Symmetric Tensor Completion from Multilinear Entries and Learning Product Mixtures over the Hypercube
This addresses an open problem in learning theory for mixtures of product distributions, offering improved algorithmic results, though it is incremental in extending tensor completion methods to this domain.
The paper tackles the problem of symmetric tensor completion from multilinear entries and applies it to learning mixtures of product distributions over the hypercube, achieving recovery of distributions with up to Ω(n) centers in polynomial time under linear independence and up to Ω̃(n) centers in quasi-polynomial time for incoherent cases.
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 $Ω(n)$ centers in polynomial time and sample complexity. In the general case, we recover distributions with as many as $\tildeΩ(n)$ 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.