MC^2: A Two-Phase Algorithm for Leveraged Matrix Completion
Leverage scores, loosely speaking, reflect the importance of the rows and columns of a matrix. Ideally, given the leverage scores of a rank-$r$ matrix $M\in\mathbb{R}^{n\times n}$, that matrix can be reliably completed from just $O(rn\log^{2}n)$ samples if the samples are chosen randomly from a nonuniform distribution induced by the leverage scores. In practice, however, the leverage scores are often unknown a priori. As such, the sample complexity in uniform matrix completion---using uniform random sampling---increases to $O(\eta(M)\cdot rn\log^{2}n)$, where $\eta(M)$ is the largest leverage score of $M$. In this paper, we propose a two-phase algorithm called MC$^2$ for matrix completion: in the first phase, the leverage scores are estimated based on uniform random samples, and then in the second phase the matrix is resampled nonuniformly based on the estimated leverage scores and then completed. For well-conditioned matrices, the total sample complexity of MC$^2$ is no worse than uniform matrix completion, and for certain classes of well-conditioned matrices---namely, reasonably coherent matrices whose leverage scores exhibit mild decay---MC$^2$ requires substantially fewer samples. Numerical simulations suggest that the algorithm outperforms uniform matrix completion in a broad class of matrices, and in particular, is much less sensitive to the condition number than our theory currently requires.