Variational Bayes method for matrix factorization to two sparse factorized matrices

Matrix factorization is the problem of factorizing a given observed matrix into two matrices, and this can be applied to various fields in information science. We use variational Bayes method for the study of this problem. In the past study of variational Bayes analysis, multivariate Gaussian prior is used for low-rank matrices for convenience of analysis. In this article, we make another assumption for sparse matrix problem: the observation matrix is the multiplication of two sparse matrices, whose priors are exponential distributions for describing sparsity and non-negativity. Under this assumption, we obtain the analytical expression of factorized matrices with several approximations. We also conduct numerical experiment to observe the property of factorized matrices via our analysis.