# Numerical comparisons between Bayesian and frequentist low-rank matrix completion: estimation accuracy and uncertainty quantification

@article{Mai2021NumericalCB,
title={Numerical comparisons between Bayesian and frequentist low-rank matrix completion: estimation accuracy and uncertainty quantification},
author={T. T. Mai},
journal={ArXiv},
year={2021},
volume={abs/2103.11749}
}
• T. T. Mai
• Published 22 March 2021
• Computer Science, Mathematics
• ArXiv
In this paper we perform numerous numerical studies for the problem of low-rank matrix completion. We compare the Bayesian approaches and a recently introduced de-biased estimator which provides a useful way to build confidence intervals of interest. From a theoretical viewpoint, the de-biased estimator comes with a sharp minimax-optimal rate of estimation error whereas the Bayesian approach reaches this rate with an additional logarithmic factor. Our simulation studies show originally… Expand
2 Citations

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#### References

SHOWING 1-10 OF 24 REFERENCES
A Bayesian approach for noisy matrix completion: Optimal rate under general sampling distribution
• Mathematics
• 2015
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 theExpand
Bayesian matrix completion: prior specification and consistency
• Mathematics
• 2014
Low-rank matrix estimation from incomplete measurements recently received increased attention due to the emergence of several challenging applications, such as recommender systems; see in particularExpand
Inference and uncertainty quantification for noisy matrix completion
• Computer Science, Mathematics
• Proceedings of the National Academy of Sciences
• 2019
A simple procedure to compensate for the bias of the widely used convex and nonconvex estimators and derive distributional characterizations for the resulting debiased estimators, which enable optimal construction of confidence intervals/regions for the missing entries and the low-rank factors. Expand
Nuclear norm penalization and optimal rates for noisy low rank matrix completion
• Mathematics
• 2010
This paper deals with the trace regression model where $n$ entries or linear combinations of entries of an unknown $m_1\times m_2$ matrix $A_0$ corrupted by noise are observed. We propose a newExpand
Bayesian Methods for Low-Rank Matrix Estimation: Short Survey and Theoretical Study
This paper reviews the different type of priors considered on matrices to favour low-rank matrix estimation and proves that the obtained Bayesian estimators enjoys the same optimality properties as the ones based on penalization. Expand
Bayesian probabilistic matrix factorization using Markov chain Monte Carlo
• Computer Science
• ICML '08
• 2008
This paper presents a fully Bayesian treatment of the Probabilistic Matrix Factorization (PMF) model in which model capacity is controlled automatically by integrating over all model parameters and hyperparameters and shows that Bayesian PMF models can be efficiently trained using Markov chain Monte Carlo methods by applying them to the Netflix dataset. Expand
1-Bit matrix completion: PAC-Bayesian analysis of a variational approximation
• Computer Science, Mathematics
• Machine Learning
• 2017
This work proposes an algorithm to compute a variational approximation of the pseudo-posterior of a (possibly) low-rank matrix with binary entries, the so-called 1-bit matrix completion problem, and derives a PAC bound on the prediction error of this algorithm. Expand
Restricted strong convexity and weighted matrix completion: Optimal bounds with noise
• Mathematics, Computer Science
• J. Mach. Learn. Res.
• 2012
The matrix completion problem under a form of row/column weighted entrywise sampling is considered, including the case of uniformentrywise sampling as a special case, and it is proved that with high probability, it satisfies a forms of restricted strong convexity with respect to weighted Frobenius norm. Expand
Sparse Bayesian Methods for Low-Rank Matrix Estimation
• Mathematics, Computer Science
• IEEE Transactions on Signal Processing
• 2012
This paper develops an approach that is very effective in determining the correct rank while providing high recovery performance and provides connections with existing methods in other similar problems and empirical results and comparisons with current state-of-the-art methods that illustrate the effectiveness. Expand
Fast Low-Rank Bayesian Matrix Completion With Hierarchical Gaussian Prior Models
• Computer Science, Mathematics
• IEEE Transactions on Signal Processing
• 2018
A variational Bayesian matrix completion method is developed, which embeds the generalized approximate massage passing technique to circumvent cumbersome matrix inverse operations and demonstrates superiority over some state-of-the-art matrix completion methods. Expand