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

2026-03-20

AI Summary

This work investigates the existence of nonnegative Gram decompositions—factorizations as $XX^\top$ with $X \geq 0$—for partially symmetric matrices subject to affine constraints. By presenting a polynomial-time reduction from the arithmetic feasibility problem ETR-AMI, the authors establish that this decision problem is $\exists\mathbb{R}$-complete even when restricted to rank 2, and extend this hardness result to any fixed rank $r \geq 2$. The core technical insight lies in encoding addition and multiplication constraints via inner products between vectors of the form $(x,1)$ and carefully chosen anchor directions in $\mathbb{R}_+^2$. This result firmly places constrained nonnegative Gram decomposition within the family of $\exists\mathbb{R}$-complete problems, while the computational complexity of the unconstrained variant remains an open question.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Non-convex Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Constrained Nonnegative Gram Feasibility
• Existential Theory of the Reals
• Matrix Factorization
• Affine Constraints
• Computational Complexity

Innovation

• ∃ℝ-completeness
• nonnegative Gram factorization
• computational complexity
• geometric encoding
• low-rank matrix representation