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This research establishes rigorous algebraic geometry certificates to determine exactly which functions shallow polynomial neural networks can or cannot represent over the complex field. By linking network architectures to Veronese secant varieties and ideal elimination, the study provides a theoretical framework for understanding fundamental approximation limits in bias-free models. The work bridges commutative algebra and machine learning theory, offering precise mathematical tools to analyze the expressive boundaries of specific neural network designs.
Algebraic Expressivity Certificates for Shallow Polynomial Neural Networks
arXiv:2609.28500v1 Announce Type: cross Abstract: 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.
Abstract: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.
Commutative Algebra (math.AC); Machine Learning (cs.LG); Algebraic Geometry (math.AG)
arXiv:2609.28500 [math.AC]
(or arXiv:2609.28500v1 [math.AC] for this version)
https://doi.org/10.48550/arXiv.2609.28500 Focus to learn more arXiv-issued DOI via DataCite
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