Probabilistic Matrix Factorization from Quantized Measurements

We consider the problem of factorizing a matrix with discrete-valued entries as a product of two low-rank matrices. Under a probabilistic framework, we seek for the minimum mean-square error estimates of these matrices, using full Bayes and empirical Bayes approaches. In the first case, we devise an integration scheme based on the Gibbs sampler that accounts also for hyperparameter and noise variance estimation. A similar technique is used also for the latter case, where we combine Gibbs sampling with the expectation-maximization (EM) algorithm to estimate the model parameters via marginal likelihood maximization. Extension to the case of missing values is also discussed. The proposed methods are evaluated on simulated data, and on a real data set for recommender systems.