# Geometric origin of adversarial vulnerability in deep learning

arXiv: https://arxiv.org/abs/2509.01235 · HTML: https://mlsanity.com/paper/2509.01235/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 36 references that are in the map.
- First posted on arXiv 1 Sep 2025.

## Builds on (most cited first, 21 of 21)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Explaining and Harnessing Adversarial Examples](https://mlsanity.com/paper/1412.6572/) (2014)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Intriguing properties of neural networks](https://mlsanity.com/paper/1312.6199/) (2013)
- [Sparks of Artificial General Intelligence: Early experiments with GPT-4](https://mlsanity.com/paper/2303.12712/) (2023)
- [Shortcut Learning in Deep Neural Networks](https://mlsanity.com/paper/2004.07780/) (2020)
- [Robustness May Be at Odds with Accuracy](https://mlsanity.com/paper/1805.12152/) (2018)
- [Prevalence of Neural Collapse during the terminal phase of deep learning training](https://mlsanity.com/paper/2008.08186/) (2020)
- [Deep supervised learning using local errors](https://mlsanity.com/paper/1711.06756/) (2017)
- [A Law of Data Separation in Deep Learning](https://mlsanity.com/paper/2210.17020/) (2022)
- [Hierarchical nucleation in deep neural networks](https://mlsanity.com/paper/2007.03506/) (2020)
- [The emergence of a concept in shallow neural networks](https://mlsanity.com/paper/2109.00454/) (2021)
- [Supervised Hebbian Learning](https://mlsanity.com/paper/2203.01304/) (2022)
- [Adversarial Examples Might be Avoidable: The Role of Data Concentration in Adversarial Robustness](https://mlsanity.com/paper/2309.16096/) (2023)
- [GanDef: A GAN based Adversarial Training Defense for Neural Network Classifier](https://mlsanity.com/paper/1903.02585/) (2019)
- [Eight challenges in developing theory of intelligence](https://mlsanity.com/paper/2306.11232/) (2023)
- [Learning credit assignment](https://mlsanity.com/paper/2001.03354/) (2020)
- [Relationship between manifold smoothness and adversarial vulnerability in deep learning with local errors](https://mlsanity.com/paper/2007.02047/) (2020)
- [Emergence of hierarchical modes from deep learning](https://mlsanity.com/paper/2208.09859/) (2022)
- [Contrastive Self-Supervised Learning As Neural Manifold Packing](https://mlsanity.com/paper/2506.13717/) (2025)
- [Fermi-Bose Machine achieves both generalization and adversarial robustness](https://mlsanity.com/paper/2404.13631/) (2024)

## Abstract (arXiv)

Balancing training accuracy and adversarial robustness has beeen a challenge since the birth of deep learning. Here, we introduce a geometry-aware deep learning framework that leverages layer-wise local training to sculpt the internal representations of deep neural networks. This framework promotes intra-class compactness and inter-class separation in feature space, leading to manifold smoothness and adversarial robustness against white or black box attacks. The performance can be explained by bluedata-dependent statistical mechanics of integrating out the network parameters, bluesupplemented by a phenomenological model with Hebbian coupling between elements of the hidden representation. Based on the current geometry-aware learning framework, the deep network can assimilate new information into existing knowledge structures while reducing representation interference.