Tubular Neighbourhoods of Pfaffian Sets and Applications to Neural Networks
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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.
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References · 15
- 2014Explaining and Harnessing Adversarial Examplescited by 7,287
- 2013Intriguing properties of neural networkscited by 5,114
- 2016Gaussian Error Linear Units (GELUs)cited by 3,431
- 2015DeepFool: a simple and accurate method to fool deep neural networkscited by 1,315
- 2014On the Number of Linear Regions of Deep Neural Networkscited by 378
- 2016On the Expressive Power of Deep Neural Networkscited by 302
- 2019Complexity of Linear Regions in Deep Networkscited by 166
- 2017Bounding and Counting Linear Regions of Deep Neural Networkscited by 161
- 2016Robustness of classifiers: from adversarial to random noisecited by 87
- 2018Adversarial vulnerability for any classifiercited by 78
- 2018The Curse of Concentration in Robust Learning: Evasion and Poisoning Attacks from Concentration of Measurecited by 44
- 2025Algebra Unveils Deep Learning -- An Invitation to Neuroalgebraic Geometrycited by 21
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