Matthew M. Lin

Low-Dimensional Polytope Approximation and Its Applications to Nonnegative Matrix Factorization

In this study, nonnegative matrix factorization is recast as the problem of approximating a polytope on the probability simplex by another polytope with fewer facets. Working on the probability simplex has the advantage that data are limited to a compact set with a known boundary, making it easier to trace the approximation procedure. In particular, the supporting hyperplane that separates a point from a disjoint polytope, a fact asserted by the Hahn-Banach theorem, can be calculated in finitely many steps. This approach leads to a convenient way of computing the proximity map which, in contrast to most existing algorithms where only an approximate map is used, finds the unique and global minimum per iteration. This paper sets up a theoretical framework, outlines a numerical algorithm, and suggests an effective implementation. Testing results strongly evidence that this approach obtains a better low rank nonnegative matrix approximation in fewer steps than conventional methods.

SEMI-DEFINITE PROGRAMMING TECHNIQUES FOR STRUCTURED QUADRATIC INVERSE EIGENVALUE PROBLEMS.

In the past decade or so, semi-definite programming (SDP) has emerged as a powerful tool capable of handling a remarkably wide range of problems. This article describes an innovative application of SDP techniques to quadratic inverse eigenvalue problems (QIEPs). The notion of QIEPs is of fundamental importance because its ultimate goal of constructing or updating a vibration system from some observed or desirable dynamical behaviors while respecting some inherent feasibility constraints well suits many engineering applications. Thus far, however, QIEPs have remained challenging both theoretically and computationally due to the great variations of structural constraints that must be addressed. Of notable interest and significance are the uniformity and the simplicity in the SDP formulation that solves effectively many otherwise very difficult QIEPs.

Preconditioned iterative methods for space-time fractional advection-diffusion equations

In this paper, we propose practical numerical methods for solving a class of initial-boundary value problems of space-time fractional advection-diffusion equations. First, we propose an implicit method based on two-sided Grunwald formulae and discuss its stability and consistency. Then, we develop the preconditioned generalized minimal residual (preconditioned GMRES) method and preconditioned conjugate gradient normal residual (preconditioned CGNR) method with easily constructed preconditioners. Importantly, because resulting systems are Toeplitz-like, fast Fourier transform can be applied to significantly reduce the computational cost. We perform numerical experiments to demonstrate the efficiency of our preconditioners, even in cases with variable coefficients.

A note on Sylvester-type equations

This work is to provide a comprehensive treatment of the relationship between the theory of the generalized (palindromic) eigenvalue problem and the theory of the Sylvester-type equations. Under a regularity assumption for a specific matrix pencil, we show that the solution of the ⋆-Sylvester matrix equation is uniquely determined and can be obtained by considering its corresponding deflating subspace. We also propose an iterative method with quadratic convergence to compute the stabilizing solution of the ⋆-Sylvester matrix equation via the well-developed palindromic doubling algorithm. We believe that our discussion is the first which implements the tactic of the deflating subspace for solving Sylvester equations and could give rise to the possibility of developing an advanced and effective solver for different types of matrix equations.

The shift techniques for a nonsymmetric algebraic Riccati equation

In this paper, we want to analyze a special instance of a nonsymmetric algebraic matrix Riccati equation arising from transport theory. Traditional approaches for finding its minimal nonnegative solution are based on fixed point iterations and the speed of the convergence is linear. Recently, iterative methods such as Newton method and the structure-preserving doubling algorithm with quadratic convergence are designed for improving the speed of convergence. But, in some case, the speed of convergence will significantly decrease so that linear convergence becomes sublinear convergence and quadratic convergence becomes linear convergence. Our contribution in this work is to provide a thorough analysis to show that after the shift techniques, the speed of linear or quadratic convergence is preserved. Finally, we apply the shift procedures to the discussion of the simple iteration algorithm, improve its speed of convergence, and reduce its total elapsed CPU time.

Discrete Eckart-Young Theorem for Integer Matrices

The well-known Eckart-Young theorem asserts that the truncated singular value decomposition, obtained by discarding all but the first

An algorithm for constructing nonnegative matrices with prescribed real eigenvalues

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Nonnegative rank factorization--a heuristic approach via rank reduction

largest singular values and their corresponding left and right singular vectors, is the best rank-

Dynamical System Characterization of the Central Path and Its Variants—A Revisit

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Inverse mode problems for real and symmetric quadratic models

approximation in the sense of least squares to the original matrix. In other words, singular values alone serve well as unambiguous indicators of proximity to the data matrix. Unlike continuous data, the decomposition of a matrix with discrete data which is subject to the requirement that its approximations have the same type of data is a harder task and it is even harder when it comes to ranking these approximations. This work generalizes the notion of singular value decomposition via a sequence of variational formulations to discrete-type data. The process itself can guarantee neither the orthogonality, as is expected of discrete data, nor the ordering of best approximations. However, at the end of the undertaking, it is shown that a quantity analogous to the singular values and a truncated low rank factorization for discrete data analogous to the truncated singular value decomposition for continuous data are attainable. Our empirical study shows the applicability of our method to cluster analysis and pattern discovery using real-life data.