# Tubular Neighbourhoods of Pfaffian Sets and Applications to Neural Networks

arXiv: https://arxiv.org/abs/2607.08370 · HTML: https://mlsanity.com/paper/2607.08370/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 45 references that are in the map.
- First posted on arXiv 9 Jul 2026.

## Builds on (most cited first, 15 of 15)

- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [DeepFool: a simple and accurate method to fool deep neural networks](https://mlsanity.com/paper/1511.04599/) (2015)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [Complexity of Linear Regions in Deep Networks](https://mlsanity.com/paper/1901.09021/) (2019)
- [Bounding and Counting Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1711.02114/) (2017)
- [Robustness of classifiers: from adversarial to random noise](https://mlsanity.com/paper/1608.08967/) (2016)
- [Adversarial vulnerability for any classifier](https://mlsanity.com/paper/1802.08686/) (2018)
- [The Curse of Concentration in Robust Learning: Evasion and Poisoning Attacks from Concentration of Measure](https://mlsanity.com/paper/1809.03063/) (2018)
- [Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometry](https://mlsanity.com/paper/2501.18915/) (2025)
- [Covering Number of Real Algebraic Varieties and Beyond: Improved Bounds and Applications](https://mlsanity.com/paper/2311.05116/) (2023)
- [VC dimension of Graph Neural Networks with Pfaffian activation functions](https://mlsanity.com/paper/2401.12362/) (2024)
- [Hausdorff approximations and volume of tubes of singular algebraic sets](https://mlsanity.com/paper/2104.05053/) (2021)

## Abstract (arXiv)

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.