# On the uncertainty principle of neural networks

arXiv: https://arxiv.org/abs/2205.01493 · HTML: https://mlsanity.com/paper/2205.01493/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 37 of its 78 references that are in the map.
- First posted on arXiv 3 May 2022.

## Builds on (most cited first, 30 of 37)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [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)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [Image-to-Image Translation with Conditional Adversarial Networks](https://mlsanity.com/paper/1611.07004/) (2016)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [PIQA: Reasoning about Physical Commonsense in Natural Language](https://mlsanity.com/paper/1911.11641/) (2019)
- [Can a Suit of Armor Conduct Electricity? A New Dataset for Open Book Question Answering](https://mlsanity.com/paper/1809.02789/) (2018)
- [Adversarial examples in the physical world](https://mlsanity.com/paper/1607.02533/) (2016)
- [Theoretically Principled Trade-off between Robustness and Accuracy](https://mlsanity.com/paper/1901.08573/) (2019)
- [Mastering Atari, Go, Chess and Shogi by Planning with a Learned Model](https://mlsanity.com/paper/1911.08265/) (2019)
- [DeepFool: a simple and accurate method to fool deep neural networks](https://mlsanity.com/paper/1511.04599/) (2015)
- [End to End Learning for Self-Driving Cars](https://mlsanity.com/paper/1604.07316/) (2016)
- [Robustness May Be at Odds with Accuracy](https://mlsanity.com/paper/1805.12152/) (2018)
- [The Limitations of Deep Learning in Adversarial Settings](https://mlsanity.com/paper/1511.07528/) (2015)
- [Square Attack: a query-efficient black-box adversarial attack via random search](https://mlsanity.com/paper/1912.00049/) (2019)
- [One pixel attack for fooling deep neural networks](https://mlsanity.com/paper/1710.08864/) (2017)
- [Adversarial Examples for Evaluating Reading Comprehension Systems](https://mlsanity.com/paper/1707.07328/) (2017)
- [Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks](https://mlsanity.com/paper/1901.06523/) (2019)
- [High Frequency Component Helps Explain the Generalization of Convolutional Neural Networks](https://mlsanity.com/paper/1905.13545/) (2019)
- [Audio Adversarial Examples: Targeted Attacks on Speech-to-Text](https://mlsanity.com/paper/1801.01944/) (2018)
- [On instabilities of deep learning in image reconstruction - Does AI come at a cost?](https://mlsanity.com/paper/1902.05300/) (2019)
- [Training behavior of deep neural network in frequency domain](https://mlsanity.com/paper/1807.01251/) (2018)
- [Video Generation From Text](https://mlsanity.com/paper/1710.00421/) (2017)
- [A Universal Law of Robustness via Isoperimetry](https://mlsanity.com/paper/2105.12806/) (2021)
- [Is Robustness the Cost of Accuracy? -- A Comprehensive Study on the Robustness of 18 Deep Image Classification Models](https://mlsanity.com/paper/1808.01688/) (2018)
- [Globally-Robust Neural Networks](https://mlsanity.com/paper/2102.08452/) (2021)

## Built on by (most cited first, 4 of 4)

- [Uncertainty in Authorship: Why Perfect AI Detection Is Mathematically Impossible](https://mlsanity.com/paper/2509.11915/) (2025)
- [Phase-space entropy at acquisition reflects downstream learnability](https://mlsanity.com/paper/2512.19223/) (2025)
- [A Geometric Probe of the Accuracy-Robustness Trade-off: Sharp Boundaries in Symmetry-Breaking Dimensional Expansion](https://mlsanity.com/paper/2602.17948/) (2026)
- [Hardware-Friendly Input Expansion for Accelerating Function Approximation](https://mlsanity.com/paper/2602.17952/) (2026)

## Abstract (arXiv)

In this study, we explore the inherent trade-off between accuracy and robustness in neural networks, drawing an analogy to the uncertainty principle in quantum mechanics. We propose that neural networks are subject to an uncertainty relation, which manifests as a fundamental limitation in their ability to simultaneously achieve high accuracy and robustness against adversarial attacks. Through mathematical proofs and empirical evidence, we demonstrate that this trade-off is a natural consequence of the sharp boundaries formed between different class concepts during training. Our findings reveal that the complementarity principle, a cornerstone of quantum physics, applies to neural networks, imposing fundamental limits on their capabilities in simultaneous learning of conjugate features. Meanwhile, our work suggests that achieving human-level intelligence through a single network architecture or massive datasets alone may be inherently limited. Our work provides new insights into the theoretical foundations of neural network vulnerability and opens up avenues for designing more robust neural network architectures.