We consider the problem of uncertainty quantification for an unknown low-rank matrix 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 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 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.
HSY and YX are supported by NSF CCF-1650913, NSF DMS-1938106, and NSF DMS-18302. SM is supported by NSF CSSI Frameworks 2004571.
We thank the editor, associate editor, and reviewers for their insightful comments on the previous version of this article. Henry Shaowu Yuchi and Yao Xie are supported by an NSF Career Award CCF-1650913, NSF DMS-2134037, NSF DMS-1938106, NSF DMS-1830210, and a SERDP grant. Simon Mak is supported by NSF CSSI Frameworks grant 2004571. The data and pictures used in the seismic sensor network recovery are provided by Sin-Mei Wu and Fan-Chi Lin.
MATLAB codes for the BayeSMG sampler can be found on GitHub.
"Bayesian Uncertainty Quantification for Low-Rank Matrix Completion." Bayesian Anal. Advance Publication 1 - 28, 2022. https://doi.org/10.1214/22-BA1317