Nonnegative matrices have a myriad of applications in the biological, social, and physical genres. Of particular importance are the primitive matrices. A nonnegative matrix, M, is primitive exactly when there is a positive integer, k, such that M[superscript k] has only positive entries; that is, all the entries in M[superscript k] are strictly greater than zero. This method of determining if a matrix is primitive uses matrix multiplication and so would require time ���(n[superscipt ��]) where ��>2.3 even if fast matrix multiplication were used. Our goal is to find a much faster algorithm. This can be achieved by viewing a nonnegative matrix, M, as the adjacency matrix for a graph, G(M). The matrix, M, is primitive if and only if G(M) is strongly connected and the greatest common divisor of the cycle lengths in G(M) is 1. We devised an algorithm based in breadth-first search which finds a set of cycle lengths whose gcd is the same as that of G(M). This algorithm has runtime O(e) where e is the number of nonzero entries in M and therefore equivalent to the number of edges in G(M). A proof is given shown the runtime of O(n + e) along with some empirical evidence that supports this finding. / Graduation date: 2003

Thesis (Ph. D.)--Washington State University, December 2009. / Title from PDF title page (viewed on Dec. 28, 2009). "Department of Mathematics." Includes bibliographical references (p. 101-104).

Thesis (Ph. D.)--Washington State University, August 2008. / Includes bibliographical references (p. 86-88).

published_or_final_version / Mathematics / Master / Master of Philosophy

Thesis (M. Phil.)--University of Hong Kong, 2008. / Includes bibliographical references (leaf 57-58) Also available in print.

No description available.

The problems we studied in this thesis are linearly constrained convex programming (LCCP) and nonnegative matrix factorization (NMF). The resolutions of these two problems are all closely related to convex programming with simple convex constraints. The work can mainly be described in the following three parts. Firstly, an interior point algorithm following a parameterized central path for linearly constrained convex programming is proposed. The convergence and polynomial-time complexity are proved under the assumption that the Hessian of the objective function is locally Lipschitz continuous. Also, an initialization strategy is proposed, and some numerical results are provided to show the efficiency of the proposed algorithm. Secondly, the path following algorithm is promoted for general barrier functions. A class of barrier functions is proposed, and their corresponding paths are proved to be continuous and converge to optimal solutions. Applying the path following algorithm to these paths provide more flexibility to interior point methods. With some adjustments, the initialization method is equipped to validate implementation and convergence. Thirdly, we study the convergence of hierarchical alternating least squares algorithm (HALS) and its fast form (Fast HALS) for nonnegative matrix factorization. The coordinate descend idea for these algorithms is restated. With a precise estimation of objective reduction, some limiting properties are illustrated. The accumulation points are proved to be stationary points, and some adjustments are proposed to improve the implementation and efficiency

Thesis (Ph.D)--Industrial and Systems Engineering, Georgia Institute of Technology, 2010. / Committee Chair: Earl Barnes; Committee Member: Arkadi Nemirovski; Committee Member: Faiz Al-Khayyal; Committee Member: Guillermo H. Goldsztein; Committee Member: Joel Sokol. Part of the SMARTech Electronic Thesis and Dissertation Collection.

In this document, we study the nonnegative least squares primal-dual method for solving linear programming problems. In particular, we investigate connections between this primal-dual method and the classical Hungarian method for the assignment problem. Firstly, we devise a fast procedure for computing the unrestricted least squares solution of a bipartite matching problem by exploiting the special structure of the incidence matrix of a bipartite graph. Moreover, we explain how to extract a solution for the cardinality matching problem from the nonnegative least squares solution. We also give an efficient procedure for solving the cardinality matching problem on general graphs using the nonnegative least squares approach. Next we look into some theoretical results concerning the minimization of p-norms, and separable differentiable convex functions, subject to linear constraints described by node-arc incidence matrices for graphs. Our main result is the reduction of the assignment problem to a single nonnegative least squares problem. This means that the primal-dual approach can be made to converge in one step for the assignment problem. This method does not reduce the primal-dual approach to one step for general linear programming problems, but it appears to give a good starting dual feasible point for the general problem.

Cette thèse vise à proposer de nouveaux modèles pour la séparation de sources dans le cadre non linéaire des méthodes à noyaux en apprentissage statistique, et à développer des algorithmes associés. Le domaine d'application privilégié est le démélange en imagerie hyperspectrale. Tout d'abord, nous décrivons un modèle original de la factorisation en matrices non négatives (NMF), en se basant sur les méthodes à noyaux. Le modèle proposé surmonte la malédiction de préimage, un problème inverse hérité des méthodes à noyaux. Dans le même cadre proposé, plusieurs extensions sont développées pour intégrer les principales contraintes soulevées par les images hyperspectrales. Pour traiter des masses de données, des algorithmes de traitement en ligne sont développés afin d'assurer une complexité calculatoire fixée. Également, nous proposons une approche de factorisation bi-objective qui permet de combiner les modèles de démélange linéaire et non linéaire, où les décompositions de NMF conventionnelle et à noyaux sont réalisées simultanément. La dernière partie se concentre sur le démélange robuste aux bandes spectrales aberrantes. En décrivant le démélange selon le principe de la maximisation de la correntropie, deux problèmes de démélange robuste sont traités sous différentes contraintes soulevées par le problème de démélange hyperspectral. Des algorithmes de type directions alternées sont utilisés pour résoudre les problèmes d'optimisation associés / This thesis aims to propose new nonlinear unmixing models within the framework of kernel methods and to develop associated algorithms, in order to address the hyperspectral unmixing problem.First, we investigate a novel kernel-based nonnegative matrix factorization (NMF) model, that circumvents the pre-image problem inherited from the kernel machines. Within the proposed framework, several extensions are developed to incorporate common constraints raised in hypersepctral images analysis. In order to tackle large-scale and streaming data, we next extend the kernel-based NMF to an online fashion, by keeping a fixed and tractable complexity. Moreover, we propose a bi-objective NMF model as an attempt to combine the linear and nonlinear unmixing models. The decompositions of both the conventional NMF and the kernel-based NMF are performed simultaneously. The last part of this thesis studies a supervised unmixing model, based on the correntropy maximization principle. This model is shown robust to outlier bands. Two correntropy-based unmixing problems are addressed, considering different constraints in hyperspectral unmixing problem. The alternating direction method of multipliers (ADMM) is investigated to solve the related optimization problems