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Abstract: In this paper we consider the Nonnegative Matrix Factorization (NMF) problem: given an (elementwise) nonnegative matrix find, for assigned k, nonnegative matrices and such that V = WH. Exact, nontrivial, nonnegative factorizations do not always exist, hence it is interesting to pose the approximate NMF problem. The criterion which is commonly employed is I-divergence between nonnegative matrices. The problem becomes that of finding, for assigned k, the factorization WH closest to V in I-divergence. An iterative algorithm, EM like, for the construction of the best pair (W, H) has been proposed in the literature. In this paper we interpret the algorithm as an alternating minimization procedure à la Csiszár–Tusnády and investigate some of its stability properties. NMF is widespreading as a data analysis method in applications for which the positivity constraint is relevant. There are other data analysis methods which impose some form of nonnegativity: we discuss here the connections between NMF and Archetypal Analysis. [Copyright &y& Elsevier]

We can define the adjacency algebra of an association scheme over an arbitrary field. It is not always semisimple over a field of positive characteristic. The structures of adjacency algebras over a field of positive characteristic have not been sufficiently studied. In this paper, we consider the structures of adjacency algebras of dual polar schemes over a field of positive characteristic and determine them when their algebras are local algebras. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we propose a new method to efficiently compute a representation of an orthogonal basis of the nullspace of a sparse matrix operator with , . We assume that B has full rank, i.e., rank. It is well-known that the last columns of the orthogonal matrix Q in a QR factorization form such a desired null basis. The orthogonal matrix Q can be represented either explicitly as a matrix, or implicitly as a matrix H of Householder vectors. Typically, the matrix H represents the orthogonal factor much more compactly than Q. We will employ this observation to design an efficient block algorithm that computes a sparse representation of the nullspace basis in almost optimal complexity. This new algorithm may, e.g., be used to construct a null space basis of the discrete divergence operator in the finite element context, and we will provide numerical results for this particular application. [Copyright &y& Elsevier]

Abstract: Koukouvinos et al. [C. Koukouvinos, M. Mitrouli, J. Seberry, Growth in Gaussian elimination for weighing matrices, W(n, n −1), Linear Algebra Appl. 306 (2000) 189–202], conjectured that the growth factor for Gaussian elimination of any completely pivoted weighing matrix of order n and weight n −1 is n −1 and that the first and last few pivots are for n >14. In the present paper we study the growth problem for skew and symmetric conference matrices. An algorithm for extending a k × k matrix with elements 0, ±1 to a skew and symmetric conference matrix of order n is described. By using this algorithm we show that the unique W(8,7) has two pivot structures. We also prove that the unique W(10,9) has three pivot patterns. [Copyright &y& Elsevier]

This paper presents a unified approach to the solution of the truncated matrix Stieltjes moment problem: Sk=∫0∞uk dσ(u) (k=0,…,m), in both the cases m=2n and m=2n−1, based on the use of the iterative Schur algorithm. [Copyright &y& Elsevier]

A novel upper bound for the permanent of (0,1)-matrices is obtained in this paper, by using an unbiased estimator of permanent [Random Structures Algorithms 5 (1994) 349]. It is a refinement of Minc’s very famous result, and apparently tighter than the current best general bound in some cases. [Copyright &y& Elsevier]