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 low-rank 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(|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.
Cite
@article{arxiv.0901.3150,
title = {Matrix Completion from a Few Entries},
author = {Raghunandan H. Keshavan and Andrea Montanari and Sewoong Oh},
journal= {arXiv preprint arXiv:0901.3150},
year = {2009}
}
Comments
30 pages, 1 figure, journal version (v1, v2: Conference version ISIT 2009)