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This paper establishes new bounds for the volume of tubular neighborhoods around smooth Pfaffian hypersurfaces, extending existing results for algebraic varieties. The authors leverage these bounds to derive tail probabilities for the condition number that assesses the robustness of neural network classifiers utilizing Pfaffian activation functions, addressing both uniform and Gaussian distributions. Notably, they provide polynomial-in-width bounds for the decision boundary of single-hidden-layer sigmoid networks with rational weights, highlighting a significant relationship between geometric properties and neural network performance.
Robustness in neural networks can be quantified through new geometric insights, revealing polynomial bounds that could enhance classifier stability.
We derive bounds for the volume of tubular neighbourhoods of smooth Pfaffian hypersurfaces, generalising known results for algebraic varieties. The bounds are given in terms of the Pfaffian format of the defining functions. As an application, we obtain tail bounds on the probability distribution of a condition number measuring the robustness of neural network classifiers with Pfaffian activation functions, in both the uniform and Gaussian settings. In the special case of single-hidden-layer sigmoid networks with rational weights, we derive polynomial-in-width bounds for tubular neighbourhoods of the decision boundary.
