PreprintCC BY 4.0arXiv
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.
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BibTeX
@misc{lezeau2026tubular,
title = {{Tubular Neighbourhoods of Pfaffian Sets and Applications to Neural Networks}},
author = {Paul Lezeau and Martin Lotz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.08370},
eprint = {2607.08370},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.08370},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Tubular Neighbourhoods of Pfaffian Sets and Applications to Neural Networks
AU - Paul Lezeau
AU - Martin Lotz
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2607.08370
UR - https://arxiv.org/abs/2607.08370
AB - 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.
ER -