Robustness Verification of Polynomial Neural Networks

2026-04-20 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Cybersecurity & Data Privacy · Depth: Expert, extended

Summary

This paper introduces an algebraic framework for formal robustness verification of neural networks, formulating it as an algebraic optimization problem. The authors use the Euclidean Distance (ED) degree, an invariant from algebraic geometry, to measure the intrinsic complexity of robustness verification, which is architecture-dependent. They define the ED discriminant to characterize input points where the number of real critical points changes, distinguishing easier or harder verification instances, and provide an algorithm for its computation. Additionally, a parameter discriminant is introduced to identify network parameters where the ED degree drops, indicating reduced algebraic complexity of the decision boundary. The work derives closed-form expressions for the ED degree for various neural architectures and formulas for the expected number of real critical points in the infinite-width limit. Finally, an exact robustness certification algorithm based on numerical homotopy continuation is presented, with an implementation available on GitHub.

Key takeaway

For AI Scientists and Research Scientists developing or deploying neural networks in safety-critical applications, this algebraic framework offers a novel way to understand and certify robustness. You should consider integrating ED degree and discriminant analysis into your model evaluation pipeline to gain intrinsic insights into verification complexity. This approach can guide architecture design towards models that are provably easier to verify, potentially by incorporating algebraic regularizers that penalize high ED degree configurations, thereby improving certification efficiency and reliability.

Key insights

Algebraic geometry tools can quantify and certify neural network robustness by analyzing decision boundary complexity.

Principles

• Robustness verification complexity is architecture-dependent.
• ED degree quantifies intrinsic verification difficulty.
• Discriminants identify easier/harder verification instances.

Method

The proposed method formulates robustness verification as an algebraic distance minimization problem, leveraging the Euclidean Distance (ED) degree and associated discriminants. It employs numerical homotopy continuation for exact robustness certification.

In practice

• Use ED degree to assess network verification complexity.
• Identify "easier" instances via ED discriminant.
• Design models with lower ED degree for cheaper verification.

Topics

• Neural Network Robustness
• Algebraic Verification
• Euclidean Distance Degree
• ED Discriminant
• Parameter Discriminant

Code references

• edwardduanhao/AlgebraicVerification

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.