User ratings on items are noisy in real-world recommender systems, which raises challenges to matrix approximation (MA)-based collaborative filtering (CF) algorithms — the learned models will be easily biased to the noisy training data and yield low generalization performance. This paper proposes a noise-resilient matrix approximation (NORMA) method, which can achieve less biased matrix approximation and thus more accurate collaborative filtering. In NORMA, an adaptive weighting strategy is proposed to decrease the gradient updates of noisy ratings, so that the learned MA models will be less prone to the noisy ratings. Theoretical analyses show that NORMA can achieve better generalization performance than standard matrix approximation methods. Experimental studies on real-world datasets demonstrate that NORMA can outperform state-of-the-art matrix approximation-based collaborative filtering methods in recommendation accuracy.