Partially occluded or illuminated faces pose a significant obstacle for robust, real-world face recognition. The problem of how to characterize the error caused by occlusion or illumination is still a challenging task. There must exist some close relationship between the error metric and error distribution. However, some metric (e.g. Z2-norm) can't characterize this error distribution completely. By some experiments, we found that nuclear norm is more suitable for characterizing the occluded or illuminated error distribution. Thus, a nuclear norm regularized sparse coding model is presented. Such a problem is solved by using ALM (or ADMM). In addition, we use nuclear norm as a metric to characterize the distance between reconstruction samples and classes. The experiments for image classification and face reconstruction demonstrate that our algorithm is robust to some face variations such as occlusion and illumination, and thus can act as a fast solver for matrix regression problem.