On Accelerated Singular Value Thresholding Algorithm for Matrix Completion

An accelerated singular value thresholding (SVT) algorithm was introduced for matrix completion in a recent paper [1], which applies an adaptive line search scheme and improves the convergence rate from O(1/N) for SVT to O(1/N2), where N is the number of iterations. In this paper, we show that it is the same as the Nemirovski’s approach, and then modify it to obtain an accelerate Nemirovski’s technique and prove the convergence. Our preliminary computational results are very favorable.

MATRIX ALGORITHMS AND ERROR FORMULA FOR BIVARIATE THIELE-TYPE RECTANGULAR MATRIX VALUED RATIONAL INTERPOLATION

A new method for the construction of bivariate matrix valued rational interpolants (BGIRI) on a rectangular grid is presented in [6]. The rational interpolants are of Thiele-type continued fraction form with scalar denominator. The generalized inverse introduced by [3]is gen-eralized to rectangular matrix case in this paper. An exact error formula for interpolation is ob-tained, which is an extension in matrix form of bivariate scalar and vector valued rational interpola-tion discussed by Siemaszko[l2] and by Gu Chuangqing [7] respectively. By defining row and col-umn-transformation in the sense of the partial inverted differences for matrices, two type matrix algorithms are established to construct corresponding two different BGIRI, which hold for the vec-tor case and the scalar case.

Solutions to the generalized Sylvester matrixequations by a singular value decomposition

In this paper, solutions to the generalized Sylvester matrix equations AX -XF = BY and MXN -X = TY with A, M ∈ R^n×n, B, T ∈ Rn×r, F, N ∈ R^p×p and the matrices N, F being in companion form, are established by a singular value decomposition of a matrix with dimensions n × （n ＋ pr）. The algorithm proposed in this paper for the euqation AX - XF = BY does not require the controllability of matrix pair （A, B） and the restriction that A, F do not have common eigenvalues. Since singular value decomposition is adopted, the algorithm is numerically stable and may provide great convenience to the computation of the solution to these equations, and can perform important functions in many design problems in control systems theory.

A SEMI-CONJUGATE MATRIX BOUNDARY VALUE PROBLEM FOR GENERAL ORTHOGONAL POLYNOMIALS ON AN ARBITRARY SMOOTH JORDAN CURVE

In this article, the author characterizes orthogonal polynomials on an arbitrary smooth Jordan curve by a semi-conjugate matrix boundary value problem, which is different from the Riemann-Hilbert problems that appear in the theory of Riemann -Hilbert approach to asymptotic analysis for orthogonal polynomials on a real interval introduced by Fokas, Its, and Kitaev and on the unit circle introduced by Baik, Deift, and Johansson. The author hopes that their characterization may be applied to asymptotic analysis for general orthogonal polynomials by combining with a new extension of steepest descent method which we are looking for.

Werner-Type Matrix Valued Rational Interpolation and Its Recurrence Algorithms

In this paper, a practical Werner-type continued fraction method for solving matrix valued rational interpolation problem is provided by using a generalized inverse of matrices. In order to reduce the continued fraction form to rational function form of the interpolants, an efficient forward recurrence algorithm is obtained.

DIRECT PERTURBATION METHOD FOR REANALYSIS OF MATRIX SINGULAR VALUE DECOMPOSITION

The perturbational reanalysis technique of matrix singular value decomposition is applicable to many theoretical and practical problems in mathematics, mechanics, control theory, engineering, etc.. An indirect perturbation method has previously been proposed by the author in this journal, and now the direct perturbation method has also been presented in this paper. The second-order perturbation results of non-repeated singular values and the corresponding left and right singular vectors are obtained. The results can meet the general needs of most problems of various practical applications. A numerical example is presented to demonstrate the effectiveness of the direct perturbation method.

PERTURBATION METHOD FOR REANALYSIS OF THE MATRIX SINGULAR VALUE DECOMPOSITION

The perturbation method for the reanalysis of the singular value decomposition (SVD) of general real matrices is presented in this paper. This is a simple but efficient reanalysis technique for the SVD, which is of great worth to enhance computational efficiency of the iterative analysis problems that require matrix singular value decomposition repeatedly. The asymptotic estimate formulas for the singular values and the corresponding left and right singular vectors up to second-order perturbation components are derived. At the end of the paper the way to extend the perturbation method to the case of general complex matrices is advanced.

