  We study the problem of low-rank tensor factorization in the presence of
missing data. We ask the following question: how many sampled entries do we
need, to efficiently and exactly reconstruct a tensor with a low-rank
orthogonal decomposition? We propose a novel alternating minimization based
method which iteratively refines estimates of the singular vectors. We show
that under certain standard assumptions, our method can recover a three-mode
$n\times n\times n$ dimensional rank-$r$ tensor exactly from $O(n^{3/2} r^5
\log^4 n)$ randomly sampled entries. In the process of proving this result, we
solve two challenging sub-problems for tensors with missing data. First, in the
process of analyzing the initialization step, we prove a generalization of a
celebrated result by Szemer\'edie et al. on the spectrum of random graphs.
Next, we prove global convergence of alternating minimization with a good
initialization. Simulations suggest that the dependence of the sample size on
dimensionality $n$ is indeed tight.
