Existence and uniqueness theorems in the theory of weighted pseudoinverses with singular weights

Necessary and sufficient conditions for the existence and uniqueness of weighted pseudoinverses with singular weights are obtained. Pseudoinverses are expanded into matrix power series and power products. A relationship is found between weighted pseudoinverses and weighted normal pseudosolutions, and iterative methods are established for calculating pseudoinverses and pseudosolutions.     

Limiting Representations of Weighted Pseudoinverse Matrices with Positive Definite Weights. Problem Regularization

Limiting representations for weighted pseudoinverse matrices with positive definite weights are derived. It is shown that regularized problems can be constructed based on such limiting representations intended for evaluation of weighted pseudoinverse matrices and weighted normal pseudosolutions with positive definite weights. The results obtained, concerning regularization of problems on evaluation of weighted normal pseudosolutions, are employed for regularization of least-squares problems with constraints.     

Representations and expansions of weighted pseudoinverse matrices, iterative methods, and problem regularization. II. Singular weights

The paper reviews studies on the representations and expansions of weighted pseudoinverse matrices with positive semidefinite weights and on the construction of iterative methods and regularized problems for the calculation of weighted pseudoinverses and weighted normal pseudosolutions based on these representations and expansions. The use of these methods to solve constrained least squares problems is examined. Continued from Cybernetics and Systems Analysis, 44, No. 1, 36–55 (2008). __________ Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 75–102, May–June 2008.     

Iterative Methods to Calculate Weighted Pseudoinverses with Mixed Weights

Cybernetics and Systems Analysis - The authors have obtained and analyzed the expansions of weighted pseudoinverses with mixed weights (one of the weight matrices is positive definite and the other...     

Iterative methods with different rates of convergence for calculating weighted pseudoinverse matrices and weighted normal pseudosolutions with positive definite weights

The authors develop and analyze iterative methods with different (linear, quadratic, or of p (p2) order) rates of convergence. The methods are used to calculate weighted pseudoinverse matrices with positive defined weights. To find weighted normal pseudosolutions with positive defined weights, iterative methods with a quadratic rate of convergence are developed and analyzed. The iterative methods for calculation of weighted normal pseudosolutions are used to solve least-square problems with constraints.Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 20–44, September–October 2004.     

Representations and expansions of weighted pseudoinverse matrices, iterative methods, and problem regularization. I. Positive definite weights

The paper reviews studies on the representations and expansions of weighted pseudoinverse matrices with positive definite weights and on iterative methods and regularized problems for calculation of weighted pseudoinverse matrices and weighted normal pseudosolutions. The use of these methods to solve constrained least-squares problems is examined. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 47–73, January–February 2008.     

Methods of computing weighted pseudoinverse matrices with singular weights

Three iterative processes are constructed and investigated for computing weighted pseudoinverse matrices with singular weights and ML-weighted pseudoinverse matrices. Two of them are based on the decompositions of the weighted pseudoinverse matrix with singular weights into matrix power series, and the third is a generalization of the Schulz method to nonsingular square matrices. Translated from Kibernetika i Sistemnyi Analiz, No. 5, pp. 150–169, September–October, 1999.     

Series expansion of weighted pseudoinverse matrices and iterative methods for calculating weighted pseudoinverse matrices and weighted normal pseudosolutions

Weighted pseudoinverse matrices are expanded into matrix power series with negative exponents and arbitrary positive parameters. Based on this expansion, iterative methods for evaluating weighted pseudoinverse matrices and weighted normal pseudosolutions are designed and analyzed. The iterative methods for weighted normal pseudosolutions are extended to solving constrained least-squares problems. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 32–62, January–February 2006.     

Expansion of weighted pseudoinverse matrices with positive definite weights into matrix power products. Iterative methods

Weighted pseudoinverse matrices with positive definite weights are expanded into matrix power products with negative exponents and arbitrary positive parameters. These expansions are used to develop and analyze iterative methods for evaluating weighted pseudoinverse matrices and weighted normal pseudosolutions and solving constrained least-squares problems. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 45–64, January–February 2007.     

Representing Weighted Pseudoinverse Matrices with Mixed Weights in Terms of Other Pseudoinverses

The paper considers weighted pseudoinverse where both weighted matrices are symmetric and one of them is positive definite matrix and the other is nonsingular and indefinite. Formulas are obtained to represent these matrices in terms of the Moore–Penrose pseudoinverse matrix and other weighted pseudoinverses.     

