Matrix Completion from Noisy Entries

TL;DR

This paper analyzes the OptSpace algorithm for low-rank matrix completion from noisy, randomly sampled entries, providing performance guarantees that are nearly optimal in various scenarios.

Contribution

It offers theoretical performance guarantees for the OptSpace algorithm, enhancing understanding of its effectiveness in noisy matrix completion tasks.

Findings

Performance guarantees are order-optimal in many cases.

OptSpace effectively reconstructs low-rank matrices from noisy samples.

Theoretical analysis supports practical applications in collaborative filtering and related fields.

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 et al.(2009), 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.