Matrix completion from very limited information is an important machine learning topic, and has received extensive attention in various scientific applications. Matrix completion aims at finding a low-rank matrix to approximate the incomplete data matrix. However, noise in the data matrix may degrade the performance of the existing matrix completion algorithms, especially if there are different types of noise. In this paper, we proposed a robust matrix completion method with column outliers and sparse noise. The incomplete matrix is iteratively divided into low-rank and sparse parts. The ${\ell }_{2,1}$-norm based objective function makes the recovered matrix keeps a low-rank structure and lets the algorithm robust to column outliers, while the regularization term based on ${\ell }_1$-norm can alleviate the influence of sparse noise. Besides, a vector completion algorithm has been proposed to help us estimate the missing entries of the out-of-sample vectors. Moreover, the proposed model can be optimized by an efficient iterative re-weighted method, without introducing any additional parameters, while the adaptive weights obtained in the optimization process can help us detect column outliers. Both theoretical analysis and experiments based on synthetic datasets and real world datasets are implemented to validate the performance of the proposed method.

从非常有限的信息中完成矩阵是一个重要的机器学习主题，并在各种科学应用中受到广泛关注。矩阵补全旨在找到一个低秩矩阵来逼近不完整的数据矩阵。然而，数据矩阵中的噪声可能会降低现有矩阵补全算法的性能，尤其是在存在不同类型的噪声时。在本文中，我们提出了一种具有列异常值和稀疏噪声的稳健矩阵完成方法。不完整矩阵被迭代地划分为低秩和稀疏部分。这${\ell }_{2,1}$-norm based 目标函数使恢复矩阵保持低秩结构并使算法对列异常值具有鲁棒性，而正则化项基于 ${\ell }_1$-norm 可以减轻稀疏噪声的影响。此外，还提出了一种向量补全算法来帮助我们估计样本外向量的缺失条目。此外，所提出的模型可以通过高效的迭代重加权方法进行优化，无需引入任何额外的参数，而优化过程中获得的自适应权重可以帮助我们检测列异常值。实施了基于合成数据集和现实世界数据集的理论分析和实验，以验证所提出方法的性能。