# The square root rank of the correlation polytope is exponential

arXiv: https://arxiv.org/abs/1411.6712 · HTML: https://mlsanity.com/paper/1411.6712/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Nov 2014.

## Built on by (most cited first, 4 of 4)

- [What is the Relationship between Tensor Factorizations and Circuits (and How Can We Exploit it)?](https://mlsanity.com/paper/2409.07953/) (2024)
- [Sum of Squares Circuits](https://mlsanity.com/paper/2408.11778/) (2024)
- [Alternating Direction Method of Multipliers for Nonlinear Matrix Decompositions](https://mlsanity.com/paper/2512.17473/) (2025)
- [On the Complexity of Low-Rank Matrix Signing and Entrywise Power Matrix Factorization](https://mlsanity.com/paper/2607.04875/) (2026)

## Abstract (arXiv)

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.