程组理论证明矩阵秩的性质

利用矩阵秩的定义证明矩阵秩的性质时,需要使用行列式的性质,证明过程较为复杂。线性方程组解的理论与矩阵秩的内在联系,使得用线性方程组解的理论证明矩阵秩的性质成为可能。应用线性方程组解的理论,可将矩阵秩的等式证明转化为线性方程组解空间相等的证明;将矩阵秩的不等式的证明转化为解空间包含的证明。从行列式性质法的证明转化为集合间关系的证明,不仅简化了矩阵秩的性质的证明,而且证明过程便于理解。