Weighted hypertree decompositions and optimal query plans

Hypertree width is a measure of the degree of cyclicity of hypergraphs. A number of relevant problems from different areas, e.g., the evaluation of conjunctive queries in database theory or the constraint satisfaction in AI, are tractable when their underlying hypergraphs have bounded hypertree width. However, in practical contexts like the evaluation of database queries, we have more information besides the structure of queries. For instance, we know the number of tuples in relations, the selectivity of attributes and so on. In fact, all commercial query-optimizers are based on quantitative methods and do not care on structural properties.In this paper, in order to combine structural decomposition methods with quantitative approaches, the notion of weighted hypertree decomposition is defined. Weighted hypertree decompositions are equipped with cost functions, that can be used for modeling many situations where there is further information on the given problem, besides its hypergraph representation. The complexity of computing hypertree decompositions having the smallest weights, called minimal hypertree decompositions, is analyzed. It is shown that in many cases tractability is lost if weights are added. However, it is proven that, under some—not very severe—restrictions on the allowed cost functions and on the target hypertrees, optimal weighted hypertree decompositions can be computed in polynomial time. For some easier hypertree weighting functions, this problem is also highly parallelizable. Then, a cost function modeling query evaluation costs is provided, and it is shown how to exploit weighted hypertree decompositions for determining (logical) query plans for answering conjunctive queries. Finally, some preliminary results of an experimental comparison of this query optimization technique with the query optimizer of a commercial DBMS are presented.     

Optimal Fusion Reduced-Order Kalman Filters Weighted by Scalars for Stochastic Singular Systems

Based on the optimal fusion algorithm weighted by scalars in the linear minimum variance sense, a distributed optimal fusion reduced-order Kalman filter with scalar weights is presented for discrete-time stochastic singular systems with multiple sensors and correlated noises. It has higher accuracy than any local filter does. Compared with the distributed fusion filter weighted by matrices, it has lower accuracy but has reduced computational burden. Computation formula of cross-covariance matrix of the filtering errors between any two sensors is given. An example with three sensors shows the effectiveness.     

A Novel Indefinite Kernel Dimensionality Reduction Algorithm: Weighted Generalized Indefinite Kernel Discriminant Analysis

Kernel methods are becoming increasingly popular for many real-world learning problems. And these methods for data analysis are frequently considered to be restricted to positive definite kernels. In practice, however, indefinite kernels arise and demand application in pattern analysis. In this paper, we present several formal extensions of kernel discriminant analysis (KDA) methods which can be used with indefinite kernels. In particular they include indefinite KDA (IKDA) based on generalized singular value decomposition (IKDA/GSVD), pseudo-inverse IKDA, null space IKDA and range space IKDA. Similar to the case of LDA-based algorithms, IKDA-based algorithms also fail to consider that different contribution of each pair of class to the discrimination. To remedy this problem, weighted schemes are incorporated into IKDA extensions in this paper and called them weighted generalized IKDA algorithms. Experiments on two real-world data sets are performed to test and evaluate the effectiveness of the proposed algorithms and the effect of weights on indefinite kernel functions. The results show that the effect of weighted schemes is very significantly.     

基于加权集成Nyström采样的谱聚类算法

针对Nyström方法在谱聚类应用中存在聚类效果不稳定、样本代表性较弱的问题,提出基于加权集成Nyström采样的谱聚类算法.首先利用统计杠杆分数区别数据间的重要程度,对数据进行加权.然后基于权重采用加权K-means中心点采样,得到多组采样点.再引入集成框架,利用集群并行运行Nyström方法构建近似核矩阵.最后利用岭回归方法组合各个近似核矩阵,产生比标准Nyström方法更准确的低秩近似.在UCI数据集上的测试实验表明,文中算法取得较理想的聚类结果.     

