The complexity of positive semidefinite matrix factorization

TL;DR

This paper investigates the computational complexity of positive semidefinite (PSD) matrix factorization, establishing that computing the PSD rank is polynomial-time equivalent to the existential theory of the reals, thus highlighting its inherent computational difficulty.

Contribution

The paper proves that determining the PSD rank of a matrix is polynomial-time equivalent to the existential theory of the reals, clarifying its complexity status.

Findings

Computing PSD rank is polynomial-time equivalent to the existential theory of the reals.

The complexity of PSD rank computation is established as inherently difficult.

The result links matrix factorization problems to fundamental complexity classes.

Abstract

Let $A$ be a matrix with nonnegative real entries. The PSD rank of $A$ is the smallest integer $k$ for which there exist $k\times k$ real PSD matrices $B_1,\ldots,B_m$, $C_1,\ldots,C_n$ satisfying $A(i|j)=\operatorname{tr}(B_iC_j)$ for all $i,j$. This paper determines the computational complexity status of the PSD rank. Namely, we show that the problem of computing this function is polynomial-time equivalent to the existential theory of the reals.