Probabilistic matrix tri-factorization

Nonnegative matrix tri-factorization (NMTF) is a 3-factor decomposition of a nonnegative data matrix, X ap USV^T, where factor matrices, U, S, and V , are restricted to be nonnegative as well. Motivated by the aspect model used for dyadic data analysis as well as in probabilistic latent semantic analysis (PLSA), we present a probabilistic model with two dependent latent variables for NMTF, referred to as probabilistic matrix tri-factorization (PMTF). Each latent variable in the model is associated with the cluster variable for the corresponding object in the dyad, leading the model suited to co-clustering. We develop an EM algorithm to learn the PMTF model, showing its equivalence to multiplicative updates derived by an algebraic approach. We demonstrate the useful behavior of PMTF in a task of document clustering. Moreover, we incorporate the likelihood in the PMTF model into existing information criteria so that the number of clusters can be detected, while the algebraic NMTF cannot.