In this paper, we derive the necessary and sufficient conditions for the quaternion matrix equation XA=B to have the least-square bisymmetric solution and give the expression of such solution when the solvability conditions are met. Futhermore, we consider the maximal and minimal inertias of the least-square bisymmetric solution to this equation. As applications, we derive sufficient and necess...

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...

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. NMF has a wide variety of applications, including bioinformatics, chemometrics, communication complexity, machine learning, polyhedral combinatorics, among many others. A longstanding open question, posed by Cohen an...

We consider the problem of constructing nonnegative matrices with prescribed extremal singular values. In particular, given 2n−1 real numbers σ ( j) 1 and σ ( j) j , j = 1, . . . , n, we construct an n×n nonnegative bidiagonal matrix B and an n×n nonnegative semi-bordered diagonal matrix C , such that σ ( j) 1 and σ ( j) j are, respectively, the minimal and the maximal singular values of certai...

An element A of the n× n copositive cone C is called irreducible with respect to the nonnegative cone N if it cannot be written as a nontrivial sum A = C + N of a copositive matrix C and an elementwise nonnegative matrix N . This property was studied by Baumert [2] who gave a characterisation of irreducible matrices. We demonstrate here that Baumert’s characterisation is incorrect and give a co...

We propose an extension of nonnegative matrix factorization (NMF) to multilayer network model for dynamic myocardial PET image analysis. NMF has been previously applied to the analysis and shown to successfully extract three cardiac components and time-activity curve from the image sequences. Here we apply triple nonnegative-matrix factorization to the dynamic PET images of dog and show details...

Nonnegative matrix factorisation and tri-factorisation Nonnegative matrix factorisation (NMF) and tri-factorisation (NMTF) methods decompose a given matrix R into two or three smaller matrices so that R ≈ UV T or R ≈ FSG , respectively. Schmidt, Winther and Hansen (2009) introduced a Bayesian version of nonnegative matrix factorisation (left), which we extend to matrix tri-factorisation (right)...

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...

In non-negative matrix factorization, it is difficult to find the optimal non-negative factor matrix in each iterative update. However, with the help of transformation matrix, it is able to derive the optimal non-negative factor matrix for the transformed cost function. Transformation matrix based nonnegative matrix factorization method is proposed and analyzed. It shows that this new method, w...

Nonnegative matrix factorization (NMF) aims at factoring a data matrix into low-rank latent factor matrices with nonnegativity constraints on (one or both of) the factors. Specifically, given a data matrix X ∈ RM×N and a target rank R, NMF seeks a factorization model X ≈WH>, W ∈ RM×R, H ∈ RN×R, to ‘explain’ the data matrix X, where W ≥ 0 and/or H ≥ 0 and R ≤ min{M,N}. At first glance, NMF is no...