As we all known, the nonnegative matrix factorization (NMF) is a dimension reduction method that has been widely used in image processing, text compressing and signal processing etc. In this paper, an algorithm for nonnegative matrix approximation is proposed. This method mainly bases on the active set and the quasi-Newton type algorithm, by using the symmetric rank-one and negative curvature d...

The so-called parallel multisplitting nonstationary iterative Model A was introduced by Bru, Elsner, and Neumann [Linear Algebra and its Applications 103:175-192 (1988)] for solving a nonsingular linear system Ax = b using a weak nonnegative multisplitting of the first type. In this paper new results are introduced when A is a monotone matrix using a weak nonnegative multisplitting of the secon...

Nonnegative matrix factorization (NMF) is a popular technique for pattern recognition, data analysis, and dimensionality reduction, the goal of which is to decompose nonnegative data matrix X into a product of basis matrix A and encoding variable matrix S with both A and S allowed to have only nonnegative elements. In this paper we consider Amari’s α-divergence as a discrepancy measure and rigo...

The notion of low rank approximations arises from many important applications. When the low rank data are further required to comprise nonnegative values only, the approach by nonnegative matrix factorization is particularly appealing. This paper intends to bring about three points. First, the theoretical Kuhn-Tucker optimality condition is described in explicit form. Secondly, a number of nume...

We investigate (0, 1)-matrices which are totally nonnegative and therefore which have all of their eigenvalues equal to nonnegative real numbers. Such matrices are characterized by four forbidden submatrices (of orders 2 and 3). We show that the maximum number of 0s in an irreducible (0, 1)-matrix of order n is (n − 1) and characterize those matrices with this number of 0s. We also show that th...

A new matrix factorization algorithm which combines two recently proposed nonnegative learning techniques is presented. Our new algorithm, α-PNMF, inherits the advantages of Projective Nonnegative Matrix Factorization (PNMF) for learning a highly orthogonal factor matrix. When the Kullback-Leibler (KL) divergence is generalized to αdivergence, it gives our method more flexibility in approximati...

The purpose of this paper is to investigate the interplay arising between max algebra, convexity and scaling problems. The latter, which have been studied in nonnegative matrix theory, are strongly related to max algebra. One problem is strict visualisation scaling, which means finding, for a given nonnegative matrix A, a diagonal matrix X such that all elements of XAX are less than or equal to...

An Alternating Rank-k Nonnegative Least Squares Framework (ARkNLS) for Matrix Factorization

A matrix is $k$-nonnegative if all its minors of size $k$ or less are nonnegative. We give a parametrized set generators and relations for the semigroup $n\times n$ invertible matrices in two special cases: when $k = n-1$ n-2$, restricted to unitriangular matrices. For these cases, we prove that can be partitioned into cells based on their factorizations generators, generalizing notion Bruhat f...

Nonnegative Matrix Factorization (NMF) is a widely used technique in many applications such as face recognition, motion segmentation, etc. It approximates the nonnegative data in an original high dimensional space with a linear representation in a low dimensional space by using the product of two nonnegative matrices. In many applications data are often partially corrupted with large additive n...