Weighted nonnegative matrix factorization

Nonnegative matrix factorization (NMF) is a widely-used method for low-rank approximation (LRA) of a nonnegative matrix (matrix with only nonnegative entries), where nonnegativity constraints are imposed on factor matrices in the decomposition. A large body of past work on NMF has focused on the case where the data matrix is complete. In practice, however, we often encounter with an incomplete data matrix where some entries are missing (e.g., a user-rating matrix). Weighted low-rank approximation (WLRA) has been studied to handle incomplete data matrix. However, there is only few work on weighted nonnegative matrix factorization (WNMF) that is WLRA with nonnegativity constraints. Existing WNMF methods are limited to a direct extension of NMF multiplicative updates, which suffer from slow convergence while the implementation is easy. In this paper we develop relatively fast and scalable algorithms for WNMF, borrowed from well-studied optimization techniques: (1) alternating nonnegative least squares; (2) generalized expectation maximization. Numerical experiments on MovieLens and Netflix prize datasets confirm the useful behavior of our methods, in a task of collaborative prediction.