基于<italic>S</italic><sub>1/2</sub>建模的稳健稀疏-低秩矩阵分解
摘要
This paper introduces the S 1/2-norm for matrices to induce their lower rank, based on which a new model for robust sparse and low-rank matrix decomposition is proposed. To the best of our knowledge, this is the first time that the S 1/2-norm for matrices is used to characterize the low-rank property. Inspired by the alternating direction method of multipliers, we propose a computationally efficient algorithm, the alternating threshold iterative algorithm, for the new model. The proposed algorithm adopts the augmented Lagrange multiplier technique and iteratively updates both the low-rank and sparse components in explicit form, making the global computation accuracy and time cost controllable. Numerous numerical simulation experiments are presented to show that the new algorithm requires much less computation time to obtain a more robust decomposition. In addition, the rank of the low-rank component and sparsity of the obtained sparse matrix are much closer to their true values compared with those of the state-of-the-art algorithm, inexact ALM, in solving these problems. When applied to a background modeling application for surveillance video, the new algorithm recovers the background matrix with a lower rank, which is sort of consistent with the prior while modeling, while the time cost is only 10% of that of the inexact ALM algorithm. All these findings confirm that the new model and algorithm can solve practical problems more effectively and efficiently.