  In this paper, we consider the statistical inference for several low-rank
tensor models. Specifically, in the Tucker low-rank tensor PCA or regression
model, provided with any estimates achieving some attainable error rate, we
develop the data-driven confidence regions for the singular subspace of the
parameter tensor based on the asymptotic distribution of an updated estimate by
two-iteration alternating minimization. The asymptotic distributions are
established under some essential conditions on the signal-to-noise ratio (in
PCA model) or sample size (in regression model). If the parameter tensor is
further orthogonally decomposable, we develop the methods and theory for
inference on each individual singular vector. For the rank-one tensor PCA
model, we establish the asymptotic distribution for general linear forms of
principal components and confidence interval for each entry of the parameter
tensor. Finally, numerical simulations are presented to corroborate our
theoretical discoveries.
  In all these models, we observe that different from many matrix/vector
settings in existing work, debiasing is not required to establish the
asymptotic distribution of estimates or to make statistical inference on
low-rank tensors. In fact, due to the widely observed
statistical-computational-gap for low-rank tensor estimation, one usually
requires stronger conditions than the statistical (or information-theoretic)
limit to ensure the computationally feasible estimation is achievable.
Surprisingly, such conditions ``incidentally" render a feasible low-rank tensor
inference without debiasing.
