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2016 Matrix completion via max-norm constrained optimization
T. Tony Cai, Wen-Xin Zhou
Electron. J. Statist. 10(1): 1493-1525 (2016). DOI: 10.1214/16-EJS1147

Abstract

Matrix completion has been well studied under the uniform sampling model and the trace-norm regularized methods perform well both theoretically and numerically in such a setting. However, the uniform sampling model is unrealistic for a range of applications and the standard trace-norm relaxation can behave very poorly when the underlying sampling scheme is non-uniform.

In this paper we propose and analyze a max-norm constrained empirical risk minimization method for noisy matrix completion under a general sampling model. The optimal rate of convergence is established under the Frobenius norm loss in the context of approximately low-rank matrix reconstruction. It is shown that the max-norm constrained method is minimax rate-optimal and yields a unified and robust approximate recovery guarantee, with respect to the sampling distributions. The computational effectiveness of this method is also discussed, based on first-order algorithms for solving convex optimizations involving max-norm regularization.

Citation

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T. Tony Cai. Wen-Xin Zhou. "Matrix completion via max-norm constrained optimization." Electron. J. Statist. 10 (1) 1493 - 1525, 2016. https://doi.org/10.1214/16-EJS1147

Information

Received: 1 December 2015; Published: 2016
First available in Project Euclid: 31 May 2016

zbMATH: 1342.62091
MathSciNet: MR3507371
Digital Object Identifier: 10.1214/16-EJS1147

Subjects:
Primary: 62H12, 62J99
Secondary: 15A83

Rights: Copyright © 2016 The Institute of Mathematical Statistics and the Bernoulli Society

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Vol.10 • No. 1 • 2016
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