Upper Bound of Bayesian Generalization Error in Stochastic Matrix Factorization

Stochastic matrix factorization (SMF) has proposed and it can be understood as a restriction to non-negative matrix factorization (NMF). SMF is useful for inference of topic models, NMF for binary matrices data, and Bayesian Network. However, it needs some strong assumption to reach unique factorization in SMF and also theoretical prediction accuracy has not yet clarified. In this paper, we study the maximum pole of zeta function of SMF and derive an upper bound of the generalization error in Bayesian inference. This results give the foundation of establishing widely applicable and rigorous factorization method for SMF and mean that the generalization error in SMF can become smaller than regular statistical models by Bayesian inference.