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}.

本文提出了一种补偿广泛使用的凸估计器偏差的简单程序，从而实现了对噪声矩阵完成的不确定性和推断，并产生了近乎精确的非渐近分布表征，进而实现了对缺失条目和低秩因子的置信区间的最优构建。