Active matrix completion with uncertainty quantification

The noisy matrix completion problem, which aims to recover a low-rank matrix $\mathbf{X}$ from a partial, noisy observation of its entries, arises in many statistical, machine learning, and engineering applications. In this paper, we present a new, information-theoretic approach for active sampling (or designing) of matrix entries for noisy matrix completion, based on the maximum entropy design principle. One novelty of our method is that it implicitly makes use of uncertainty quantification (UQ) -- a measure of uncertainty for unobserved matrix entries -- to guide the active sampling procedure. The proposed framework reveals several novel insights on the role of compressive sensing (e.g., coherence) and coding design (e.g., Latin squares) on the sampling performance and UQ for noisy matrix completion. Using such insights, we develop an efficient posterior sampler for UQ, which is then used to guide a closed-form sampling scheme for matrix entries. Finally, we illustrate the effectiveness of this integrated sampling / UQ methodology in simulation studies and two applications to collaborative filtering.