Let M be an n2n matrix of rank r,andassumethat a uniformly random subset E of its entries is observed. We describe an efficient algorithm, which we call OPTSPACE, that reconstructs M from jEj = O(rn) observed entries with relative root mean square error RMSE C() nr jEj 1=2 with probability larger than 1 0 1=n 3 . Further, if r = O(1) and M is sufficiently unstructured, then OPTSPACE reconstructs it exactly from jEj = O(nlogn) entries with probability larger than 1 0 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(jEjr 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.
For a bounded rank r = O(1), the performance bound of OPTSPACE is order optimal REF .
Matrix Completion From a Few Entries
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Computer Science Mathematics