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 ∈ Rn×n, that matrix can be reliably completed from just O(rn log 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(η(M) · rn log n), where η(M) is the largest leverage score of M . In this paper, we propose a twophase algorithm called MC 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. The total sample complexity of MC is provably smaller than uniform matrix completion—substantially so for reasonably coherent but wellconditioned matrices whose leverage scores exhibit mild decay. Numerical simulations suggest that the algorithm outperforms uniform matrix completion in a much broader class of matrices, and in particular, is much less sensitive to the condition number than our theory currently requires. At the same time, the two-phase algorithm is never worse in sample complexity than standard matrix completion by more than a constant factor, since most of the samples are drawn uniformly.