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We consider the problem of exact recovery of any $m\times n$ matrix of rank$\varrho$ from a small number of observed entries via the standard nuclear normminimization framework. Such low-rank matrices have degrees of freedom$(m+n)\varrho - \varrho^2$. We show that any arbitrary low-rank matrices can berecovered exactly from a $\Theta\left(((m+n)\varrho -\varrho^2)\log^2(m+n)\right)$ randomly sampled entries, thus matching the lowerbound on the required number of entries (in terms of degrees of freedom), withan additional factor of $O(\log^2(m+n))$. To achieve this bound on sample sizewe observe each entry with probabilities proportional to the sum ofcorresponding row and column leverage scores, minus their product. We show thatthis relaxation in sampling probabilities (as opposed to sum of leverage scoresin Chen et al, 2014) can give us an $O(\varrho^2\log^2(m+n))$ additiveimprovement on the (best known) sample size obtained by Chen et al, 2014, forthe nuclear norm minimization. Experiments on real data corroborate thetheoretical improvement on sample size. Further, exact recovery of $(a)$incoherent matrices (with restricted leverage scores), and $(b)$ matrices withonly one of the row or column spaces to be incoherent, can be performed usingour relaxed leverage score sampling, via nuclear norm minimization, withoutknowing the leverage scores a priori. In such settings also we can achieveimprovement on sample size.

Relaxed Leverage Sampling for Low-rank Matrix Completion