Weighted Nuclear Norm Minimization and Its Applications to Low Level Vision

As a convex relaxation of the rank minimization model, the nuclear norm minimization (NNM) problem has been attracting significant research interest in recent years. The standard NNM regularizes each singular value equally, composing an easily calculated convex norm. However, this restricts its capability and flexibility in dealing with many practical problems, where the singular values have clear physical meanings and should be treated differently. In this paper we study the weighted nuclear norm minimization (WNNM) problem, which adaptively assigns weights on different singular values. As the key step of solving general WNNM models, the theoretical properties of the weighted nuclear norm proximal (WNNP) operator are investigated. Albeit nonconvex, we prove that WNNP is equivalent to a standard quadratic programming problem with linear constrains, which facilitates solving the original problem with off-the-shelf convex optimization solvers. In particular, when the weights are sorted in a non-descending order, its optimal solution can be easily obtained in closed-form. With WNNP, the solving strategies for multiple extensions of WNNM, including robust PCA and matrix completion, can be readily constructed under the alternating direction method of multipliers paradigm. Furthermore, inspired by the reweighted sparse coding scheme, we present an automatic weight setting method, which greatly facilitates the practical implementation of WNNM. The proposed WNNM methods achieve state-of-the-art performance in typical low level vision tasks, including image denoising, background subtraction and image inpainting.     

时段划分的多向主元分析间歇过程监测及故障变量追溯

针对间歇过程的多时段特性,提出一种生产过程操作时段划分方法.该方法利用反映过程特性变化的负载矩阵以及主成份矩阵的变化实现了间歇过程子时段的两步划分.提出了基于加权负载向量夹角余弦的负载矩阵相似性度量以及基于加权奇异值变化的奇异值矩阵相似性度量方法,以更客观的反映负载矩阵以及奇异值矩阵的相似性,进而更准确的判断过程特性的变化.根据同一操作子时段的过程特性,其负载矩阵和奇异值矩阵相似性较大的特点,实现了生产过程的子时段划分.将基于子时段划分的多向主元分析(MPCA)建模应用于三水箱系统的在线监测和故障变量追溯,实验结果验证了该方法的有效性.     

赋权图上优化问题的DNA计算方法研究(英文)

对赋权图上经典优化问题的DNA计算方法进行探讨,改进原有DNA计算模型中的权值编码方法,提出一些新的DNA编码方法及DNA算法．具体地说,通过设计赋权无向图的相对长度图给出了旅行商问题的一种相对长度DNA编码方法及DNA算法,通过设计赋权无向图的广义边图给出了中国邮递员问题的一种广义边图DNA编码方法及DNA算法,通过选取DNA序列的最佳逆补比对给出了最小生成树问题的一种基于逆补比对的DNA编码方法及DNA算法,通过设计从顶点覆盖问题到Hamilton回路问题的一种改进多项式变换给出了顶点覆盖问题的一种基于多项式变换的DNA编码方法及DNA算法．所设计的DNA计算方法提高了DNA计算中表示数值和处理数值的能力．     

时间序列稳定协和性的数值分析

§1.引言 许多观测到的时间序列常显示出非平稳性.当时间序列的一阶差分为白噪声时,就产生了非平稳性的简单形式,在这种情况下,称该序列,比如说yt为一阶求和的,记为yt-Ⅰ(1),而对其一阶差分记为△yt-Ⅰ(0).在求和序列的统计分析中,重要的一步是实现对Ⅰ(1)变量的线性组合使之成为Ⅰ(0)就可能实现了,此时称这些变量为协和的.     

基于证据理论的多传感器加权融合改进方法

针对多传感器系统中高冲突证据合成问题,考虑到传感器同时具有固有可靠性与实时可靠性的特点,基于证据理论提出了一种新的多传感器加权融合方法;该方法通过计算证据间的相似矩阵获取证据的后验权重,并结合根据传感器固有可靠性预先分配的先验权重,得到证据体的复合权重,然后据此对原始证据进行加权平均,最后利用D-S证据组合规则合成加权平均后的证据;实例仿真表明,与D-S、Yager、Murphy等方法相比,该方法能够更好地处理高冲突证据,且收敛速度更快。     

图的赋权路径矩阵与所有点对最短路径问题

给出了二维元素矩阵的概念,对于赋权图对应的赋权矩阵,定义了二维元素初始赋权路径矩阵和二维元素一般赋权路径矩阵,在通常赋权矩阵“乘法”运算基础上定义了路径“乘法”运算,从而得到了二维元素一般赋权路径矩阵的“乘法”运算,通过其“乘法”运算来求出所有点对的最短距离与对应路径,在得到最短距离的同时也得到对应的路径,结果显示在最终的一般赋权路径矩阵上。该算法易于通过计算机编程实现,对于大规模有向图或无向图,更有优势。