  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.
