  We give an algorithm for completing an order-$m$ symmetric low-rank tensor
from its multilinear entries in time roughly proportional to the number of
tensor entries. We apply our tensor completion algorithm to the problem of
learning mixtures of product distributions over the hypercube, obtaining new
algorithmic results. If the centers of the product distribution are linearly
independent, then we recover distributions with as many as $\Omega(n)$ centers
in polynomial time and sample complexity. In the general case, we recover
distributions with as many as $\tilde\Omega(n)$ centers in quasi-polynomial
time, answering an open problem of Feldman et al. (SIAM J. Comp.) for the
special case of distributions with incoherent bias vectors.
  Our main algorithmic tool is the iterated application of a low-rank matrix
completion algorithm for matrices with adversarially missing entries.
