Matrix Completion from a Few Entries
Abstract
Let M be a random (alpha n) x 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(rn) observed entries with relative root mean square error RMSE <= C(rn/E)^0.5 . Further, if r=O(1), M can be reconstructed exactly from E = O(n log(n)) entries. These results apply beyond random matrices to general lowrank incoherent matrices. 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(Er 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 FriedmanKahnSzemeredi and FeigeOfek on the spectrum of sparse random matrices.
 Publication:

arXiv eprints
 Pub Date:
 January 2009
 arXiv:
 arXiv:0901.3150
 Bibcode:
 2009arXiv0901.3150K
 Keywords:

 Computer Science  Machine Learning;
 Statistics  Machine Learning
 EPrint:
 30 pages, 1 figure, journal version (v1, v2: Conference version ISIT 2009)