Partial Matrix Completion

In the matrix completion problem, one wishes to reconstruct a low-rank matrix based on a revealed set of (possibly noisy) entries. Prior work considers completing the entire matrix, which may be highly inaccurate in the common case where the distribution over entries is non-uniform. We formalize the problem of Partial Matrix Completion where the goal is to complete a large subset of the entries, or equivalently to complete the entire matrix and specify an accurate subset of the entries. Interestingly, even though the distribution is unknown and arbitrarily complex, our efficient algorithm is able to guarantee: (a) high accuracy over all completed entries, and (b) high coverage, meaning that it covers at least as much of the matrix as the distribution of observations.