  We study the distribution and uncertainty of nonconvex optimization for noisy
tensor completion -- the problem of estimating a low-rank tensor given
incomplete and corrupted observations of its entries. Focusing on a two-stage
estimation algorithm proposed by Cai et al. (2019), we characterize the
distribution of this nonconvex estimator down to fine scales. This
distributional theory in turn allows one to construct valid and short
confidence intervals for both the unseen tensor entries and the unknown tensor
factors. The proposed inferential procedure enjoys several important features:
(1) it is fully adaptive to noise heteroscedasticity, and (2) it is data-driven
and automatically adapts to unknown noise distributions. Furthermore, our
findings unveil the statistical optimality of nonconvex tensor completion: it
attains un-improvable $\ell_{2}$ accuracy -- including both the rates and the
pre-constants -- when estimating both the unknown tensor and the underlying
tensor factors.
