A Converse to Low-Rank Matrix Completion

In many practical applications, one is given a subset Omega of the entries in a d x N data matrix X, and aims to infer all the missing entries. Existing theory in low-rank matrix completion (LRMC) provides conditions on X (e.g., bounded coherence or genericity) and Omega (e.g., uniform random sampling or deterministic combinatorial conditions) to guarantee that if X is rank-r, then X is the only rank-r matrix that agrees with the observed entries, and hence X can be uniquely recovered by some method (e.g., nuclear norm or alternating minimization). In many situations, though, one does not know beforehand the rank of X, and depending on X and Omega, there may be rank-r matrices that agree with the observed entries, even if X is not rank-r. Hence one can be deceived into thinking that X is rankrwhen it really is not. In this paper we give conditions on X (genericity) and a deterministic condition on Omega to guarantee that if there is a rank-r matrix that agrees with the observed entries, then X is indeed rank-r. While our condition on Omega is combinatorial, we provide a deterministic efficient algorithm to verify whether the condition is satisfied. Furthermore, this condition is satisfied with high probability under uniform random sampling schemes with only O(max {r; log d}) samples per column. This strengthens existing results in LRMC, allowing to drop the assumption that X is known a priori to be low-rank.