Low-rank matrix completion has achieved great success in many real-world data applications. A matrix factorization model that learns latent features is usually employed and, to improve prediction performance, the similarities between latent variables can be exploited by pairwise learning using the graph regularized matrix factorization (GRMF) method. However, existing GRMF approaches often use the squared loss to measure the pairwise differences, which may be overly influenced by dissimilar pairs and lead to inferior prediction. To fully empower pairwise learning for matrix completion, we propose a general optimization framework that allows a rich class of (non-)convex pairwise penalty functions. A new and efficient algorithm is developed to solve the proposed optimization problem. We conduct extensive experiments on real recommender datasets to