Constrained Nonnegative Gram Feasibility is $\exists\mathbb{R}$-Complete

TL;DR

This paper proves that determining the existence of a constrained nonnegative Gram factorization is $orall ext{R}$-complete, revealing its high computational complexity even for rank 2, with implications for geometric and algebraic problems.

Contribution

The paper establishes $orall ext{R}$-completeness of constrained nonnegative Gram feasibility for rank 2, using a geometric encoding of arithmetic constraints and extending to all ranks ≥ 2.

Findings

Feasibility problem is $orall ext{R}$-complete for rank 2.

Reduction from ETR-AMI encodes arithmetic constraints geometrically.

Hardness extends to all fixed ranks ≥ 2.

Abstract

We study the computational complexity of constrained nonnegative Gram feasibility. Given a partially specified symmetric matrix together with affine relations among selected entries, the problem asks whether there exists a nonnegative matrix $H \in \mathbb{R}_+^{n\times r}$ such that $W = HH^\top$ satisfies all specified entries and affine constraints. Such factorizations arise naturally in structured low-rank matrix representations and geometric embedding problems. We prove that this feasibility problem is $\exists\mathbb{R}$-complete already for rank $r=2$. The hardness result is obtained via a polynomial-time reduction from the arithmetic feasibility problem \textsc{ETR-AMI}. The reduction exploits a geometric encoding of arithmetic constraints within rank-$2$ nonnegative Gram representations: by fixing anchor directions in $\mathbb{R}_+^2$ and representing variables through vectors of the form $(x,1)$, addition and multiplication constraints can be realized through inner-product relations. Combined with the semialgebraic formulation of the feasibility conditions, this establishes $\exists\mathbb{R}$-completeness. We further show that the hardness extends to every fixed rank $r\ge 2$. Our results place constrained symmetric nonnegative Gram factorization among the growing family of geometric feasibility problems that are complete for the existential theory of the reals. Finally, we discuss limitations of the result and highlight the open problem of determining the complexity of unconstrained symmetric nonnegative factorization feasibility.