# A Geometric Framework for Adversarial Vulnerability in Machine Learning

arXiv: https://arxiv.org/abs/2407.11029 · HTML: https://mlsanity.com/paper/2407.11029/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- First posted on arXiv 3 Jul 2024.

## Abstract (arXiv)

This work starts with the intention of using mathematics to understand the intriguing vulnerability observed by ~citetszegedy2013 within artificial neural networks. Along the way, we will develop some novel tools with applications far outside of just the adversarial domain. We will do this while developing a rigorous mathematical framework to examine this problem. Our goal is to build out theory which can support increasingly sophisticated conjecture about adversarial attacks with a particular focus on the so called ``Dimpled Manifold Hypothesis'' by ~citetshamir2021dimpled. Chapter one will cover the history and architecture of neural network architectures. Chapter two is focused on the background of adversarial vulnerability. Starting from the seminal paper by ~citetszegedy2013 we will develop the theory of adversarial perturbation and attack. Chapter three will build a theory of persistence that is related to Ricci Curvature, which can be used to measure properties of decision boundaries. We will use this foundation to make a conjecture relating adversarial attacks. Chapters four and five represent a sudden and wonderful digression that examines an intriguing related body of theory for spatial analysis of neural networks as approximations of kernel machines and becomes a novel theory for representing neural networks with bilinear maps. These heavily mathematical chapters will set up a framework and begin exploring applications of what may become a very important theoretical foundation for analyzing neural network learning with spatial and geometric information. We will conclude by setting up our new methods to address the conjecture from chapter 3 in continuing research.