Completing any low-rank matrix, provably

Matrix completion concerns the recovery of a low-rank matrix from a subset of its revealedentries, and nuclear norm minimization has emerged as an effective surrogate for this combina-torial problem. Here, we show that nuclear norm minimization can recover an arbitrary n × nmatrix of rank r from O(nr log(n)) revealed entries, provided that revealed entries are drawnproportionally to the local row and column coherences (closely related to leverage scores) of theunderlying matrix. Our results are order-optimal up to logarithmic factors, and extend existingresults for nuclear norm minimization which require strong incoherence conditions on the typesof matrices that can be recovered, due to assumed uniformly distributed revealed entries. Wefurther provide extensive numerical evidence that a proposed two-phase sampling algorithm canperform nearly as well as local-coherence sampling and without requiring a priori knowledge ofthe matrix coherence structure. Finally, we apply our results to quantify how weighted nuclearnorm minimization can improve on unweighted minimization given an arbitrary set of sampledentries.