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Matrix Completion with Noise
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| Citations: | 255 - 13 self |
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@MISC{Candès_matrixcompletion,
author = {Emmanuel J. Candès and Yaniv Plan},
title = {Matrix Completion with Noise},
year = {}
}
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Abstract
On the heels of compressed sensing, a remarkable new field has very recently emerged. This field addresses a broad range of problems of significant practical interest, namely, the recovery of a data matrix from what appears to be incomplete, and perhaps even corrupted, information. In its simplest form, the problem is to recover a matrix from a small sample of its entries, and comes up in many areas of science and engineering including collaborative filtering, machine learning, control, remote sensing, and computer vision to name a few. This paper surveys the novel literature on matrix completion, which shows that under some suitable conditions, one can recover an unknown low-rank matrix from a nearly minimal set of entries by solving a simple convex optimization problem, namely, nuclear-norm minimization subject to data constraints. Further, this paper introduces novel results showing that matrix completion is provably accurate even when the few observed entries are corrupted with a small amount of noise. A typical result is that one can recover an unknown n × n matrix of low rank r from just about nr log 2 n noisy samples with an error which is proportional to the noise level. We present numerical results which complement our quantitative analysis and show that, in practice, nuclear norm minimization accurately fills in the many missing entries of large low-rank matrices from just a few noisy samples. Some analogies between matrix completion and compressed sensing are discussed throughout.
Keyphrases
matrix completion noisy sample broad range large low-rank matrix simple convex optimization problem minimal set unknown matrix small amount many area novel literature quantitative analysis typical result suitable condition noise level compressed sensing data matrix computer vision unknown low-rank matrix present numerical result nr log novel result remarkable new field small sample data constraint significant practical interest collaborative filtering low rank machine learning nuclear-norm minimization subject nuclear norm minimization observed entry
