Bayesian Matrix Factorization and Applications

Nonnegative matrix factorization (NMF) reduces the observed nonnegative matrix into a product of two nonnegative matrices. Nonnegativity entails two major implications: non-negative components and purely additive combination. These characteristics made this method useful in a wide range of applications. In this thesis, we propose two novel Bayesian nonnegative matrix factorization techniques. First, we propose a model dedicated to semi-bounded data where each entry of the observed matrix is supposed to follow an Inverted Beta distribution. Latent variables of the factorized parameter matrices follow a Gamma prior. Variational Bayesian inference and lower bound approximation for the objective function are used to find an analytically tractable solution for the model. An online extension of the algorithm is also proposed for more scalability. Both models are evaluated on five different applications. Second, we propose a Bayesian NMF that can be specifically useful for non intrusive load monitoring (NILM). NILM can be formulated as a source separation problem where the aggregated signal is expressed as linear combination of basis vectors in a matrix factorization framework. The model achieves superior performance by imposing sparsity on the activation matrix using Dirichlet priors. To estimate the parameters of the model, variational Bayesian inference is used. A novel optimization approach is proposed to find an analytically tractable solution for the model. We evaluate the model with three data sets: REDD, AMPds and IRISE, and with multiple experimental setups. The proposed model provides interpretability, flexibility and high performance