MC^2: A Two-Phase Algorithm for Leveraged Matrix Completion

TL;DR

The paper introduces MC^2, a two-phase algorithm that estimates leverage scores and uses them for efficient matrix completion, reducing sample complexity especially for well-conditioned and coherent matrices.

Contribution

It proposes a novel two-phase approach for matrix completion that estimates leverage scores and improves sampling efficiency over uniform methods.

Findings

MC^2 outperforms uniform matrix completion in simulations.

The algorithm is less sensitive to matrix condition number.

For well-conditioned matrices, MC^2 matches the sample complexity of ideal leverage-based methods.

Abstract

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.