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...

Recommender system has attracted lots of attentions since it helps users alleviate the information overload problem. Matrix factorization technique is one of the most widely employed collaborative filtering techniques in the research of recommender systems due to its effectiveness and efficiency in dealing with very large user-item rating matrices. Recently, based on the intuition that addition...

Non-negative Matrix Factorization (NMF) is a part-based image representation method which adds a non-negativity constraint to matrix factorization. NMF is compatible with the intuitive notion of combining parts to form a whole face. In this paper, we propose a framework of face recognition by adding NMF constraint and classifier constraints to matrix factorization to get both intuitive features...

This paper introduces the interpolative butterfly factorization for nearly optimal implementation of several transforms in harmonic analysis, when their explicit formulas satisfy certain analytic properties and the matrix representations of these transforms satisfy a complementary low-rank property. A preliminary interpolative butterfly factorization is constructed based on interpolative low-ra...

Recently, variational Bayesian (VB) techniques have been applied to probabilistic matrix factorization and shown to perform very well in experiments. In this paper, we theoretically elucidate properties of the VB matrix factorization (VBMF) method. Through finite-sample analysis of the VBMF estimator, we show that two types of shrinkage factors exist in the VBMF estimator: the positive-part Jam...

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...

Matrix factorization-based approaches have proven to be efficient for recommender systems. However, due to the time complexity in composing recommendations, matrix factorization-based approaches are inefficient in dealing with large scale datasets. In this paper, we present a new similar user index-based matrix factorization approach for large scale recommender systems. Finding similar users is...

We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the positive semidefinite rank, and their symmetric analogues: the completely positive rank and the completely positive semidefinite rank. We study the convergence properties of o...

Chordal graphs play a central role in techniques for exploiting sparsity in large semidefinite optimization problems and in related convex optimization problems involving sparse positive semidefinite matrices. Chordal graph properties are also fundamental to several classical results in combinatorial optimization, linear algebra, statistics, signal processing, machine learning, and nonlinear op...

Nonnegative matrix factorization NMF is a popular tool for analyzing the latent structure of nonnegative data. For a positive pairwise similarity matrix, symmetric NMF SNMF and weighted NMF WNMF can be used to cluster the data. However, both of them are not very efficient for the ill-structured pairwise similarity matrix. In this paper, a novel model, called relationship matrix nonnegative deco...