Improving nonnegative matrix factorization with advanced graph regularization

Abstract

Nonnegative Matrix Factorization (NMF) produces interpretable solutions for many applications including collaborative filtering. Typically, regularization is needed to address issues such as overfitting and interpretability, especially for collaborative filtering where the rating matrices are sparse. However, the existing regularizers are typically constructed from the factorization results instead of the rating matrices. Intuitively, we regard these existing regularizers as representing either user factors or item factors and anticipate that a more holistic regularizer could improve the effectiveness of NMF. To this end, we propose a graph regularizer based on a linear projection of the rating matrix, and call the resulting method: Linear Projection and Graph Regularized Nonnegative Matrix Factorization (LPGNMF). We develop two iterative methods to minimize the cost function and derive two update rules named LPGNMF and F-LPGNMF. Additionally, we prove the value of the objective function decreases with LPGNMF and converges to a fixed point with F-LPGNMF. Finally, we test these methods against a number of NMF algorithms on different data sets and show both LPGNMF and F-LPGNMF always achieve smaller errors based on two different error measures.