Tubular Neighbourhoods of Pfaffian Sets and Applications to Neural Networks

2026-07-09

AI Summary

This work investigates the robustness of decision boundaries in neural network classifiers equipped with Pfaffian activation functions, introducing Pfaffian set theory into neural network analysis for the first time. By establishing a connection between the volume of tubular neighborhoods of smooth Pfaffian hypersurfaces and the Pfaffian complexity of their defining functions, and by integrating tools from differential geometry, real analytic geometry, and probabilistic tail bounds, the authors derive tail probability bounds for the classifier’s condition number under both uniform and Gaussian input distributions. For single-hidden-layer sigmoid networks with rational weights, they further obtain a polynomial upper bound on the tubular neighborhood volume in terms of network width, thereby providing geometric-probabilistic guarantees for networks employing non-algebraic activation functions.

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.

Problem

• tubular neighbourhoods
• Pfaffian sets
• neural networks
• condition number
• robustness

Innovation

• Pfaffian sets
• tubular neighbourhoods
• neural network robustness
• condition number
• decision boundary