Recovering a low rank matrix from a subset of its entries, some of which may be corrupted, is known as the robust matrix completion (RMC) problem. Existing RMC methods have several limitations: they require a relatively large number of observed entries; they may fail under overparametrization, when their assumed rank is higher than the correct one; and many of them fail to recover even mildly ill-conditioned matrices. In this paper we propose a novel RMC method, denoted $\texttt{RGNMR}$, which overcomes these limitations. $\texttt{RGNMR}$ is a simple factorization-based iterative algorithm, which combines a Gauss-Newton linearization with removal of entries suspected to be outliers. On the theoretical front, we prove that under suitable assumptions, $\texttt{RGNMR}$ is guaranteed exact recovery of the underlying low rank matrix. Our theoretical results improve upon the best currently known for factorization-based methods. On the empirical front, we show via several simulations the advantages of $\texttt{RGNMR}$ over existing RMC methods, and in particular its ability to handle a small number of observed entries, overparameterization of the rank and ill-conditioned matrices.

翻译：从矩阵的部分条目（其中部分可能已损坏）中恢复低秩矩阵的问题被称为鲁棒矩阵补全（RMC）问题。现有RMC方法存在若干局限：需要相对较多的观测条目；在过参数化（即假设秩高于真实秩）时可能失效；且许多方法甚至无法恢复轻度病态矩阵。本文提出一种新型RMC方法，记为$\\texttt{RGNMR}$，该方法克服了上述局限。$\\texttt{RGNMR}$是一种基于因子化的简单迭代算法，结合了高斯-牛顿线性化与疑似异常值条目剔除机制。在理论层面，我们证明在适当假设下，$\\texttt{RGNMR}$能保证精确恢复潜在低秩矩阵，该理论结果改进了当前基于因子化方法的最佳已知结论。在实证层面，通过多组仿真实验展示了$\\texttt{RGNMR}$相较于现有RMC方法的优势，特别是其处理少量观测条目、秩过参数化及病态矩阵的能力。