# Understanding and Quantifying Adversarial Examples Existence in Linear Classification

arXiv: https://arxiv.org/abs/1910.12163 · HTML: https://mlsanity.com/paper/1910.12163/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Oct 2019.

## Built on by (most cited first, 3 of 3)

- [Adversarial Fine-Tuning of Language Models: An Iterative Optimisation Approach for the Generation and Detection of Problematic Content](https://mlsanity.com/paper/2308.13768/) (2023)
- [Understanding Robustness in Teacher-Student Setting: A New Perspective](https://mlsanity.com/paper/2102.13170/) (2021)
- [Finding Dynamics Preserving Adversarial Winning Tickets](https://mlsanity.com/paper/2202.06488/) (2022)

## Abstract (arXiv)

State-of-art deep neural networks (DNN) are vulnerable to attacks by adversarial examples: a carefully designed small perturbation to the input, that is imperceptible to human, can mislead DNN. To understand the root cause of adversarial examples, we quantify the probability of adversarial example existence for linear classifiers. Previous mathematical definition of adversarial examples only involves the overall perturbation amount, and we propose a more practical relevant definition of strong adversarial examples that separately limits the perturbation along the signal direction also. We show that linear classifiers can be made robust to strong adversarial examples attack in cases where no adversarial robust linear classifiers exist under the previous definition. The quantitative formulas are confirmed by numerical experiments using a linear support vector machine (SVM) classifier. The results suggest that designing general strong-adversarial-robust learning systems is feasible but only through incorporating human knowledge of the underlying classification problem.