Bayesian Uncertainty Quantification for Low-rank Matrix Completion

We consider the problem of uncertainty quantification (UQ) 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 largely focused on the completion (i.e., point estimation) of the matrix $\mathbf{X}$, with little work on exploring the uncertainty of such estimates. 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 allows for efficient posterior inference on matrix subspaces (which represent interpretable phenomena in many applications), which can then be leveraged for improved matrix recovery. We demonstrate the effectiveness of BayeSMG over existing Bayesian matrix recovery methods in extensive numerical experiments and a seismic sensor network application.