Robert J. Plemmons

Nonnegative matrices in the mathematical sciences

1. Matrices which leave a cone invariant 2. Nonnegative matrices 3. Semigroups of nonnegative matrices 4. Symmetric nonnegative matrices 5. Generalized inverse- Positivity 6. M-matrices 7. Iterative methods for linear systems 8. Finite Markov Chains 9. Input-output analysis in economics 10. The Linear complementarity problem 11. Supplement 1979-1993 References Index.

Algorithms and applications for approximate nonnegative matrix factorization

The development and use of low-rank approximate nonnegative matrix factorization (NMF) algorithms for feature extraction and identification in the fields of text mining and spectral data analysis are presented. The evolution and convergence properties of hybrid methods based on both sparsity and smoothness constraints for the resulting nonnegative matrix factors are discussed. The interpretability of NMF outputs in specific contexts are provided along with opportunities for future work in the modification of NMF algorithms for large-scale and time-varying data sets.

Document clustering using nonnegative matrix factorization

A methodology for automatically identifying and clustering semantic features or topics in a heterogeneous text collection is presented. Textual data is encoded using a low rank nonnegative matrix factorization algorithm to retain natural data nonnegativity, thereby eliminating the need to use subtractive basis vector and encoding calculations present in other techniques such as principal component analysis for semantic feature abstraction. Existing techniques for nonnegative matrix factorization are reviewed and a new hybrid technique for nonnegative matrix factorization is proposed. Performance evaluations of the proposed method are conducted on a few benchmark text collections used in standard topic detection studies.

Parallel algorithms for dense linear algebra computations

Scientific and engineering research is becoming increasingly dependent upon the development and implementation of efficient parallel algorithms on modern high-performance computers. Numerical linear algebra is an indispensable tool in such research and this paper attempts to collect and describe a selection of some of its more important parallel algorithms. The purpose is to review the current status and to provide an overall perspective of parallel algorithms for solving dense, banded, or block-structured problems arising in the major areas of direct solution of linear systems, least squares computations, eigenvalue and singular value computations, and rapid elliptic solvers. A major emphasis is given here to certain computational primitives whose efficient execution on parallel and vector computers is essential in order to obtain high performance algorithms.

M-matrix characterizations.I—nonsingular M-matrices

Abstract The purpose of this survey is to classify systematically a widely ranging list of characterizations of nonsingular M -matrices from the economics and mathematics literatures. These characterizations are grouped together in terms of their relations to the properties of (1) positivity of principal minors, (2) inverse-positivity and splittings, (3) stability and (4) semipositivity and diagonal dominance. A list of forty equivalent conditions is given for a square matrix A with nonpositive off-diagonal entries to be a nonsingular M -matrix. These conditions are grouped into classes in order to identify those that are equivalent for arbitrary real matrices A . In addition, other remarks relating nonsingular M -matrices to certain complex matrices are made, and the recent literature on these general topics is surveyed.

Positive diagonal solutions to the Lyapunov equations

We study various stability type conditions on a matrix A related to the consistency of the Lyapunov equation AD+DAt positive definite, where D is a positive diagonal matrix. Such problems arise in mathematical economics, in the study of time-invariant continuous-time systems and in the study of predator-prey systems. Using a theorem of the alternative, a characterization is given for all A satisfying the above equation. In addition, some necessary conditions for consistency and some related ideas are discussed. Finally, a method for constructing a solution D to the equation is given for matrices A satisfying certain conditions.

Structured low rank approximation

This paper concerns the construction of a structured low rank matrix that is nearest to a given matrix. The notion of structured low rank approximation arises in various applications, ranging from signal enhancement to protein folding to computer algebra, where the empirical data collected in a matrix do not maintain either the specified structure or the desirable rank as is expected in the original system. The task to retrieve useful information while maintaining the underlying physical feasibility often necessitates the search for a good structured lower rank approximation of the data matrix. This paper addresses some of the theoretical and numerical issues involved in the problem. Two procedures for constructing the nearest structured low rank matrix are proposed. The procedures are flexible enough that they can be applied to any lower rank, any linear structure, and any matrix norm in the measurement of nearness. The techniques can also be easily implemented by utilizing available optimization packages. The special case of symmetric Toeplitz structure using the Frobenius matrix norm is used to exemplify the ideas throughout the discussion. The concept, rather than the implementation details, is the main emphasis of the paper.

Least squares modifications with inverse factorizations: Parallel implications

Abstract The process of modifying least squares computations by updating the covariance matrix has been used in control and signal processing for some time in the context of linear sequential filtering. Here we give an alternative derivation of the process and provide extensions to downdating. Our purpose is to develop algorithms that are amenable to implementation on modern multiprocessor architectures. In particular, the inverse Cholesky factor R −1 is considered and it is shown that R −1 can be updated (downdated) by applying the same sequence of orthogonal (hyperbolic) plane rotations that are used to update (downdate) R . We have attempted to provide some new insights into least squares modification processes and to suggest parallel algorithms for implementing Kalman type sequential filters in the analysis and solution of estimation problems in control and signal processing.

FFT-based preconditioners for Toeplitz-block least squares problems

Discretized two-dimensional deconvolution problems arising, e.g., in image restoration and seismic tomography, can be formulated as least squares computations,

Large scale geodetic least squares adjustment by dissection and orthogonal decomposition

\min \| {b - Tx} \|_2