Coherent Matrix Completion

Abstract

Matrix completion concerns the recovery of a low-rank matrix from a subset of its revealed entries, and nuclear norm minimization has emerged as an effective surrogate for this combinatorial problem. Here, we show that nuclear norm minimization can recover an arbitrary n \times n matrix of rank r from O(nr log^2(n)) revealed entries, provided that revealed entries are drawn proportionally to the local row and column coherences (closely related to leverage scores) of the underlying matrix. Our results are order-optimal up to logarithmic factors, and extend existing results for nuclear norm minimization which require strong incoherence conditions on the types of matrices that can be recovered, due to assumed uniformly distributed revealed entries. We further provide extensive numerical evidence that a proposed two-phase sampling algorithm can perform nearly as well as local-coherence sampling and without requiring a priori knowledge of the matrix coherence structure. Finally, we apply our results to quantify how weighted nuclear norm minimization can improve on unweighted minimization given an arbitrary set of sampled entries.