Matrix Completion from Noisy Entries

Raghunandan H. Keshavan, Andrea Montanari, Sewoong Oh.

Year: 2010, Volume: 11, Issue: 69, Pages: 2057−2078


Given a matrix M of low-rank, we consider the problem of reconstructing it from noisy observations of a small, random subset of its entries. The problem arises in a variety of applications, from collaborative filtering (the 'Netflix problem') to structure-from-motion and positioning. We study a low complexity algorithm introduced by Keshavan, Montanari, and Oh (2010), based on a combination of spectral techniques and manifold optimization, that we call here OPTSPACE. We prove performance guarantees that are order-optimal in a number of circumstances.