Matrix Completion under Gaussian Models Using MAP and EM Algorithms

Completing a partially-known matrix (matrix completion) is an important problem in the field of data mining and signal processing, and has been successfully applied to sensor localization and recommendation system. Low-rank and factorization models are the two most popular and successful classes of models used for matrix completion. In this paper, we investigate another approach based on statistical estimation which has previously been used for matrix completion. In an initial work involving Gaussian models (GM), the formulation was inaccurate necessitating an ad-hoc empirical diagonal loading to a covariance matrix, requiring additional tuning, and making the final estimate of model parameters difficult to interpret. An accurate formulation using a correct objective function based on likelihood estimation already exists in statistical literature, which we utilize here to learn the model parameters using an Expectation Maximization (EM) algorithm. This approach no longer needs tuning and performs better in the numerical experiments. Owing to the difference that stems from the difference in choice of objective function, we note that the original method leads to an underestimated covariance matrix necessitating an artificial diagonal loading, while the method we use provides a Maximum Likelihood (ML) estimate of the model parameters. We also validate our approach using realworld data from MovieLens, EachMovie and Netflix.