Two Dual Expansions for Generalized Bivariate Thiele-Type Matrix Valued Interpolating Continued Fractions

A new method for the construction of bivariate matrix valued rational interpolants on a rectangulargrid is introduced. The rational interpolants are expressed in the continued fraction form with scalardenominator. Tile matrix quotients are based oil the generalized inverse for a matrix, Which is found to beeffective in continued fraction interpolation. In this paper, tWo dual expansions for bivariate matrix valuedThiele-type interpolating continued fractions are presented, then, tWo dual rational interpolants are definedout of them.

PROBLEMS AND METHODS IN MATRIX VALUED RATIONAL INTERPOLATION

A variety of matrix rational interpolation problems include the partial realizationproblem for matrix power series and the minimal rational interpolation problem for generalmatrix functions.Several problems in circuit theory and digital filter design can also be re-duced to the solution of matrix rational interpolation problems[1—4].By means of thereachability and the observability indices of defined pairs of matrices,Antoulas,Ball,Kang and Willems solved the minimal matrix rational interpolation problem in[1].On

Matrix Boundary Value Problem on Hyperbola

We study a special class of lower trigonometric matrix value boundary value problems on hyperbolas. Firstly, the pseudo-orthogonal polynomial on hyperbola is given in bilinear form and it is shown that it is the only one. Secondly, a special boundary value problem of lower triangular matrix is presented and transformed into four related boundary value problems. Finally, Liouville theorem and Painlevé theorem and pseudo-orthogonal polynomials are used to give solutions.

Orthogonal Matrix-Valued Wavelet Packets

In this paper, we introduce matrix-valued multiresolution analysis and matrix- valued wavelet packets. A procedure for the construction of the orthogonal matrix-valued wavelet packets is presented. The properties of the matrix-valued wavelet packets are investigated. In particular, a new orthonormal basis of L2(R, Cs×s) is obtained from the matrix-valued wavelet packets.

THE NOTE ON MATRIX-VALUED RATIONAL INTERPOLATION

In [3], a kind of matrix-valued rational interpolants (MRIs) in the form of Rn(x) = M(x)/D(x) with the divisibility condition D(x) | ‖M(x)‖2, was defined, and the characterization theorem and uniqueness theorem for MRIs were proved. However this divisibility condition is found not necessary in some cases. In this paper, we remove this restricted condition, define the generalized matrix-valued rational interpolants (GMRIs) and establish the characterization theorem and uniqueness theorem for GMRIs. One can see that the characterization theorem and uniqueness theorem for MRIs are the special cases of those for GMRIs. Moreover, by defining a kind of inner product,we succeed in unifying the Samelson inverses for a vector and a matrix.

ON HASEMAN BOUNDARY VALUE PROBLEM FOR A CLASS OF META-ANALYTIC FUNCTIONS

In this article, Haseman boundary value problem for a class of meta-analytic functions is studied. The expression of solution and the condition of solvability for Haseman boundary value problem are obtained by changing the problem discussed into the equivalent Haseman boundary value problem of bi-analytic function. And the expression of solution and the condition of solvability depend on the canonical matrix.

NEW RESULTS ON ESTIMATES FOR SINGULAR VALUES

WT5,5”BX]New results are provided to estimate matrix singular values in terms of partial absolute deleted row sums and column sums. Illustrative examples are presented to show comparisons with results in literature.[WT5,5”HZ]

Application and Generalization of Eigenvalues Perturbation Bounds for Hermitian Block Tridiagonal Matrices

The paper contains two parts. First, by applying the results about the eigenvalue perturbation bounds for Hermitian block tridiagonal matrices in paper [1], we obtain a new efficient method to estimate the perturbation bounds for singular values of block tridiagonal matrix. Second, we consider the perturbation bounds for eigenvalues of Hermitian matrix with block tridiagonal structure when its two adjacent blocks are perturbed simultaneously. In this case, when the eigenvalues of the perturbed matrix are well-separated from the spectrum of the diagonal blocks, our eigenvalues perturbation bounds are very sharp. The numerical examples illustrate the efficiency of our methods.

