  Tensor completion exhibits an interesting computational-statistical gap in
terms of the number of samples needed to perform tensor estimation. While there
are only $\Theta(tn)$ degrees of freedom in a $t$-order tensor with $n^t$
entries, the best known polynomial time algorithm requires $O(n^{t/2})$ samples
in order to guarantee consistent estimation. In this paper, we show that weak
side information is sufficient to reduce the sample complexity to $O(n)$. The
side information consists of a weight vector for each of the modes which is not
orthogonal to any of the latent factors along that mode; this is significantly
weaker than assuming noisy knowledge of the subspaces. We provide an algorithm
that utilizes this side information to produce a consistent estimator with
$O(n^{1+\kappa})$ samples for any small constant $\kappa > 0$.
