Decomposing Overcomplete 3rd Order Tensors using Sum-of-Squares Algorithms

TL;DR

This paper introduces the first quasi-polynomial time algorithm for decomposing overcomplete 3rd order tensors with high rank using sum-of-squares methods, advancing tensor decomposition capabilities.

Contribution

It presents a novel quasi-polynomial time algorithm for overcomplete tensor decomposition and a polynomial-time method for certifying tensor injective norms, leveraging sum-of-squares techniques.

Findings

Decomposes random 3rd order tensors with rank up to n^{3/2}/polylog n

Provides a polynomial-time algorithm for tensor injective norm certification

Introduces new matrix concentration bounds via decoupling techniques

Abstract

Tensor rank and low-rank tensor decompositions have many applications in learning and complexity theory. Most known algorithms use unfoldings of tensors and can only handle rank up to $n^{\lfloor p/2 \rfloor}$ for a $p$-th order tensor in $\mathbb{R}^{n^p}$. Previously no efficient algorithm can decompose 3rd order tensors when the rank is super-linear in the dimension. Using ideas from sum-of-squares hierarchy, we give the first quasi-polynomial time algorithm that can decompose a random 3rd order tensor decomposition when the rank is as large as $n^{3/2}/\textrm{polylog} n$. We also give a polynomial time algorithm for certifying the injective norm of random low rank tensors. Our tensor decomposition algorithm exploits the relationship between injective norm and the tensor components. The proof relies on interesting tools for decoupling random variables to prove better matrix concentration bounds, which can be useful in other settings.