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Nonnegative matrix factorization (NMF) is a popular model in the field of pattern recognition. The aim is to find a low rank approximation for nonnegative matrix M by a product of two nonnegative matrices W and H. In general, NMF is NP-hard to solve while it can be solved efficiently under a separability assumption, which requires that the columns of the factor matrix are some columns of the input matrix M. In this paper, we generalize the separability assumption based on 3-factor NMF (M = P1SP2), and require that S is a submatrix of the input matrix M. We refer to this NMF as a Coseparable NMF (CoS-NMF). In the paper, we discuss and study mathematical properties of CoS-NMF, and present its relationships with other matrix factorizations such as generalized separable NMF, tri-symNMF, biorthogonal trifactorization and CUR decomposition. An optimization method for CoS-NMF is proposed, and an alternating fast gradient method is employed to determine the rows and the columns of M for the submatrix S. Numerical experiments on synthetic data sets, document data sets, and facial data sets are conducted to verify the effectiveness of the proposed CoS-NMF model. By comparison with state-of-the-art methods, the CoS-NMF model performs very well in a coclustering task by finding useful features, and keeps a good approximation to the input data matrix as well. [ABSTRACT FROM AUTHOR]

In this paper, we first give new model formulations for computing arbitrary generalized singular value of a Grassmann matrix pair or a real matrix pair. In these new formulations, we need to solve matrix optimization problems with unitary constraints or orthogonal constraints. We propose a geometric inexact Newton--conjugate gradient (Newton-CG) method for solving the resulting matrix optimization problems. Under some mild assumptions, we establish the global and quadratic convergence of the proposed method for the complex case. Some numerical examples are given to illustrate the effectiveness and high accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

In practical applications, incremental tensors are very common: only a portion of tensor data is available, and new data are arriving in the next time step or continuously over time. To handle this type of tensors time-saving algorithms are required for online computation. In this paper, we consider incremental CP ( CANDECOMP/PARAFAC ) decomposition, which requires one to update the CP decomposition after new tensor data, together with the exiting tensor, are ready for analysis. There exist several incremental CP decomposition algorithms, but almost all of these algorithms assume that the number of CP decomposition components remains fixed in the incremental process. The main contribution of this paper is the study of how to add components in the incremental CP decomposition. We derive the coordinate representation of the incremental CP decomposition with respect to a special basis and show related properties of tensor rank and the uniqueness of such incremental CP decomposition. Under the framework of this representation, the proposed method can be solved by using an alternating minimization algorithm. Numerical examples are presented to show the good performance of the proposed algorithms in terms of computational time and data fitting compared with existing methods. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to develop the quaternion generalized minimal residual method (QGMRES) for solving quaternion linear systems. Quaternion linear systems arise fromthree-dimensional or color imaging filtering problems. The proposed quaternion Arnoldi procedure can preserve quaternion Hessenberg form during the iterations. The main advantage is that the storage of the proposed iterative method can be reduced by comparing with the Hessenberg form constructed by the classical GMRES iterations for the real representation of quaternion linear systems. The convergence of the proposed QGMRES is also established. Numerical examples are presented to demonstrate the effectiveness of the proposed QGMRES compared with the traditional GMRES in terms of storage and computing time [ABSTRACT FROM AUTHOR]

Quaternion matrices are employed successfully in many color image processing applications. In particular, a pure quaternion matrix can be used to represent red, green, and blue channels of color images. A low-rank approximation for a pure quaternion matrix can be obtained by using the quaternion singular value decomposition. However, this approximation is not optimal in the sense that the resulting low-rank approximation matrix may not be pure quaternion, i.e., the low-rank matrix contains a real component which is not useful for the representation of a color image. The main contribution of this paper is to find an optimal rank-r pure quaternion matrix approximation for a pure quaternion matrix (a color image). Our idea is to use a pro jection on a low-rank quaternion matrix manifold and a projection on a quaternion matrix with zero real component, and develop an alternating pro jections algorithm to find such optimal low-rank pure quaternion matrix approximation. The convergence of the pro jection algorithm can be established by showing that the low-rank quaternion matrix manifold and the zero real component quaternion matrix manifold has a nontrivial intersection point. Numerical examples on synthetic pure quaternion matrices and color images are presented to illustrate the pro jection algorithm can find optimal low-rank pure quaternion approximation for pure quaternion matrices or color images. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse problem with quasi-boundary value regularization for recovering a source term of the time fractional diffusion equations from the final observation data. In particular, a two-by-two block linear system arising from the problem is studied. We propose a fast preconditioning technique by approximating the Schur complement in the system using a product of some factors, motivated by an approximate diagonalization of one of the blocks. The eigenvalues of the preconditioned system are shown to be clustered around 1, and the fast convergence of the methods is guaranteed theoretically. We also present an approach for selecting the regularization parameter of the quasi-boundary value method. Numerical experiments are carried out to demonstrate the effectiveness of our method. [ABSTRACT FROM AUTHOR]

In this paper, we study orthogonal nonnegative matrix factorization. We demonstrate the coefficient matrix can be sparse and low-rank in the orthogonal nonnegative matrix factorization. By using these properties, we propose to use a sparsity and nuclear norm minimization for the factorization and develop a convex optimization model for finding the coefficient matrix in the factorization. Numerical examples including synthetic and real-world data sets are presented to illustrate the effectiveness of the proposed algorithm and demonstrate that its performance is better than other testing methods. [ABSTRACT FROM AUTHOR]

In this paper, we study the Birkhoff-von Neumann theorem for a class of multistochastic tensors. In particular, we give a necessary and sufficient condition such that a multistochastic tensor is a convex combination of finitely many permutation tensors. It is well-known that extreme points in the set of doubly stochastic matrices are just permutation matrices. However, we find that extreme points in the set of multistochastic tensors are not just permutation tensors. We provide the other types of tensors contained in the set of extreme points. [ABSTRACT FROM AUTHOR]