## Matrix Completion from Noisy Entries

Raghunandan H. Keshavan, Andrea Montanari, Sewoong Oh; 11(69):2057−2078, 2010.

### Abstract

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.

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