The square root rank of the correlation polytope is exponential

TL;DR

This paper proves that the square root rank of the correlation polytope's slack matrix grows exponentially, providing insights into the complexity of related matrix factorizations and their bounds.

Contribution

It introduces a novel technique to lower bound matrix rank under sign changes, establishing exponential lower bounds for the square root rank of the correlation polytope.

Findings

Square root rank of the correlation polytope's slack matrix is exponential.

New method for lower bounding matrix rank using polynomial roots in number fields.

Implications for positive semidefinite rank and matrix factorization complexity.

Abstract

The square root rank of a nonnegative matrix $A$ is the minimum rank of a matrix $B$ such that $A=B \circ B$, where $\circ$ denotes entrywise product. We show that the square root rank of the slack matrix of the correlation polytope is exponential. Our main technique is a way to lower bound the rank of certain matrices under arbitrary sign changes of the entries using properties of the roots of polynomials in number fields. The square root rank is an upper bound on the positive semidefinite rank of a matrix, and corresponds the special case where all matrices in the factorization are rank-one.