Learning from Incomplete Ratings Using Non-negative Matrix Factorization

We use a low-dimensional linear model to describe the user rating matrix in a recommendation system. A non-negativity constraint is enforced in the linear model to ensure that each user's rating profile can be represented as an additive linear combination of canonical coordinates. In order to learn such a constrained linear model from an incomplete rating matrix, we introduce two variations on Non-negative Matrix Factorization (NMF): one based on the Expectation-Maximization (EM) procedure and the other a Weighted Nonnegative Matrix Factorization (WNMF). Based on our experiments, the EM procedure converges well empirically and is less susceptible to the initial starting conditions than WNMF, but the latter is much more computationally efficient. Taking into account the advantages of both algorithms, a hybrid approach is presented and shown to be effective in real data sets. Overall, the NMF-based algorithms obtain the best prediction performance compared with other popular collaborative filtering algorithms in our experiments; the resulting linear models also contain useful patterns and features corresponding to user communities.