Recovering an unknown high dimensional low rank matrix from a small set of entries is widely spread in the fields of machine learning, system identification and image restoration, etc. In many practical applications, the few observations are always corrupted by noise and the noise level is also unknown. A novel model with nuclear norm and square root type estimator has been proposed, which does not rely on the knowledge or on an estimation of the standard deviation of the noise. In this paper, we firstly reformulate the problem to an equivalent variable separated form by introducing an auxiliary variable. Then we propose an efficient alternating direction method of multipliers(ADMM) for solving it. Both of resulting subproblems admit an explicit solution, which makes our algorithm have a cheap computing. Finally, the numerical results show the benefits of the model and the efficiency of the proposed method.

Recovering an unknown high dimensional low rank matrix from a small set of entries is widely spread in the fields of machine learning, system identification and image restoration, etc. In many practical applications, the few observations are always corrupted by noise and the noise level is also unknown. A novel model with nuclear norm and square root type estimator has been proposed, which does not rely on the knowledge or on an estimation of the standard deviation of the noise. In this paper, we firstly reformulate the problem to an equivalent variable separated form by introducing an auxiliary variable. Then we propose an efficient alternating direction method of multipliers(ADMM) for solving it. Both of resulting subproblems admit an explicit solution, which makes our algorithm have a cheap computing. Finally, the numerical results show the benefits of the model and the efficiency of the proposed method.