Matrix Completion from a Few Entries

Let M be a random n*alpha by n matrix of rank r, and assume that a uniformly random subset E of its entries is observed. We describe an efficient algorithm that reconstructs M from |E| = O(rn) observed entries. More precisely, for any d>0, there exists finite C(d,alpha) such that, if |E|>C(d,alpha)rn, then M can be reconstructed with root mean square error smaller than d. Further, if |E|>C'(alpha)rnlog(n), M can be reconstructed exactly with high probability. This settles a question left open by Candes and Recht and improves over the guarantees for their reconstruction algorithm. The complexity of our algorithm is O(|E|rlog(n)), which opens the way to its use for massive data sets. In the process of proving these statements, we obtain a generalization of a celebrated result by Friedman-Kahn-Szemeredi and Feige-Ofek on the spectrum of sparse random matrices.