Bayesian Uncertainty Quantification for Low-Rank Matrix Completion

TL;DR

This paper introduces BayeSMG, a Bayesian framework for quantifying uncertainty in low-rank matrix completion, enabling better interpretation and decision-making in applications like imaging and seismology.

Contribution

The paper proposes a novel Bayesian subspace parametrization method, BayeSMG, for uncertainty quantification in low-rank matrix completion, which improves inference and interpretability.

Findings

BayeSMG outperforms existing Bayesian methods in numerical experiments.

Effective in image inpainting and seismic sensor network applications.

Provides interpretable subspace representations for matrix recovery.

Abstract

We consider the problem of uncertainty quantification for an unknown low-rank matrix $\mathbf{X}$, given a partial and noisy observation of its entries. This quantification of uncertainty is essential for many real-world problems, including image processing, satellite imaging, and seismology, providing a principled framework for validating scientific conclusions and guiding decision-making. However, existing literature has mainly focused on the completion (i.e., point estimation) of the matrix $\mathbf{X}$, with little work on investigating its uncertainty. To this end, we propose in this work a new Bayesian modeling framework, called BayeSMG, which parametrizes the unknown $\mathbf{X}$ via its underlying row and column subspaces. This Bayesian subspace parametrization enables efficient posterior inference on matrix subspaces, which represents interpretable phenomena in many applications. This can then be leveraged for improved matrix recovery. We demonstrate the effectiveness of BayeSMG over existing Bayesian matrix recovery methods in numerical experiments, image inpainting, and a seismic sensor network application.