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

TL;DR

This paper introduces a Bayesian estimator for noisy low-rank matrix completion under general sampling, demonstrating its theoretical optimality and providing an oracle inequality with near-minimax convergence rates.

Contribution

It proposes a Bayesian approach for matrix completion with general sampling distributions and establishes its minimax-optimal convergence rate.

Findings

Estimator achieves near-minimax rate of convergence

Oracle inequality proven for the Bayesian estimator

Simulation study supports theoretical results

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.