Matrix completion from a few entries

Let M be an n¿ × n matrix of rank r ¿ n, and assume that a uniformly random subset E of its entries is observed. We describe an efficient algorithm that reconstructs M from |E| = O(r n) observed entries with relative root mean square error RMSE ¿ C(¿) (nr/|E|)^1/2. Further, if r = O(1) and M is sufficiently unstructured, then it can be reconstructed exactly from |E| = O(n log n) entries. This settles (in the case of bounded rank) 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|r log 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.