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

TL;DR

This paper introduces an efficient tensor completion algorithm for symmetric low-rank tensors and applies it to learn mixtures of product distributions over the hypercube, achieving near-linear and quasi-polynomial time results.

Contribution

It presents a novel tensor completion method that enables learning high-dimensional mixture models with improved efficiency and theoretical guarantees.

Findings

Successfully completes symmetric tensors from multilinear entries.

Learns mixtures of product distributions with many centers in polynomial time.

Extends to distributions with incoherent bias vectors in quasi-polynomial time.

Abstract

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.