Computing nonnegative rank factorizations

The existence of nonnegative generalized inverses in terms of nonnegative rank factorizations is considered. An algorithm is presented which computes a nonnegative rank factorization of a nonnegative matrix when a nonnegative 1-inverse exists.     

The nonnegative rank factorizations of nonnegative matrices

Let A ∈ P^{m × n}_r, the set of all m × n nonnegative matrices having the same rank r. For matrices A in P^{m × n}_n, we introduce the concepts of “A has only trivial nonnegative rank factorizations” and “A can have nontrivial nonnegative rank factorizations.” Correspondingly, the set P^{m × n}_n is divided into two disjoint subsets P₍₁₎ and P₍₂₎ such that P₍₁₎∪P₍₂₎ = P^{m × n}_n. It happens that the concept of “A has only trivial nonnegative rank factorizations” is a generalization of “A is prime in P^{n × n}_n.” We characterize the sets P₍₁₎ and P₍₂₎. Some of our results generalize some theorems in the paper of Daniel J. Richman and Hans Schneider [9].     

Real rank versus nonnegative rank

We consider the set of m×n nonnegative real matrices and define the nonnegative rank of a matrix A to be the minimum k such that A=BC where B is m×k and C is k×n. Given that the real rank of A is j for some j, we give bounds on the nonnegative rank of A and A².     

A note on factorizations of singular M-matrices

Supposing that M is a singular M-matrix, we show that there exists a permutation matrix P such that PMP^T = LU, where L is a lower triangular M-matrix and U is an upper triangular singular M-matrix. An example is given to illustrate that the above result is the best possible one.     

A note on the p-local rank

In this paper, we give an inductive definition of p-local rank of a p-block in a finite group G with p||G| and show a necessary and sufficient condition for a p-block B such that plr(B) = 2.Received: 10 March 2004     

A note on comparison theorems for nonnegative matrices

Summary In a recent paper, [4], Csordas and Varga have unified and extended earlier theorems, of Varga in [10] and Wonicki in [11], on the comparison of the asymptotic rates of convergence of two iteration matrices induced by two regular splittings. The main purpose of this note is to show a connection between the Csordas-Varga paper and a paper by Beauwens, [1], in which a comparison theorem is developed for the asymptotic rate of convergence of two nonnegative iteration matrices induced by two splittings which are not necessarily regular. Monotonic norms already used in [1] play an important role in our work here.Research supported in part by NSF grant number DMS-8400879     

Alternative gradient algorithms with applications to nonnegative matrix factorizations

Three nonnegative matrix factorization (NMF) algorithms are discussed and employed to three real-world applications. Based on the alternative gradient algorithm with the iteration steps being determined columnwisely without projection, and columnwisely and elementwisely with projections, three algorithms are developed respectively. Also, the computational costs and the convergence properties of the new algorithms are given. The numerical examples show the advantage of our algorithms over the multiplicative update algorithm proposed by Lee and Seung [11].     

A note on the rank of diagonals

For each natural number k?4, we construct a Tychonoff space with a rank k-diagonal but without a rank (k+1)-diagonal. This example proves a conjecture on rank of diagonal given by A.V. Arhangel'skii and R.Z. Buzyakova (2006) in [1] and answers some questions raised by them in the same paper.     

A note on the rank parity function

We provide some further theorems on the partitions generated by the rank parity function. New Bailey pairs are established, which are of independent interest.     

A note on the rank of semigroups

The rank of a semigroup $\mathcal{A}The rank of a semigroup A\mathcal{A} of functions from a finite set X to X is the minimum of |f(X)| over f ? Af\in \mathcal{A}. Given a finite set X and a subset Y of X, we show that if A\mathcal{A} is a semigroup of functions from X to X and ℬ a transitive semigroup of functions from Y to Y, then the rank of A\mathcal{A} divides that of ℬ provided that f(X)⊆Y for some f ? Af\in \mathcal{A} and that each function in ℬ is the restriction of a function in A\mathcal{A} to Y. To prove this, we generalize a result of Friedman which says that one can partition Y into q subsets of equal weight where q is the rank of ℬ. When one extends a transitive automaton by adding new states and letters, a similar condition guarantees that the rank of the extension divides the original rank.     

A note on the generalized rank reduction

We give constructive necessary and sufficient conditions for the validity of the generalized rank reduction formula of Ouellette [12]:

A note on characterizations of irreducibility of nonnegative matrices

Three sufficient conditions for the irreducibility of a matrix A are given, which for nonnegative A are also necessary.     

A note on rank functions and integer programming

An attempt is made, with only partial success, to obtain integer analogues of some linear-programming results that are useful in combinatorics.     

A note on the postulation of maximal rank curves

Summary Lets be the lowest degree of a surface containing a maximal rank curveY. We want to compare the dimension of the space of the surfaces of degrees containingY with the dimension of the space of the surfaces of degrees+1. We apply the bound that we find for getting a bound for the third Chern class of a rank-two reflexive sheaf onP _k ³ with seminatural cohomology, thus answering to a conjecture proposed byR. Hartshorne.

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Riassunto Sias il più basso grado di una superficie contenente una curva di rango massimoY. Si vuole confrontare la dimensione dello spazio delle superfici di grados contenentiY con la dimensione dello spazio delle superfici di grados+1. Applichiamo la limitazione trovata per ottenere una limitazione per la terza classe di Chern di un fascio riflessivo di rango 2 suP _k ³ con coomologia seminaturale, rispondendo così ad una congettura posta daR. Hartshorne.

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Résumé Soits le plus petit degré d’une surface contenante une curveY de rang maximum. Nous comparons la dimension de l’espace des surfaces de degrés contenantesY avec la dimension de l’espace des surfaces de degrés+1. Nous appliquons la borne trouvée pour obtenir une borne pour la troisième classe de Chern d’un faisceau réflexif de rang deux surP _k ³ à cohomologie seminaturelle. Ceci répond à une conjecture posée parR. Hartshorne.

     

A note on the split rank of intersection cuts

In this note, we present a simple geometric argument to determine a lower bound on the split rank of intersection cuts. As a first step of this argument, a polyhedral subset of the lattice-free convex set that is used to generate the intersection cut is constructed. We call this subset the restricted lattice-free set. It is then shown that élog₂ (l)ù{\lceil \log_2 (l)\rceil} is a lower bound on the split rank of the intersection cut, where l is the number of integer points lying on the boundary of the restricted lattice-free set satisfying the condition that no two points lie on the same facet of the restricted lattice-free set. The use of this result is illustrated by obtaining a lower bound of élog₂( n+1) ù{\lceil \log_2( n+1) \rceil} on the split rank of n-row mixing inequalities.     

A note on the characteristic rank of null-cobordant manifolds

We give an upper bound for the characteristic rank of smooth closed null-cobordant manifolds and apply it to the Grassmann manifolds of oriented vector subspaces in Euclidean space. The bound turns out to be sharp: it coincides with the exact value of characteristic rank for the Grassmann manifolds of oriented 2-dimensional vector subspaces in odd-dimensional Euclidean space.