Electronic Journal of Statistics

Matrix completion via max-norm constrained optimization

T. Tony Cai and Wen-Xin Zhou

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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.

Article information

Electron. J. Statist., Volume 10, Number 1 (2016), 1493-1525.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12: Estimation 62J99: None of the above, but in this section
Secondary: 15A83: Matrix completion problems

Compressed sensing low-rank matrix matrix completion max-norm constrained minimization minimax optimality non-uniform sampling sparsity


Cai, T. Tony; Zhou, Wen-Xin. Matrix completion via max-norm constrained optimization. Electron. J. Statist. 10 (2016), no. 1, 1493--1525. doi:10.1214/16-EJS1147. https://projecteuclid.org/euclid.ejs/1464710239

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