Tubular Neighbourhoods of Pfaffian Sets and Applications to Neural Networks

The authors derive bounds for the volume of tubular neighborhoods of smooth Pfaffian hypersurfaces, generalizing known results for algebraic varieties in terms of the Pfaffian format of defining functions.

As an application, they obtain tail bounds on the probability distribution of a condition number measuring the robustness of neural network classifiers with Pfaffian activation functions in both uniform and Gaussian settings. For single-hidden-layer sigmoid networks with rational weights, polynomial-in-width bounds for tubular neighborhoods of the decision boundary are derived.