We propose a second-order accurate method to estimate the eigenvectors of extremely large matrices thereby addressing a problem of relevance to statisticians working in the analysis of very large datasets. More specifically, we show that averaging eigenvectors of randomly subsampled matrices efficiently approximates the true eigenvectors of the original matrix under certain conditions on the incoherence of the spectral decomposition. This incoherence assumption is typically milder than those made in matrix completion and allows eigenvectors to be sparse. We discuss applications to spectral methods in dimensionality reduction and information retrieval.
In the context of matrix displacement decomposition, Bozzo and Di Fiore introduced the so-called $τ_{\varepsilon,φ}$ algebra, a generalization of the more known $τ$ algebra originally proposed by Bini and Capovani. We study the properties of eigenvalues and eigenvectors of the generator $T_{n,\varepsilon,φ}$ of the $τ_{\varepsilon,φ}$ algebra. In particular, we derive the asymptotics for the outliers of $T_{n,\varepsilon,φ}$ and the associated eigenvectors; we obtain equations for the eigenvalues of $T_{n,\varepsilon,φ}$, which provide also the eigenvectors of $T_{n,\varepsilon,φ}$; and we com…
Iterative Refinement for a Subset of Eigenvectors of Symmetric Matrices via Matrix Multiplications
Research Institute for Science and Engineering, Waseda University（早稻田大学理工学研究科）；Department of Mathematical Sciences, Shibaura Institute of Technology（埼玉工业大学数学科学系）；RIKEN Center for Computational Science（理化学研究所计算科学中心）；Department of Applied Mathematics, Faculty of Science and Engineering, Waseda University（早稻田大学理工学部应用数学系）
We consider the problem of computing ratings using the results of games played between a set of n players, and show how this problem can be reduced to computing the positive eigenvectors corresponding to the dominant eigenvalues of certain n by n matrices. There is a close connection with the stationary probability distributions of certain Markov chains. In practice, if n is large, then the matrices involved will be sparse, and the power method may be used to solve the eigenvalue problems efficiently. We give an algorithm based on the power method, and also derive the same algorithm by an inde…
In this note we present a parameterized class of lower triangular matrices. The components of the eigenvectors grow rapidly and will exceed the representational range of any finite number system. The eigenvalues and the eigenvectors are well-conditioned with respect to componentwise relative perturbations of the matrix. This class of matrices is well suited for testing software for computing eigenvectors as these routines must be able to handle overflow successfully.
Matrix functions play an important role in applied mathematics. In network analysis, in particular, the exponential of the adjacency matrix associated with a network provides valuable information about connectivity, as well as about the relative importance or centrality of nodes. Another popular approach to rank the nodes of a network is to compute the left Perron vector of the adjacency matrix for the network. The present article addresses the problem of evaluating matrix functions, as well as computing an approximation to the left Perron vector, when only some of the columns and/or some of t…
In a series of recent papers the spectral behavior of the matrix sequence $\{Y_nT_n(f)\}$ is studied in the sense of the spectral distribution, where $Y_n$ is the main antidiagonal (or flip matrix) and $T_n(f)$ is the Toeplitz matrix generated by the function $f$, with $f$ being Lebesgue integrable and with real Fourier coefficients. This kind of study is also motivated by computational purposes for the solution of the related large linear systems using the (preconditioned) MINRES algorithm. Here we complement the spectral study with more results holding both asymptotically and for a fixed dim…
Motivated by the conjectures formulated in 2003 by Tunçel et al., we study interlacing properties of the eigenvalues of $A\otimes B + B\otimes A$ for pairs of $n$-by-$n$ matrices $A, B$. We prove that for every pair of symmetric matrices (and skew-symmetric matrices) with one of them at most rank two, the \emph{odd spectrum} (those eigenvalues determined by skew-symmetric eigenvectors) of $A\otimes B + B\otimes A$ interlaces its \emph{even spectrum} (those eigenvalues determined by symmetric eigenvectors). Using this result, we also show that when $n \leq 3$, the odd spectrum of $A\otimes B + …
We present a new formulation of the hyperbolic singular value decomposition (HSVD) for an arbitrary complex (or real) matrix without hyperexchange matrices and redundant invariant parameters. In our formulation, we use only the concept of pseudo-unitary (or pseudo-orthogonal) matrices. We show that computing the HSVD in the general case is reduced to calculation of eigenvalues, eigenvectors, and generalized eigenvectors of some auxiliary matrices. The new formulation is more natural and useful for some applications. It naturally includes the ordinary singular value decomposition.
This paper is concerned with the interplay between statistical asymmetry and spectral methods. Suppose we are interested in estimating a rank-1 and symmetric matrix $\mathbf{M}^{\star}\in \mathbb{R}^{n\times n}$, yet only a randomly perturbed version $\mathbf{M}$ is observed. The noise matrix $\mathbf{M}-\mathbf{M}^{\star}$ is composed of zero-mean independent (but not necessarily homoscedastic) entries and is, therefore, not symmetric in general. This might arise, for example, when we have two independent samples for each entry of $\mathbf{M}^{\star}$ and arrange them into an {\em asymmetric}…
