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 in (\ref{eqn:main_problem}) (\cite{CR09}). 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, \textit{minus their product} (see (\ref{eqn:main_probability})). We show that this relaxation in sampling probabilities (as opposed to sum of leverage scores in \cite{BCSW14}) gives us an \textit{additive improvement} on the (best known) sample size obtained by \cite{BCSW14} for the optimization problem in (\ref{eqn:main_problem}). 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 \textit{relaxed leverage score sampling}, via (\ref{eqn:main_problem}), {without knowing the leverage scores a priori}. In such settings also we achieve additive improvement on sample size.