# High Dimensional Spaces, Deep Learning and Adversarial Examples

arXiv: https://arxiv.org/abs/1801.00634 · HTML: https://mlsanity.com/paper/1801.00634/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jan 2018.

## Built on by (most cited first, 5 of 5)

- [Disrupting Adversarial Transferability in Deep Neural Networks](https://mlsanity.com/paper/2108.12492/) (2021)
- [Physical Passive Patch Adversarial Attacks on Visual Odometry Systems](https://mlsanity.com/paper/2207.05729/) (2022)
- [Random Projections for Improved Adversarial Robustness](https://mlsanity.com/paper/2102.09230/) (2021)
- [A novel network training approach for open set image recognition](https://mlsanity.com/paper/2109.12756/) (2021)
- [Decorrelative Network Architecture for Robust Electrocardiogram Classification](https://mlsanity.com/paper/2207.09031/) (2022)

## Abstract (arXiv)

In this paper, we analyze deep learning from a mathematical point of view and derive several novel results. The results are based on intriguing mathematical properties of high dimensional spaces. We first look at perturbation based adversarial examples and show how they can be understood using topological and geometrical arguments in high dimensions. We point out mistake in an argument presented in prior published literature, and we present a more rigorous, general and correct mathematical result to explain adversarial examples in terms of topology of image manifolds. Second, we look at optimization landscapes of deep neural networks and examine the number of saddle points relative to that of local minima. Third, we show how multiresolution nature of images explains perturbation based adversarial examples in form of a stronger result. Our results state that expectation of L2-norm of adversarial perturbations is Oleft(frac1√nright) and therefore shrinks to 0 as image resolution n becomes arbitrarily large. Finally, by incorporating the parts-whole manifold learning hypothesis for natural images, we investigate the working of deep neural networks and root causes of adversarial examples and discuss how future improvements can be made and how adversarial examples can be eliminated.