Electrical Data Matrix Decomposition in Smart Grid

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.

Randomized Generalized Singular Value Decomposition

The generalized singular value decomposition(GSVD)of two matrices with the same number of columns is a very useful tool in many practical applications.However,the GSVD may suffer from heavy computational time and memory requirement when the scale of the matrices is quite large.In this paper,we use random projections to capture the most of the action of the matrices and propose randomized algorithms for computing a low-rank approximation of the GSVD.Serval error bounds of the approximation are also presented for the proposed randomized algorithms.Finally,some experimental results show that the proposed randomized algorithms can achieve a good accuracy with less computational cost and storage requirement.

Least-Squares Solutions of the Matrix Equation A^TXA=B Over Bisymmetric Matrices and its Optimal Approximation

A real n×n symmetric matrix X=(x_(ij))_(n×n)is called a bisymmetric matrix if x_(ij)=x_(n+1-j,n+1-i).Based on the projection theorem,the canonical correlation de- composition and the generalized singular value decomposition,a method useful for finding the least-squares solutions of the matrix equation A^TXA=B over bisymmetric matrices is proposed.The expression of the least-squares solutions is given.Moreover, in the corresponding solution set,the optimal approximate solution to a given matrix is also derived.A numerical algorithm for finding the optimal approximate solution is also described.

Parallel Active Subspace Decomposition for Tensor Robust Principal Component Analysis

Tensor robust principal component analysis has received a substantial amount of attention in various fields.Most existing methods,normally relying on tensor nuclear norm minimization,need to pay an expensive computational cost due to multiple singular value decompositions at each iteration.To overcome the drawback,we propose a scalable and efficient method,named parallel active subspace decomposition,which divides the unfolding along each mode of the tensor into a columnwise orthonormal matrix(active subspace)and another small-size matrix in parallel.Such a transformation leads to a nonconvex optimization problem in which the scale of nuclear norm minimization is generally much smaller than that in the original problem.We solve the optimization problem by an alternating direction method of multipliers and show that the iterates can be convergent within the given stopping criterion and the convergent solution is close to the global optimum solution within the prescribed bound.Experimental results are given to demonstrate that the performance of the proposed model is better than the state-of-the-art methods.

Low-Rank Positive Approximants of Symmetric Matrices

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.

Cyclic Solution and Optimal Approximation of the Quaternion Stein Equation

In this paper, two different methods are used to study the cyclic structure solution and the optimal approximation of the quaternion Stein equation AXB - X = F . Firstly, the matrix equation equivalent to the target structure matrix is constructed by using the complex decomposition of the quaternion matrix, to obtain the necessary and sufficient conditions for the existence of the cyclic solution of the equation and the expression of the general solution. Secondly, the Stein equation is converted into the Sylvester equation by adding the necessary parameters, and the condition for the existence of a cyclic solution and the expression of the equation’s solution are then obtained by using the real decomposition of the quaternion matrix and the Kronecker product of the matrix. At the same time, under the condition that the solution set is non-empty, the optimal approximation solution to the given quaternion circulant matrix is obtained by using the property of Frobenius norm property. Numerical examples are given to verify the correctness of the theoretical results and the feasibility of the proposed method. .

Truncated sparse approximation property and truncated q-norm minimization

This paper considers approximately sparse signal and low-rank matrix’s recovery via truncated norm minimization minx∥xT∥q and minX∥XT∥Sq from noisy measurements.We first introduce truncated sparse approximation property,a more general robust null space property,and establish the stable recovery of signals and matrices under the truncated sparse approximation property.We also explore the relationship between the restricted isometry property and truncated sparse approximation property.And we also prove that if a measurement matrix A or linear map A satisfies truncated sparse approximation property of order k,then the first inequality in restricted isometry property of order k and of order 2k can hold for certain different constantsδk andδ2k,respectively.Last,we show that ifδs(k+|T^c|)<√(s-1)/s for some s≥4/3,then measurement matrix A and linear map A satisfy truncated sparse approximation property of order k.It should be pointed out that when Tc=Ф,our conclusion implies that sparse approximation property of order k is weaker than restricted isometry property of order sk.

The Reflexive Solutions of the Matrix Equations (AX,XB^H )= (C,D^H )

The matrix equations (AX, XBH)=(C, DH) have been widely used in structural design, parameter identification, linear optimal control, and so on. But few researches studied the reflexive solutions. A new approach for the reflexive solutions to the matrix equations was proposed. By applying the canonical correlation decomposition (CCD) of matrix pairs, the necessary and sufficient conditions for the existence and the general expression for the reflexive solutions of the matrix equations (AX, XBH)=(C, DH) were established. In addition, by using the methods of space decomposition, the expression of the optimal approximation solution to a given matrix was derived.

