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.

翻译：我们研究约束非负Gram可行性的计算复杂性。给定一个部分指定的对称矩阵以及选定条目之间的仿射关系，该问题询问是否存在一个非负矩阵$H \in \mathbb{R}_+^{n\times r}$，使得$W = HH^\top$满足所有指定条目和仿射约束。此类因子分解自然出现在结构化低秩矩阵表示和几何嵌入问题中。我们证明，即使对于秩$r=2$，该可行性问题已经是$\exists\mathbb{R}$-完全的。该难度结果通过从算术可行性问题\textsc{ETR-AMI}的多项式时间归约得到。该归约利用了秩$2$非负Gram表示中算术约束的几何编码：通过在$\mathbb{R}_+^2$中固定锚定方向，并将变量表示为$(x,1)$形式的向量，加法和乘法约束可通过内积关系实现。结合可行性条件的半代数公式，这确立了$\exists\mathbb{R}$-完全性。我们进一步证明，该难度可推广到每个固定秩$r\ge 2$。我们的结果将约束对称非负Gram因子分解置于日益增长的、对于实数的存在性理论完全的几何可行性问题家族中。最后，我们讨论该结果的局限性，并强调确定无约束对称非负因子分解可行性的复杂性这一开放问题。