# Topological Approaches to Deep Learning

arXiv: https://arxiv.org/abs/1811.01122 · HTML: https://mlsanity.com/paper/1811.01122/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Nov 2018.

## Built on by (most cited first, 29 of 29)

- [Z-GCNETs: Time Zigzags at Graph Convolutional Networks for Time Series Forecasting](https://mlsanity.com/paper/2105.04100/) (2021)
- [Topological Deep Learning: A Review of an Emerging Paradigm](https://mlsanity.com/paper/2302.03836/) (2023)
- [Topological Uncertainty: Monitoring trained neural networks through persistence of activation graphs](https://mlsanity.com/paper/2105.04404/) (2021)
- [Activation Landscapes as a Topological Summary of Neural Network Performance](https://mlsanity.com/paper/2110.10136/) (2021)
- [Topology and geometry of data manifold in deep learning](https://mlsanity.com/paper/2204.08624/) (2022)
- [Tensor-view Topological Graph Neural Network](https://mlsanity.com/paper/2401.12007/) (2024)
- [Predicting the generalization gap in neural networks using topological data analysis](https://mlsanity.com/paper/2203.12330/) (2022)
- [Topological Deep Learning](https://mlsanity.com/paper/2101.05778/) (2021)
- [Deep neural networks architectures from the perspective of manifold learning](https://mlsanity.com/paper/2306.03406/) (2023)
- [UGformer for Robust Left Atrium and Scar Segmentation Across Scanners](https://mlsanity.com/paper/2210.05151/) (2022)
- [Tensor-Fused Multi-View Graph Contrastive Learning](https://mlsanity.com/paper/2410.15247/) (2024)
- [Classification based on Topological Data Analysis](https://mlsanity.com/paper/2102.03709/) (2021)
- [Abstraction, Reasoning and Deep Learning: A Study of the "Look and Say" Sequence](https://mlsanity.com/paper/2109.12755/) (2021)
- [Zeroth-Order Topological Insights into Iterative Magnitude Pruning](https://mlsanity.com/paper/2206.06563/) (2022)
- [Topology Estimation of Simulated 4D Image Data by Combining Downscaling and Convolutional Neural Networks](https://mlsanity.com/paper/2306.14442/) (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)
- [TMetaNet: Topological Meta-Learning Framework for Dynamic Link Prediction](https://mlsanity.com/paper/2506.00453/) (2025)
- [Bayesian Topological Convolutional Neural Nets](https://mlsanity.com/paper/2510.11704/) (2025)
- [From augmented microscopy to the topological transformer: a new approach in cell image analysis for Alzheimer's research](https://mlsanity.com/paper/2108.01625/) (2021)
- [Leveraging The Topological Consistencies of Learning in Deep Neural Networks](https://mlsanity.com/paper/2111.15651/) (2021)
- [Decorrelating neurons using persistence](https://mlsanity.com/paper/2308.04870/) (2023)
- [Pool-Based Active Learning with Proper Topological Regions](https://mlsanity.com/paper/2310.01597/) (2023)
- [Topology-enhanced machine learning for speech signal processing](https://mlsanity.com/paper/2311.15210/) (2023)
- [Topology-Aware Exploration of Energy-Based Models Equilibrium: Toric QC-LDPC Codes and Hyperbolic MET QC-LDPC Codes](https://mlsanity.com/paper/2401.14749/) (2024)
- [Towards an Algebraic Framework For Approximating Functions Using Neural Network Polynomials](https://mlsanity.com/paper/2402.01058/) (2024)
- [Topological Deep Learning for Speech Data](https://mlsanity.com/paper/2505.21173/) (2025)
- [Graph Lineages and Skeletal Graph Products](https://mlsanity.com/paper/2508.00197/) (2025)
- [Reparameterization through Coverings and Topological Weight Priors](https://mlsanity.com/paper/2604.23804/) (2026)
- [Graph Representation via Elements of Discrete Morse and Cobordism Theories](https://mlsanity.com/paper/2610.01937/) (2026)

## Abstract (arXiv)

We perform topological data analysis on the internal states of convolutional deep neural networks to develop an understanding of the computations that they perform. We apply this understanding to modify the computations so as to (a) speed up computations and (b) improve generalization from one data set of digits to another. One byproduct of the analysis is the production of a geometry on new sets of features on data sets of images, and use this observation to develop a methodology for constructing analogues of CNN's for many other geometries, including the graph structures constructed by topological data analysis.