# On Space Folds of ReLU Neural Networks

arXiv: https://arxiv.org/abs/2502.09954 · HTML: https://mlsanity.com/paper/2502.09954/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 59 references that are in the map.
- First posted on arXiv 14 Feb 2025.

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

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [XNOR-Net: ImageNet Classification Using Binary Convolutional Neural Networks](https://mlsanity.com/paper/1603.05279/) (2016)
- [Exploring Generalization in Deep Learning](https://mlsanity.com/paper/1706.08947/) (2017)
- [BinaryConnect: Training Deep Neural Networks with binary weights during propagations](https://mlsanity.com/paper/1511.00363/) (2015)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [On the Number of Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1402.1869/) (2014)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [Parseval Networks: Improving Robustness to Adversarial Examples](https://mlsanity.com/paper/1704.08847/) (2017)
- [Benefits of depth in neural networks](https://mlsanity.com/paper/1602.04485/) (2016)
- [Deep Learning: A Critical Appraisal](https://mlsanity.com/paper/1801.00631/) (2018)
- [Sorting out Lipschitz function approximation](https://mlsanity.com/paper/1811.05381/) (2018)
- [Lipschitz-Margin Training: Scalable Certification of Perturbation Invariance for Deep Neural Networks](https://mlsanity.com/paper/1802.04034/) (2018)
- [Complexity of Linear Regions in Deep Networks](https://mlsanity.com/paper/1901.09021/) (2019)
- [Bounding and Counting Linear Regions of Deep Neural Networks](https://mlsanity.com/paper/1711.02114/) (2017)
- [Deep ReLU Networks Have Surprisingly Few Activation Patterns](https://mlsanity.com/paper/1906.00904/) (2019)
- [ReLU Strikes Back: Exploiting Activation Sparsity in Large Language Models](https://mlsanity.com/paper/2310.04564/) (2023)
- [Representation Benefits of Deep Feedforward Networks](https://mlsanity.com/paper/1509.08101/) (2015)
- [The Unreasonable Effectiveness of Deep Learning in Artificial Intelligence](https://mlsanity.com/paper/2002.04806/) (2020)
- [A Study on ReLU and Softmax in Transformer](https://mlsanity.com/paper/2302.06461/) (2023)
- [On the Number of Linear Regions of Convolutional Neural Networks](https://mlsanity.com/paper/2006.00978/) (2020)
- [On the Effectiveness of Lipschitz-Driven Rehearsal in Continual Learning](https://mlsanity.com/paper/2210.06443/) (2022)
- [Hidden symmetries of ReLU networks](https://mlsanity.com/paper/2306.06179/) (2023)
- [XNOR Neural Engine: a Hardware Accelerator IP for 21.6 fJ/op Binary Neural Network Inference](https://mlsanity.com/paper/1807.03010/) (2018)
- [A Survey on Face Data Augmentation](https://mlsanity.com/paper/1904.11685/) (2019)
- [Deep ReLU Networks Preserve Expected Length](https://mlsanity.com/paper/2102.10492/) (2021)
- [Are All Linear Regions Created Equal?](https://mlsanity.com/paper/2202.11749/) (2022)
- [On the Lipschitz Constant of Deep Networks and Double Descent](https://mlsanity.com/paper/2301.12309/) (2023)
- [Origami in N dimensions: How feed-forward networks manufacture linear separability](https://mlsanity.com/paper/2203.11355/) (2022)
- [Understanding the Evolution of Linear Regions in Deep Reinforcement Learning](https://mlsanity.com/paper/2210.13611/) (2022)
- [Trajectory growth lower bounds for random sparse deep ReLU networks](https://mlsanity.com/paper/1911.10651/) (2019)

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

- [The Resurrection of the ReLU](https://mlsanity.com/paper/2505.22074/) (2025)
- [The Space Between: On Folding, Symmetries and Sampling](https://mlsanity.com/paper/2503.08502/) (2025)
- [Bounding Box Anomaly Scoring for simple and efficient Out-of-Distribution detection](https://mlsanity.com/paper/2603.22660/) (2026)

## Abstract (arXiv)

Recent findings suggest that the consecutive layers of ReLU neural networks can be understood geometrically as space folding transformations of the input space, revealing patterns of self-similarity. In this paper, we present the first quantitative analysis of this space folding phenomenon in ReLU neural networks. Our approach focuses on examining how straight paths in the Euclidean input space are mapped to their counterparts in the Hamming activation space. In this process, the convexity of straight lines is generally lost, giving rise to non-convex folding behavior. To quantify this effect, we introduce a novel measure based on range metrics, similar to those used in the study of random walks, and provide the proof for the equivalence of convexity notions between the input and activation spaces. Furthermore, we provide empirical analysis on a geometrical analysis benchmark (CantorNet) as well as an image classification benchmark (MNIST). Our work advances the understanding of the activation space in ReLU neural networks by leveraging the phenomena of geometric folding, providing valuable insights on how these models process input information.