We study efficient iterative methods for the large sparse non-Hermitian positive definite system of linear equations based on the Hermitian and skew-Hermitian splitting of the coefficient matrix. These methods include a Hermitian/skew-Hermitian splitting (HSS) iteration and its inexact variant, the inexact Hermitian/skew-Hermitian splitting (IHSS) iteration, which employs some Krylov subspace methods as its inner iteration processes at each step of the outer HSS iteration. Theoretical analyses show that the HSS method converges unconditionally to the unique solution of the system of linear equations. Moreover, we derive an upper bound of the contraction factor of the HSS iteration which is dependent solely on the spectrum of the Hermitian part and is independent of the eigenvectors of the matrices involved. Numerical examples are presented to illustrate the effectiveness of both HSS and IHSS iterations. In addition, a model problem of a three-dimensional convection-diffusion equation is used to illustrate...

This paper presents a new k-means type algorithm for clustering high-dimensional objects in sub-spaces. In high-dimensional data, clusters of objects often exist in subspaces rather than in the entire space. For example, in text clustering, clusters of documents of different topics are categorized by different subsets of terms or keywords. The keywords for one cluster may not occur in the documents of other clusters. This is a data sparsity problem faced in clustering high-dimensional data. In the new algorithm, we extend the k-means clustering process to calculate a weight for each dimension in each cluster and use the weight values to identify the subsets of important dimensions that categorize different clusters. This is achieved by including the weight entropy in the objective function that is minimized in the k-means clustering process. An additional step is added to the k-means clustering process to automatically compute the weights of all dimensions in each cluster. The experiments on both synthetic and real data have shown that the new algorithm can generate better clustering results than other subspace clustering algorithms. The new algorithm is also scalable to large data sets.

We address the minimization of regularized convex cost functions which are customarily used for edge-preserving restoration and reconstruction of signals and images. In order to accelerate computation, the multiplicative and the additive half-quadratic reformulation of the original cost-function have been pioneered in Geman and Reynolds [IEEE Trans. Pattern Anal. Machine Intelligence, 14 (1992), pp. 367--383] and Geman and Yang IEEE Trans. Image Process., 4 (1995), pp. 932--946]. The alternate minimization of the resultant (augmented) cost-functions has a simple explicit form. The goal of this paper is to provide a systematic analysis of the convergence rate achieved by these methods. For the multiplicative and additive half-quadratic regularizations, we determine their upper bounds for their root-convergence factors. The bound for the multiplicative form is seen to be always smaller than the bound for the additive form. Experiments show that the number of iterations required for convergence for the multi...

Blur removal is an important problem in signal and image processing. The blurring matrices obtained by using the zero boundary condition (corresponding to assuming dark background outside the scene) are Toeplitz matrices for one-dimensional problems and block-Toeplitz--Toeplitz-block matrices for two-dimensional cases. They are computationally intensive to invert especially in the block case. If the periodic boundary condition is used, the matrices become (block) circulant and can be diagonalized by discrete Fourier transform matrices. In this paper, we consider the use of the Neumann boundary condition (corresponding to a reflection of the original scene at the boundary). The resulting matrices are (block) Toeplitz-plus-Hankel matrices. We show that for symmetric blurring functions, these blurring matrices can always be diagonalized by discrete cosine transform matrices. Thus the cost of inversion is significantly lower than that of using the zero or periodic boundary conditions. We also show that the us...

In this paper we propose an iterative method for calculating the largest eigenvalue of an irreducible nonnegative tensor. This method is an extension of a method of Collatz (1942) for calculating the spectral radius of an irreducible nonnegative matrix. Numerical results show that our proposed method is promising. We also apply the method to studying higher-order Markov chains.