Bayesian methods for low-rank matrix estimation: short survey and theoretical study

TL;DR

This paper reviews Bayesian approaches for low-rank matrix estimation, comparing them to penalization methods, and provides theoretical results showing their optimality under certain conditions.

Contribution

It offers a comprehensive survey of priors used in Bayesian low-rank estimation and proves their estimators can achieve optimality similar to penalized methods.

Findings

Bayesian estimators can attain the same optimality as penalized methods.

Different priors for low-rank matrices are systematically reviewed.

Theoretical guarantees are established for Bayesian estimators.

Abstract

The problem of low-rank matrix estimation recently received a lot of attention due to challenging applications. A lot of work has been done on rank-penalized methods and convex relaxation, both on the theoretical and applied sides. However, only a few papers considered Bayesian estimation. In this paper, we review the different type of priors considered on matrices to favour low-rank. We also prove that the obtained Bayesian estimators, under suitable assumptions, enjoys the same optimality properties as the ones based on penalization.