Tubular Neighbourhoods of Pfaffian Sets and Applications to Neural Networks

Abstract: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.

Algebraic Geometry (math.AG); Machine Learning (cs.LG)

arXiv:2607.08370 [math.AG]

(or arXiv:2607.08370v2 [math.AG] for this version)

https://doi.org/10.48550/arXiv.2607.08370 Focus to learn more arXiv-issued DOI via DataCite

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