Fast nonnegative tensor ring decomposition based on the modulus method and low-rank approximation

Nonnegative tensor ring(NTR) decomposition is a powerful tool for capturing the significant features of tensor objects while preserving the multi-linear structure of tensor data. The existing algorithms rely on frequent reshaping and permutation operations in the optimization process and use a shrinking step size or projection techniques to ensure core tensor nonnegativity, which leads to a slow convergence rate, especially for large-scale problems. In this paper, we first propose an NTR algorithm based on the modulus method(NTR-MM), which constrains core tensor nonnegativity by modulus transformation. Second, a low-rank approximation(LRA) is introduced to NTR-MM(named LRA-NTR-MM), which not only reduces the computational complexity of NTR-MM significantly but also suppresses the noise. The simulation results demonstrate that the proposed LRA-NTR-MM algorithm achieves higher computational efficiency than the state-of-the-art algorithms while preserving the effectiveness of feature extraction.

四元数Lyapunov方程的酉结构解及最佳逼近

针对四元数Lyapunov方程AX+XA*=C,在A、C为正规矩阵条件下,利用四元数矩阵的Frobenius范数酉乘积不变性和矩阵的右特征值分解,得到了该方程存在酉结构解的充分必要条件及其通解表达式。同时对预先给定的酉矩阵M,应用四元数矩阵的迹不等式和矩阵的分块方法,在该方程的酉结构解集中得到与M的最佳逼近解。最后给出求解步骤,通过数值算例和四元数矩阵的复表示运算,检验了所得理论结果的正确性及所给方法的可行性。

Hermite R-反对称矩阵的二次特征值反问题

研究了Hermite R-反对称矩阵的二次特征值反问题.利用矩阵分块法、奇异值分解、向量拉直和Moore-Penrose逆,证明了该问题Hermite R-反对称解的存在性,给出了Hermite R-反对称解的一般表达式,讨论了最佳逼近问题.并给出了算例验证理论的正确性.

基于局部优化奇异值分解和K-means聚类的协同过滤算法

为了克服传统协同过滤(CF)推荐方法数据稀疏和可扩展性差的不足,该文提出1种基于局部优化降维和聚类的协同过滤算法。采用局部优化的奇异值分解(SVD)降维技术和K-均值(K-means)聚类技术对用户-项目评分矩阵中的相似用户进行聚类并降低维度。利用近似差分矩阵表示评分矩阵的局部结构,实现局部优化。局部优化的SVD降维技术可以利用更少的迭代次数缓解CF中数据稀疏和算法可扩展性差的问题。K-means聚类技术可以缩小邻居集查找范围,提高推荐速度。将该文算法与基于Pearson相关系数的协同过滤算法、基于SVD的协同过滤算法、基于K-means聚类的协同过滤算法相比较。在MovieLens数据集上的实验结果表明,该算法的平均绝对误差(MAE)较其他算法降低了大约12%,准确性(Precision)提高了7%。

阶矩阵及其在传统预处理方法中的应用

本文应用矩阵元素阶和阶矩阵概念,讨论了ICCG和MICCG这两种传统的预处理方法在实用中的一些问题。为什么ICCG(s,t)在s+t固定时取(s,t)=(1,1),(1,2),(1,3),(2,4),(3,5),…有较高的收敛速度?为什么MICCG(m)当m>3时迭代次数不变?ICCG和MICCG的填入方式如何系统化?MICCG是否总比ICCG收敛速度高?本文拟作一个初步的讨论。通过LU分解的阶矩阵,本文给出了按阶递增的填入原则,将ICCG和MICCG系统化为P阶ICCG和P阶MICCG,并讨论了MICCG原有填入方式存在的问题。应用误差阵的阶矩阵,本文讨沦了MICCG迭代参数选取中存在的问题,给出了合理的参数选取方法。通过不同算例,本文还比较了ICCG和MICCG的计算效率。

基于非局部双边随机投影低秩逼近图像去噪算法

该文提出一种基于非局部双边随机投影的低秩逼近图像去噪新方法。首先,对每个图像块通过非局部搜索寻找相似匹配块簇,然后对相似匹配块簇进行双边随机投影,用投影后的低秩结构恢复原图像。实验结果表明,所提方法比奇异值分解方法有较低的计算复杂度,比单边随机投影方法有较小的重构误差。特别是和3维块匹配方法相比,所提方法能保持相近的信噪比和较好的视觉质量。

