Electronic Journal of Statistics

A Bayesian approach for noisy matrix completion: Optimal rate under general sampling distribution

The Tien Mai and Pierre Alquier

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Bayesian methods for low-rank matrix completion with noise have been shown to be very efficient computationally [3, 18, 19, 24, 28]. While the behaviour of penalized minimization methods is well understood both from the theoretical and computational points of view (see [7, 9, 16, 23] among others) in this problem, the theoretical optimality of Bayesian estimators have not been explored yet. In this paper, we propose a Bayesian estimator for matrix completion under general sampling distribution. We also provide an oracle inequality for this estimator. This inequality proves that, whatever the rank of the matrix to be estimated, our estimator reaches the minimax-optimal rate of convergence (up to a logarithmic factor). We end the paper with a short simulation study.

Article information

Electron. J. Statist., Volume 9, Number 1 (2015), 823-841.

First available in Project Euclid: 2 April 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H12: Estimation
Secondary: 62J12: Generalized linear models 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 68T05: Learning and adaptive systems [See also 68Q32, 91E40]

Matrix completion Bayesian Analysis PAC-Bayesian bounds oracle inequality low-rank matrix Gibbs sampler


Mai, The Tien; Alquier, Pierre. A Bayesian approach for noisy matrix completion: Optimal rate under general sampling distribution. Electron. J. Statist. 9 (2015), no. 1, 823--841. doi:10.1214/15-EJS1020. https://projecteuclid.org/euclid.ejs/1427990076

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