发表机构
Lake Forest College（莱克福里斯特学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
利用代数几何与理想消元，为浅层多项式神经网络生成不可表示性证书，并推导秩二二次网络在球面上的精确损失下限。
AI 中文摘要
我们利用代数几何研究无偏置浅层多项式神经网络的精确可表示性。在复数域$\mathbb{C}$上，一个宽度为$r$、激活函数为$z\mapsto z^d$的网络计算$r$个线性形式的$d$次幂之和，其Zariski闭包是Veronese割线簇。因此，理想消元可产生不可表示性的多项式证书。我们将此构造实现为从架构到证书的通用流水线。对于二次函数，我们恢复了精确的对称行列式描述，并通过正交对称性解释其维数。在更高次数下，该实现恢复了经典的催化张量（catalecticant）和割线方程，并在有限架构扫描中绘制了直接消元的实际可达范围。我们还推导了球面上秩二二次网络的确切总体损失下限，说明代数障碍如何导致不可约的逼近误差。
英文摘要
We study exact representability by bias-free shallow polynomial neural networks using algebraic geometry. Over $\mathbb{C}$, a width-$r$ network with activation $z\mapsto z^d$ computes a sum of $r$ $d$-th powers of linear forms, whose Zariski closure is a Veronese secant variety. Ideal elimination therefore yields polynomial certificates of nonrepresentability. We implement this construction as a generic architecture-to-certificate pipeline. For quadratics, we recover the exact symmetric determinantal description and explain its dimension through orthogonal symmetry. In higher degree, the implementation recovers classical catalecticant and secant equations and maps the practical reach of direct elimination across a finite architecture sweep. We also derive the exact population loss floor for a rank-two quadratic network on the sphere, illustrating how an algebraic obstruction induces irreducible approximation error.
Comments9 pages, 1 figure
