Hierarchical variational Bayesian matrix co-factorization

Matrix co-factorization involves jointly decomposing several data matrices to approximate each data matrix as a product of two factor matrices, sharing some factor matrices in the factorization. We have recently developed variational Bayesian matrix co-factorization where factor matrices are inferred by computing variational posterior distributions in the case of Gaussian likelihood with Gaussian prior placed on factor matrices. Empirical Bayesian method was used, so hyperparameters are set to specific values determined by maximizing marginal likelihood. In this paper we present a hierarchical Bayesian model for matrix co-factorization in which we derive a variational inference algorithm to approximately compute posterior distributions over factor matrices as well as hyperparameters, placing Gaussian-Wishart prior on hyperparameters. Numerical experiments on MovieLens data demonstrate that the hierarchical variational Bayesian matrix co-factorization alleviates the over-fitting better than the empirical variational Bayesian matrix co-factorization, leading to the improved performance in terms of MAE and RMSE.