Matrix Completion with Noise via Leveraged Sampling

Many matrix completion methods assume that the data follows the uniform distribution. To address the limitation of this assumption, Chen et al. \cite{Chen20152999} propose to recover the matrix where the data follows the specific biased distribution. Unfortunately, in most real-world applications, the recovery of a data matrix appears to be incomplete, and perhaps even corrupted information. This paper considers the recovery of a low-rank matrix, where some observed entries are sampled in a \emph{biased distribution} suitably dependent on \emph{leverage scores} of a matrix, and some observed entries are uniformly corrupted. Our theoretical findings show that we can provably recover an unknown $n\times n$ matrix of rank $r$ from just about $O(nr\log^2 n)$ entries even when the few observed entries are corrupted with a small amount of noisy information. Empirical studies verify our theoretical results.