Matrix completion from a few entries

Let M be an nα × 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(r n) observed entries with relative root mean square error RMSE ≤ C (α) (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/n ³. This settles (in the case of bounded rank) a question left open by Candès 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-Szemerédi and Feige-Ofek on the spectrum of sparse random matrices.