Simulation comparisons between Bayesian and de-biased estimators in low-rank matrix completion

TL;DR

This paper compares Bayesian and de-biased estimators for low-rank matrix completion, revealing that while de-biased estimators are theoretically optimal, Bayesian methods are more stable and perform better with small samples.

Contribution

The study provides a comprehensive simulation comparison of Bayesian and de-biased estimators, highlighting their relative strengths and limitations in low-rank matrix completion.

Findings

De-biased estimator achieves minimax-optimal error rate.

Bayesian methods are more stable and outperform in small samples.

Empirical coverage of de-biased confidence intervals is lower.

Abstract

In this paper, we study the low-rank matrix completion problem, a class of machine learning problems, that aims at the prediction of missing entries in a partially observed matrix. Such problems appear in several challenging applications such as collaborative filtering, image processing, and genotype imputation. 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 interesting results that the de-biased estimator is just as good as the Bayesian estimators. Moreover, Bayesian approaches are much more stable and can outperform the de-biased estimator in the case of small samples. In addition, we also find that the empirical coverage rate of the confidence intervals obtained by the de-biased estimator for an entry is absolutely lower than of the considered credible interval. These results suggest further theoretical studies on the estimation error and the concentration of Bayesian methods as they are quite limited up to present.