In this paper, we consider the problem of Robust Matrix Completion (RMC) where the goal is to recover a low-rank matrix by observing a small number of its entries out of which a few can be arbitrarily corrupted. We propose a simple projected gradient descent method to estimate the low-rank matrix that alternately performs a projected gradient descent step and cleans up a few of the corrupted entries using hard-thresholding. Our algorithm solves RMC using nearly optimal number of observations as well as nearly optimal number of corruptions. Our result also implies significant improvement over the existing time complexity bounds for the low-rank matrix completion problem. Finally, an application of our result to the robust PCA problem (low-rank+sparse matrix separation) leads to nearly linear time (in matrix dimensions) algorithm for the same; existing state-of-the-art methods require quadratic time. Our empirical results corroborate our theoretical results and show that even for moderate sized problems, our method for robust PCA is an an order of magnitude faster than the existing methods.

本文提出了一种简单的投影梯度下降方法来估计低秩矩阵，用于解决鲁棒矩阵完成问题，并且包括清除一些受损条目的步骤，并在低秩矩阵完成问题中获得了最优观测次数和最优破坏次数的解决方法。同时，本文的结果还意味着，对于低秩矩阵完成问题的时间复杂度界限，取得了重要的改进。最后，通过将结果应用于鲁棒PCA问题，得到了高效的解决方案

近乎最优的鲁棒矩阵完成