This paper studies noisy low-rank matrix completion: given partial and corrupted entries of a large low-rank matrix, the goal is to estimate the underlying matrix faithfully and efficiently. Arguably one of the most popular paradigms to tackle this problem is convex relaxation, which achieves remarkable efficacy in practice. However, the theoretical support of this approach is still far from optimal in the noisy setting, falling short of explaining the empirical success. We make progress towards demystifying the practical efficacy of convex relaxation vis-\`a-vis random noise. When the rank of the unknown matrix is a constant, we demonstrate that the convex programming approach achieves near-optimal estimation errors --- in terms of the Euclidean loss, the entrywise loss, and the spectral norm loss --- for a wide range of noise levels. All of this is enabled by bridging convex relaxation with the nonconvex Burer-Monteiro approach, a seemingly distinct algorithmic paradigm that is provably robust against noise. More specifically, we show that an approximate critical point of the nonconvex formulation serves as an extremely tight approximation of the convex solution, allowing us to transfer the desired statistical guarantees of the nonconvex approach to its convex counterpart.

本文研究了针对大规模低秩矩阵的部分和带噪声数据中的矩阵补全问题，采用凸松弛和Burer-Monteiro方法，成功地将凸松弛的实践与非凸方法的统计保证相结合，取得了近乎最优的估计误差。

噪声矩阵填充：通过非凸优化理解对凸松弛的统计保证