It is known that positive operators Ω on a Hilbert space admit a factorization of the form Ω = W∗W, where W is an outer operator whose matrix representation is upper. As upper Hilbert space operators have an interpretation of transfer operators of linear time-varying systems in discrete time, this proves the existence of a spectral factorization for time-varying systems. In this paper, the abov...

We derive analytical expression of matrix factorization/completion solution by variational Bayes method, under the assumption that observed matrix is originally the product of low-rank dense and sparse matrices with additive noise. We assume the prior of sparse matrix is Laplace distribution by taking matrix sparsity into consideration. Then we use several approximations for derivation of matri...

Matrix factorization is one of the best approaches for collaborative filtering, because of its high accuracy in presenting users and items latent factors. The main disadvantages of matrix factorization are its complexity, and being very hard to be parallelized, specially with very large matrices. In this paper, we introduce a new method for collaborative filtering based on Matrix Factorization ...

We analyze skip-gram with negative-sampling (SGNS), a word embedding method introduced by Mikolov et al., and show that it is implicitly factorizing a word-context matrix, whose cells are the pointwise mutual information (PMI) of the respective word and context pairs, shifted by a global constant. We find that another embedding method, NCE, is implicitly factorizing a similar matrix, where each...

Matrix factorization or factor analysis is an important task helpful in the analysis of high dimensional real world data. There are several well known methods and algorithms for factorization of real data but many application areas including information retrieval, pattern recognition and data mining often require processing of binary rather than real data. Unfortunately, the methods used for re...

Matrix factorization problems over various semirings naturally arise in different contexts of modern pure and applied mathematics. These problems are very hard in general and cause computational difficulties in applications. We give a survey of what is known on the algorithmic complexity of Boolean, fuzzy, tropical, nonnegative, and positive semidefinite factorizations, and we examine the behav...

In this paper, we propose a data distortion strategy based on dual sparseness constrained Nonnegative Matrix Factorization (NMF). The dual sparseness constrained nonnegative matrix factorization model incorporates attached term constrain and positive symmetric matrix into NMF, which is different from the previous approaches. The goal of our study is data perturbation and we study the distortion...

Let Ax = b be a large, sparse, nonsymmetric system of linear equations. A new sparse approximate inverse preconditioning technique for such a class of systems is proposed. We show how the matrix A−1 0 −A−1, where A0 is a nonsingular matrix whose inverse is known or easy to compute, can be factorized in the form UΩV T using the Sherman–Morrison formula. When this factorization process is done in...

We use a multiscale approach to reduce the time to produce the nonnegative matrix factorization (NMF) of a matrix A, that is, A ≈ WH. We also investigate QR factorization as a method for initializing W during the iterative process for producing the nonnegative matrix factorization of A. Finally, we use our approach to produce nonnegative matrix factorizations for classifying images and compare ...

The Cayley transform of A, F = (I+A)−1 (I−A), is studied when A is a P -matrix, an M -matrix, an inverse M-matrix, a positive definite matrix, or a totally nonnegative matrix. Given a matrix A in each of these positivity classes and using the fact that the Cayley transform is an involution, properties of the ensuing factorization A = (I+F )−1 (I−F ) are examined. Specifically, it is investigate...