Exact tensor completion with sum-of-squares

TL;DR

This paper introduces the first polynomial-time algorithm for exact tensor completion that surpasses previous bounds, leveraging sum-of-squares techniques to recover 3-tensors efficiently from fewer observations.

Contribution

It extends sum-of-squares methods from matrix to tensor completion, achieving improved bounds and polynomial-time guarantees for exact recovery of orthogonal tensor components.

Findings

Achieves exact tensor recovery with $r\cdot \tilde O(n^{1.5})$ samples

Improves previous bounds from $r\cdot \tilde O(n^{2})$

Extends matrix completion techniques to tensors using sum-of-squares

Abstract

We obtain the first polynomial-time algorithm for exact tensor completion that improves over the bound implied by reduction to matrix completion. The algorithm recovers an unknown 3-tensor with $r$ incoherent, orthogonal components in $\mathbb R^n$ from $r\cdot \tilde O(n^{1.5})$ randomly observed entries of the tensor. This bound improves over the previous best one of $r\cdot \tilde O(n^{2})$ by reduction to exact matrix completion. Our bound also matches the best known results for the easier problem of approximate tensor completion (Barak & Moitra, 2015). Our algorithm and analysis extends seminal results for exact matrix completion (Candes & Recht, 2009) to the tensor setting via the sum-of-squares method. The main technical challenge is to show that a small number of randomly chosen monomials are enough to construct a degree-3 polynomial with precisely planted orthogonal global optima over the sphere and that this fact can be certified within the sum-of-squares proof system.