A reusable framework generates verification instances with provably known robustness labels, revealing numeric tolerance issues and bugs in five verifiers while introducing difficulty profiles to diagnose failure modes.
Robustness Verification of Polynomial Neural Networks
3 Pith papers cite this work. Polarity classification is still indexing.
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abstract
We study robustness verification of neural networks via metric algebraic geometry. For polynomial neural networks, certifying a robustness radius amounts to computing the distance to the algebraic decision boundary. We use the Euclidean distance (ED) degree as an intrinsic measure of the complexity of this problem, analyze the associated ED discriminant, and introduce a parameter discriminant that detects parameter values at which the ED degree drops. We derive formulas for the ED degree for several network architectures and characterize the expected number of real critical points in the infinite-width limit. We develop symbolic elimination methods to compute these quantities and homotopy-continuation methods for exact robustness certification. Finally, experiments on lightning self-attention modules reveal decision boundaries with strictly smaller ED degree than generic cubic hypersurfaces of the same ambient dimension.
years
2026 3representative citing papers
For large monomial activation degree, critical points in deep fully-connected networks coincide exactly with subnetwork configurations where neurons are inactive or redundant.
Lightning self-attention coefficients are coordinates on an algebraic variety obeying Chow-type, low-rank, Veronese-type, and Sylvester-resultant invariants.
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Singular Learning and Occam's Razor in Deep Monomial Networks
For large monomial activation degree, critical points in deep fully-connected networks coincide exactly with subnetwork configurations where neurons are inactive or redundant.
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Algebraic Invariants of Lightning Self-Attention
Lightning self-attention coefficients are coordinates on an algebraic variety obeying Chow-type, low-rank, Veronese-type, and Sylvester-resultant invariants.