Continuous-Time and Discrete-Time Singular Value Decomposition of an Impulse Response Function

This paper proposes the continuous-time singular value decomposition (SVD) for the impulse response function, a special kind of Green’s functions, in order to find a set of singular functions and singular values so that the convolutions of such function with the set of singular functions on a specified domain are the solutions to the inhomogeneous differential equations for those singular functions. A numerical example was illustrated to verify the proposed method. Besides the continuous-time SVD, a discrete-time SVD is also presented for the impulse response function, which is modeled using a Toeplitz matrix in the discrete system. The proposed method has broad applications in signal processing, dynamic system analysis, acoustic analysis, thermal analysis, as well as macroeconomic modeling.

On the Inequalities for Traces and Singular Values of the Quaternion Matrices

In this paper we derive some inequalities for traces and singular values of the quaternion matrices,extend and improve some of the corresponding results appeared in other papers we know.

惩罚矩阵T混合模型及其在省域经济分类中的应用

为充分考虑矩阵数据的特性和数据内部的关联性,文章基于矩阵T分布建立矩阵T混合模型及其惩罚模型来研究聚类问题。在矩阵T混合模型的似然函数上对均值矩阵分量施加自适应核范数低秩惩罚,应用ECM算法提出惩罚似然估计算法,同时提出了一种改进的BIC模型选择准则来选择最优的混合模型数量和调节参数,进而通过自适应核范数阈值自动实现低秩估计,实现准确聚类。最后,通过数值模拟研究及与已有方法的对比验证了该方法的有用性,且将所建立的惩罚混合模型应用于中国省域经济发展水平划分研究,得到了比较准确的聚类结果。

突发性耳聋患者血清VCAM-1、ENA-78、MMP-2水平变化及检测意义

目的:探讨突发性耳聋患者血清血管细胞黏附分子-1(VCAM-1)、上皮中性粒细胞活化肽-78(ENA-78)、基质金属蛋白酶-2(MMP-2)水平变化及检测意义。方法:选取接受治疗的突发性耳聋患者136例为观察组,另选取同期体检健康者136例为对照组。采用ELISA法测定血清VCAM-1、ENA-78、MMP-2水平。根据听力损失程度将患者分为轻度组(46例)、中度组(51例)和重度组(39例)。根据治疗效果将患者分为疗效良好组(82例)和疗效不良组(54例)。比较观察组和对照组血清VCAM-1、ENA-78、MMP-2水平。比较不同听力损失程度及治疗效果患者血清VCAM-1、ENA-78、MMP-2水平。绘制受试者工作特征(ROC)曲线分析血清VCAM-1、ENA-78、MMP-2对突发性耳聋患者疗效不良的预测价值。结果:观察组血清VCAM-1、ENA-78、MMP-2水平高于对照组(均P<0.05)。与轻度组比较,中度和重度组血清VCAM-1、ENA-78、MMP-2水平逐渐升高(均P<0.05)。疗效良好组和疗效不良组患者临床资料比较差异无统计学意义(均P>0.05)。与疗效良好组比较,疗效不良组患者血清VCAM-1、ENA-78、MMP-2水平升高(均P<0.05)。血清VCAM-1、ENA-78、MMP-2三者联合预测突发性耳聋患者疗效不良的AUC高于血清单独指标预测的AUC(均P<0.05)。结论:突发性耳聋患者血清VCAM-1、ENA-78、MMP-2水平升高,与听力损失程度有关,三者联合对突发性耳聋患者治疗效果的预测价值较高。

不协调广义多尺度序决策信息系统的知识获取

针对不协调广义多尺度序决策信息系统中的最优尺度组合和IF-THEN知识获取问题,首先,提出了集值优势矩阵的概念,给出了其判断序决策信息系统是否协调的条件。然后,在不协调的广义多尺度序决策信息系统中,引入基于优势关系的广义决策概念,同时通过定义区间值优势关系的方法,构造了新的协调广义多尺度序决策信息系统,并设计了通过集值优势矩阵求最优尺度组合的算法,挖掘了隐藏在不协调广义多尺度序决策信息系统中的决策规则。最后,通过实验验证了所提广义决策最优尺度组合的有效性。