基于自适应交叉逼近技术的边界元法在3维静电场中的应用

常规边界元法(boundary element method,BEM)因形成的方程矩阵稠密非对称而难以应用于大规模问题,BEM矩阵的存储限制了BEM的工程应用。该文介绍了递阶矩阵(hierarchical matrix,H-matrix)的基本概念,并引入了一种新颖的H-matrix压缩技术:自适应交叉逼近(adaptive cross approximation,ACA)。通过对普法BEM的节点根据拓扑结构重新分区和编号,可以将方程矩阵分解为若干具有递阶特性的子矩阵,应用自适应交叉逼近技术将这些子矩阵进行低秩分解可大大降低BEM矩阵对内存的消耗。为了对基于ACA技术的BEM内存消耗和有效性进行验证,建立了三维球形电极模型,通过与普通BEM比较,证实了该文方法可显著减小边界元法的内存使用,并验证了该方法的有效性。

二次特征值反问题的中心斜对称解及其最佳逼近

利用矩阵的奇异值分解，讨论构造n阶中心斜对称矩阵M，C和K，使得二次束Q（λ）=λ^2M＋λC＋K具有给定特征值和特征向量的特征值反问题．首先证明反问题是可解的，并给出了解集SMCK的通式．然后考虑从解集SMCK中求给定矩阵[M^-，C^-，K^-]的最佳逼近问题，给出了最佳逼近解的存在唯一性及表达式．

Lanczos双对角化:一种快速的非负矩阵初始化方法

对于大型的非负矩阵,利用Lanczos双对角化得到了一个低秩近似.类似于Boutsidis Gallopoulos的方法,可以进一步得到它的非负近似,由此得到了非负矩阵分解的一种新的初始化方法.它虽然带有一点随意性,但可以和已有的非负矩阵分解方法相结合.从数值试验可以看出,与基于奇异值分解的初始化方法相比较,该初始化方法更加有效.

运用增广矩阵束方法稀布优化平面阵

基于增广矩阵束方法(Matrix Enhancement and Matrix Pencil,MEMP),以使用尽可能少的阵元逼近期望的方向图为目标,提出了一种求解阵元位置和设计激励幅度的新方法.首先对期望平面阵的方向图进行采样得到离散的数据集,再构造增广矩阵,对此增广矩阵进行奇异值分解(Singular Value Decomposition,SVD),确定逼近期望方向图所需的最小阵元数目;基于广义特征值分解求解两组特征值,并根据类基于旋转不变技术的信号参数估计(Estimating Signal Parameters Via RotationalInvariance Techniques,ESPRIT)对这两组特值配对;在最小二乘准则下求解稀布面阵的阵元位置和激励.仿真试验验证了该方法在稀布平面阵优化问题中的高效性和数值精度.

低秩重检测的多特征时空上下文的视觉跟踪

时空上下文跟踪算法充分地利用空间上下文中包含的结构信息能够有效地对目标进行跟踪,实时性优良.但该算法仅利用单一的灰度信息,使得目标的表观表达缺乏判别性,而且该方法在由于遮挡等问题造成的跟踪漂移后无法进行初始化.针对时空上下文算法存在的弱点,提出了一种基于低秩重检测的多特征时空上下文跟踪方法.首先,利用多特征对时空上下文进行多方面的提取,构建复合时空上下文信息,充分利用目标周围的特征信息,提高目标表观表达的有效性.其次,利用简单、有效的矩阵分解方式将跟踪到的历史跟踪信息进行低秩表达,将其引入有效的在线重检测器中来保持跟踪结构的一致稳定性,解决了跟踪方法在跟踪失败后的重定位问题,在一系列跟踪数据集上的实验结果表明,该算法与原始算法及当前的主流算法相比有更好的跟踪精度与鲁棒性,且满足实时性要求.

二次特征值反问题的对称次反对称解及其最佳逼近

利用矩阵的奇异值分解和矩阵的Kronecker乘积,讨论构造对称次反对称矩阵M,C和K,使得二次约束Q（λ）=λ^2M＋λC＋K具有给定特征值和特征向量的特征值反问题.首先证明反问题是可解的,并给出了解集SMCK的通式.进而考虑了解集SMCK中对给定矩阵（M,C,K）的最佳逼近问题,得到了最佳逼近解.