Relaxed Leverage Sampling for Low-rank Matrix Completion

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 norm minimization framework. Such low-rank matrices have degrees of freedom $(m+n)\varrho - \varrho^2$. We show that such arbitrary low-rank matrices can be recovered exactly from a small subset of $\Theta\left(((m+n)\varrho - \varrho^2)\log^2(m+n)\right)$ randomly sampled entries, thus matching the lower bound on the required number of entries (in terms of degrees of freedom), with an additional factor of $O(\log^2(m+n))$. The above bound on sample size is achieved if each entry is observed according to probabilities proportional to the sum of corresponding row and column leverage scores, minus their product. We show that this relaxation in sampling probabilities (as opposed to sum of leverage scores in Chen et al, 2014) gives us an additive improvement on the (best known) sample size obtained by Chen et al, 2014, for the nuclear norm minimization. Further, exact recovery of $(a)$ incoherent matrices (with restricted leverage scores), and $(b)$ matrices with only one of the row or column spaces to be incoherent, can be performed using our relaxed leverage score sampling, via nuclear norm minimization, without knowing the leverage scores a priori. In such settings also we achieve additive improvement on sample size.