Lower bounds on matrix factorization ranks via noncommutative polynomial optimization

TL;DR

This paper develops hierarchies of semidefinite programming lower bounds for various matrix factorization ranks using noncommutative polynomial optimization, providing convergence analysis and numerical comparisons.

Contribution

It introduces a novel approach to bounding matrix factorization ranks via noncommutative polynomial optimization hierarchies, extending existing methods.

Findings

Hierarchies converge to true ranks

Numerical examples demonstrate effectiveness

Comparison with known bounds shows improvements

Abstract

We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the positive semidefinite rank, and their symmetric analogues: the completely positive rank and the completely positive semidefinite rank. We study the convergence properties of our hierarchies, compare them extensively to known lower bounds, and provide some (numerical) examples.