  In this article, we develop methods for estimating a low rank tensor from
noisy observations on a subset of its entries to achieve both statistical and
computational efficiencies. There have been a lot of recent interests in this
problem of noisy tensor completion. Much of the attention has been focused on
the fundamental computational challenges often associated with problems
involving higher order tensors, yet very little is known about their
statistical performance. To fill in this void, in this article, we characterize
the fundamental statistical limits of noisy tensor completion by establishing
minimax optimal rates of convergence for estimating a $k$th order low rank
tensor under the general $\ell_p$ ($1\le p\le 2$) norm which suggest
significant room for improvement over the existing approaches. Furthermore, we
propose a polynomial-time computable estimating procedure based upon power
iteration and a second-order spectral initialization that achieves the optimal
rates of convergence. Our method is fairly easy to implement and numerical
experiments are presented to further demonstrate the practical merits of our
estimator.
