Probabilistic Latent Tensor Factorization (PLTF) is a recently proposed probabilistic framework for modelling multi-way data. Not only the common tensor factorization models but also any arbitrary tensor factorization structure can be realized by the PLTF framework. This paper presents full Bayesian inference via variational Bayes that facilitates more powerful modelling and allows more sophisticated inference on the PLTF framework. We illustrate our approach on model order selection and link prediction.
Matrix factorization is a popular method for collaborative prediction, where unknown ratings are predicted by user and item factor matrices which are determined to approximate a user-item matrix as their product. Bayesian matrix factorization is preferred over other methods for collaborative filtering, since Bayesian approach alleviates overfitting, integrating out all model parameters using variational inference or sampling methods. However, Bayesian matrix factorization still suffers from the cold-start problem where predictions of ratings for new items or of new users' preferences are required. In this paper we present Bayesian matrix co-factorization as an approach to exploiting side information such as content information and demographic user data, where multiple data matrices are jointly decomposed, i.e., each Bayesian decomposition is coupled by sharing some factor matrices. We derive variational inference algorithm for Bayesian matrix co-factorization. In addition, we compute Bayesian Cramer-Rao bound in the case of Gaussian likelihood, showing that Bayesian matrix co-factorization indeed improves the reconstruction over Bayesian factorization of single data matrix. Numerical experiments demonstrate the useful behavior of Bayesian matrix co-factorization in the case of cold-start problems. Bayesian methods of matrix factorization (MF) have been actively explored recently as promising alternatives to classical singular value decomposition. In this paper, we show that, despite the fact that the optimization problem is non-convex, the global optimal solution of variational Bayesian (VB) MF can be computed analytically by solving a quartic equation. This is highly advantageous over a popular VBMF algorithm based on iterated conditional modes since it can only find a local optimal solution after iterations. We further show that the global optimal solution of empirical VBMF (hyperparameters are also learned from data) can also be analytically computed. We illustrate the usefulness of our results through experiments.