  Noisy matrix completion aims at estimating a low-rank matrix given only
partial and corrupted entries. Despite substantial progress in designing
efficient estimation algorithms, it remains largely unclear how to assess the
uncertainty of the obtained estimates and how to perform statistical inference
on the unknown matrix (e.g.~constructing a valid and short confidence interval
for an unseen entry).
  This paper takes a step towards inference and uncertainty quantification for
noisy matrix completion. We develop a simple procedure to compensate for the
bias of the widely used convex and nonconvex estimators. The resulting
de-biased estimators admit nearly precise non-asymptotic distributional
characterizations, which in turn enable optimal construction of confidence
intervals\,/\,regions for, say, the missing entries and the low-rank factors.
Our inferential procedures do not rely on sample splitting, thus avoiding
unnecessary loss of data efficiency. As a byproduct, we obtain a sharp
characterization of the estimation accuracy of our de-biased estimators, which,
to the best of our knowledge, are the first tractable algorithms that provably
achieve full statistical efficiency (including the preconstant). The analysis
herein is built upon the intimate link between convex and nonconvex
optimization --- an appealing feature recently discovered by
\cite{chen2019noisy}.
