Matrix Completion From a Few Entries

Let M be an n¿ × n matrix of rank r, and assume that a uniformly random subset E of its entries is observed. We describe an efficient algorithm, which we call OptSpace, that reconstructs M from |E| = O(rn) observed entries with relative root mean square error 1/2 RMSE ¿ C(¿) (nr/|E|)^1/2 with probability larger than 1 - 1/n³. Further, if r = O(1) and M is sufficiently unstructured, then OptSpace reconstructs it exactly from |E| = O(n log n) entries with probability larger than 1 - 1/n³. 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.