A Bayesian Approach for Noisy Matrix Completion: Optimal Rate under General Sampling Distribution

Abstract

Bayesian methods for low-rank matrix completion with noise have been shown to be very efficient computationally. While the behaviour of penalized minimization methods is well understood both from the theoretical and computational points of view 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.

References in corpus (4)

• Noisy low-rank matrix completion with general sampling distribution
• Aggregation by exponential weighting, sharp PAC-Bayesian bounds and sparsity
• Bayesian methods for low-rank matrix estimation: short survey and theoretical study
• Pac-bayesian bounds for sparse regression estimation with exponential weights

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