Eigenvalue Problem Of Large Matrices Research Articles

Suppression of self-excited vibrations by a random parametric excitation

Previous theoretical and experimental studies have shown that some vibrating systems can be stabilized by zero-averaged periodic parametric excitations. It is shown in this paper that some zero-mean random parametric excitations can also be useful for this stabilization. Under some conditions, they can be even more efficient compared to the periodic ones. Two-mass mechanical system with self-excited vibrations is considered for this comparison. The so-called bounded noise is used as a model of the random parametric excitation. The mean-square stability diagrams are obtained numerically by considering an eigenvalue problem for large matrices.

An efficient algorithm for generalized discriminant analysis using incomplete Cholesky decomposition

Generalized discriminant analysis (GDA) has provided an extremely powerful approach to extracting nonlinear features via kernel trick. And it has been suggested for a number of applications, such as classification problem. Whereas the GDA could be solved by the utilization of Mercer kernels, a drawback of the standard GDA is that it may suffer from computational problem for large scale data set. Besides, there is still attendant problem of numerical accuracy when computing the eigenvalue problem of large matrices. Also, the GDA would occupy large memory (to store the kernel matrix). To overcome these deficiencies, we use Gram–Schmidt orthonormalization and incomplete Cholesky decomposition to find a basis for the entire training samples, and then formulate GDA as another eigenvalue problem of matrix whose size is much smaller than that of the kernel matrix by using the basis, while still working out the optimal discriminant vectors from all training samples. The theoretical analysis and experimental results on both artificial and real data set have shown the superiority of the proposed method for performing GDA in terms of computational efficiency and even the recognition accuracy, especially when the training samples size is large.

Class-Incremental Generalized Discriminant Analysis

Generalized discriminant analysis (GDA) is the nonlinear extension of the classical linear discriminant analysis (LDA) via the kernel trick. Mathematically, GDA aims to solve a generalized eigenequation problem, which is always implemented by the use of singular value decomposition (SVD) in the previously proposed GDA algorithms. A major drawback of SVD, however, is the difficulty of designing an incremental solution for the eigenvalue problem. Moreover, there are still numerical problems of computing the eigenvalue problem of large matrices. In this article, we propose another algorithm for solving GDA as for the case of small sample size problem, which applies QR decomposition rather than SVD. A major contribution of the proposed algorithm is that it can incrementally update the discriminant vectors when new classes are inserted into the training set. The other major contribution of this article is the presentation of the modified kernel Gram-Schmidt (MKGS) orthogonalization algorithm for implementing the QR decomposition in the feature space, which is more numerically stable than the kernel Gram-Schmidt (KGS) algorithm. We conduct experiments on both simulated and real data to demonstrate the better performance of the proposed methods.

Class-Incremental Generalized Discriminant Analysis

Generalized discriminant analysis (GDA) is the nonlinear extension of the classical linear discriminant analysis (LDA) via the kernel trick. Mathematically, GDA aims to solve a generalized eigenequation problem, which is always implemented by the use of singular value decomposition (SVD) in the previously proposed GDA algorithms. A major drawback of SVD, however, is the difficulty of designing an incremental solution for the eigenvalue problem. Moreover, there are still numerical problems of computing the eigenvalue problem of large matrices. In this article, we propose another algorithm for solving GDA as for the case of small sample size problem, which applies QR decomposition rather than SVD. A major contribution of the proposed algorithm is that it can incrementally update the discriminant vectors when new classes are inserted into the training set. The other major contribution of this article is the presentation of the modified kernel Gram-Schmidt (MKGS) orthogonalization algorithm for implementing the QR decomposition in the feature space, which is more numerically stable than the kernel Gram-Schmidt (KGS) algorithm. We conduct experiments on both simulated and real data to demonstrate the better performance of the proposed methods.