# Exposition and Interpretation of the Topology of Neural Networks

arXiv: https://arxiv.org/abs/1810.03234 · HTML: https://mlsanity.com/paper/1810.03234/

## Key facts

- Cited by 20 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Oct 2018.

## Built on by (most cited first, 20 of 20)

- [Activation Landscapes as a Topological Summary of Neural Network Performance](https://mlsanity.com/paper/2110.10136/) (2021)
- [Experimental Observations of the Topology of Convolutional Neural Network Activations](https://mlsanity.com/paper/2212.00222/) (2022)
- [Understanding Mode Connectivity via Parameter Space Symmetry](https://mlsanity.com/paper/2505.23681/) (2025)
- [Predicting the generalization gap in neural networks using topological data analysis](https://mlsanity.com/paper/2203.12330/) (2022)
- [Quantitative Performance Assessment of CNN Units via Topological Entropy Calculation](https://mlsanity.com/paper/2103.09716/) (2021)
- [Revisiting Point Cloud Completion: Are We Ready For The Real-World?](https://mlsanity.com/paper/2411.17580/) (2024)
- [Singular value decomposition based matrix surgery](https://mlsanity.com/paper/2302.11446/) (2023)
- [Wasserstein Auto-Encoders of Merge Trees (and Persistence Diagrams)](https://mlsanity.com/paper/2307.02509/) (2023)
- [Do Neural Networks Trained with Topological Features Learn Different Internal Representations?](https://mlsanity.com/paper/2211.07697/) (2022)
- [Mixup Barcodes: Quantifying Geometric-Topological Interactions between Point Clouds](https://mlsanity.com/paper/2402.15058/) (2024)
- [Texture synthesis via projection onto multiscale, multilayer statistics](https://mlsanity.com/paper/2105.10825/) (2021)
- [Deep Augmentation: Dropout as Augmentation for Self-Supervised Learning](https://mlsanity.com/paper/2303.14537/) (2023)
- [Addressing caveats of neural persistence with deep graph persistence](https://mlsanity.com/paper/2307.10865/) (2023)
- [Spherical and Hyperbolic Toric Topology-Based Codes On Graph Embedding for Ising MRF Models: Classical and Quantum Topology Machine Learning](https://mlsanity.com/paper/2307.15778/) (2023)
- [A Quotient Homology Theory of Representation in Neural Networks](https://mlsanity.com/paper/2502.01360/) (2025)
- [Topological Regularization for Dense Prediction](https://mlsanity.com/paper/2111.10984/) (2021)
- [Leveraging The Topological Consistencies of Learning in Deep Neural Networks](https://mlsanity.com/paper/2111.15651/) (2021)
- [Topology-enhanced machine learning for speech signal processing](https://mlsanity.com/paper/2311.15210/) (2023)
- [PI-Att: Topology Attention for Segmentation Networks through Adaptive Persistence Image Representation](https://mlsanity.com/paper/2408.08038/) (2024)
- [Persistent Entropy as a Detector of Phase Transitions](https://mlsanity.com/paper/2602.09058/) (2026)

## Abstract (arXiv)

Convolutional neural networks (CNN's) are powerful and widely used tools. However, their interpretability is far from ideal. One such shortcoming is the difficulty of deducing a network's ability to generalize to unseen data. We use topological data analysis to show that the information encoded in the weights of a CNN can be organized in terms of a topological data model and demonstrate how such information can be interpreted and utilized. We show that the weights of convolutional layers at depths from 1 through 13 learn simple global structures. We also demonstrate the change of the simple structures over the course of training. In particular, we define and analyze the spaces of spatial filters of convolutional layers and show the recurrence, among all networks, depths, and during training, of a simple circle consisting of rotating edges, as well as a less recurring unanticipated complex circle that combines lines, edges, and non-linear patterns. We also demonstrate that topological structure correlates with a network's ability to generalize to unseen data and that topological information can be used to improve a network's performance. We train over a thousand CNN's on MNIST, CIFAR-10, SVHN, and ImageNet.