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

Symmetric Tensor Completion from Multilinear Entries and Learning Product Mixtures over the Hypercube