Variational Bayesian Matrix Factorization for Bounded Support Data

A novel Bayesian matrix factorization method for bounded support data is presented. Each entry in the observation matrix is assumed to be beta distributed. As the beta distribution has two parameters, two parameter matrices can be obtained, which matrices contain only nonnegative values. In order to provide low-rank matrix factorization, the nonnegative matrix factorization (NMF) technique is applied. Furthermore, each entry in the factorized matrices, i.e., the basis and excitation matrices, is assigned with gamma prior. Therefore, we name this method as beta-gamma NMF (BG-NMF). Due to the integral expression of the gamma function, estimation of the posterior distribution in the BG-NMF model can not be presented by an analytically tractable solution. With the variational inference framework and the relative convexity property of the log-inverse-beta function, we propose a new lower-bound to approximate the objective function. With this new lower-bound, we derive an analytically tractable solution to approximately calculate the posterior distributions. Each of the approximated posterior distributions is also gamma distributed, which retains the conjugacy of the Bayesian estimation. In addition, a sparse BG-NMF can be obtained by including a sparseness constraint to the gamma prior. Evaluations with synthetic data and real life data demonstrate the good performance of the proposed method.