Given a finite set {M0, . . . ,Md−1} of nonnegative 2× 2 matrices and a nonnegative column-vector V , we associate to each (ωn) ∈ {0, . . . , d− 1}N the sequence of the column-vectors Mω1 . . .MωnV ‖Mω1 . . .MωnV ‖ . We give the necessary and sufficient condition on the matrices Mk and the vector V for this sequence to converge for all (ωn) ∈ {0, . . . , d− 1}N such that ∀n, Mω1 . . .MωnV 6= ( ...

The nonnegative rank of an entrywise nonnegative matrix A ∈ Rm×n + is the smallest integer r such that A can be written as A = UV where U ∈ Rm×r + and V ∈ Rr×n + are both nonnegative. The nonnegative rank arises in different areas such as combinatorial optimization and communication complexity. Computing this quantity is NP-hard in general and it is thus important to find efficient bounding tec...

Let V ∈ R be a nonnegative matrix. The nonnegative matrix factorization (NNMF) problem consists of finding nonnegative matrix factors W ∈ R and H ∈ R such that V ≈ WH. Lee and Seung proposed two algorithms which find nonnegative W and H such that ‖V −WH‖F is minimized. After examining the case in which r = 1 about which a complete characterization of the solution is possible, we consider the ca...

Let η = (η1, η2, . . . , ηt) and ν = (ν1, ν2, . . . , νt) be two sequences of nonnegative integers. (Append zeros if necessary to the end of the shorter sequence so that they are the same length.) We say that ν is majorized by η if ∑j l=1 νl ≤ ∑j l=1 ηl for all 1 ≤ j ≤ t and tl=1 νl = ∑t l=1 ηl. We write ν 1 η. Let Γ = (V,E) be a graph where V is a finite vertex set and E ⊆ V ×V is an edge set....

We show that there is p 0 > 0 and p 1 ; : : :; p N non-negative integers, such that 0 p(x) = p 0 + p 1 cos x + + p N cos Nx and p 0 s 1=3 for s = P N j=0 p j , improving on a result of Odlyzko who showed the existence of such a polynomial p that satisses p 0 (s log s) 1=3. Our result implies an improvement of the best known estimate for a problem of Erdd os and Szekeres. As our method is non-co...

Nonnegative Matrix Factorization (NMF) is an efficient tool for clustering and supervised classification of various objects, including text document, musical recordings, gene expressions, and images. In this paper, we are concerned with supervised classification of texture patterns. NMF is used for creating localized nonnegative feature vectors and low-dimensional nonnegative encoding vectors f...

Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative n×m matrix M into a product of a nonnegative n × d matrix W and a nonnegative d ×m matrix H. A longstanding open question, posed by Cohen and Rothblum in 1993, is whether a rational matrix M always has an NMF of minimal inner dimension d whose factors W and H are also rational. We answer this question negat...

In this work, we analyze the behavior of the active-set method for the nonnegative regularization of discrete ill-posed problems. In many applications, the solution of a linear ill-posed problem is known to be nonnegative. Standard Tikhonov regularization often provides an approximated solution with negative entries. We apply the activeset method to find a nonnegative approximate solution of th...

Nonnegative matrix factorization (NMF) is the problem of decomposing a given nonnegative n×m matrix M into a product of a nonnegative n× d matrix W and a nonnegative d ×m matrix H . A longstanding open question, posed by Cohen and Rothblum in 1993, is whether a rational matrix M always has an NMF of minimal inner dimension d whose factors W and H are also rational. We answer this question negat...

Nonnegative matrix factorization (NMF) is a popular method for multivariate analysis of nonnegative data, the goal of which is decompose a data matrix into a product of two factor matrices with all entries in factor matrices restricted to be nonnegative. NMF was shown to be useful in a task of clustering (especially document clustering). In this paper we present an algorithm for orthogonal nonn...