Matrix Completion from a Few Entries

TL;DR

This paper presents an efficient algorithm for reconstructing low-rank matrices from a small subset of entries, achieving near-optimal error bounds and exact recovery in certain cases, applicable to large-scale data.

Contribution

It introduces a new algorithm for matrix completion that improves guarantees over previous methods and extends results to general low-rank incoherent matrices.

Findings

Reconstructs matrices with O(rn) entries with low error

Exact recovery from O(n log n) entries when r=O(1)

Algorithm runs in O(|E|r log(n)) time

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.