# Topological Deep Learning: Classification Neural Networks

arXiv: https://arxiv.org/abs/2102.08354 · HTML: https://mlsanity.com/paper/2102.08354/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 35 references that are in the map.
- First posted on arXiv 16 Feb 2021.

## Builds on (most cited first, 14 of 14)

- [Grad-CAM: Visual Explanations from Deep Networks via Gradient-based Localization](https://mlsanity.com/paper/1610.02391/) (2016)
- [Visualizing and Understanding Convolutional Networks](https://mlsanity.com/paper/1311.2901/) (2013)
- [Understanding Neural Networks Through Deep Visualization](https://mlsanity.com/paper/1506.06579/) (2015)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Approximating Continuous Functions by ReLU Nets of Minimal Width](https://mlsanity.com/paper/1710.11278/) (2017)
- [Deep Learning with Topological Signatures](https://mlsanity.com/paper/1707.04041/) (2017)
- [Topology of deep neural networks](https://mlsanity.com/paper/2004.06093/) (2020)
- [A Topology Layer for Machine Learning](https://mlsanity.com/paper/1905.12200/) (2019)
- [Geometric Understanding of Deep Learning](https://mlsanity.com/paper/1805.10451/) (2018)
- [Deep, Skinny Neural Networks are not Universal Approximators](https://mlsanity.com/paper/1810.00393/) (2018)
- [Neural Networks Should Be Wide Enough to Learn Disconnected Decision Regions](https://mlsanity.com/paper/1803.00094/) (2018)
- [What can topology tell us about the neural code?](https://mlsanity.com/paper/1605.01905/) (2016)
- [A Topological Framework for Deep Learning](https://mlsanity.com/paper/2008.13697/) (2020)
- [Using Contour Trees in the Analysis and Visualization of Radio Astronomy Data Cubes](https://mlsanity.com/paper/1704.04561/) (2017)

## Built on by (most cited first, 5 of 5)

- [Demystifying Higher-Order Graph Neural Networks](https://mlsanity.com/paper/2406.12841/) (2024)
- [ICML Topological Deep Learning Challenge 2024: Beyond the Graph Domain](https://mlsanity.com/paper/2409.05211/) (2024)
- [Defining Neural Network Architecture through Polytope Structures of Dataset](https://mlsanity.com/paper/2402.02407/) (2024)
- [Estimating class separability of text embeddings with persistent homology](https://mlsanity.com/paper/2305.15016/) (2023)
- [Data Topology-Dependent Upper Bounds of Neural Network Widths](https://mlsanity.com/paper/2305.16375/) (2023)

## Abstract (arXiv)

Topological deep learning is a formalism that is aimed at introducing topological language to deep learning for the purpose of utilizing the minimal mathematical structures to formalize problems that arise in a generic deep learning problem. This is the first of a sequence of articles with the purpose of introducing and studying this formalism. In this article, we define and study the classification problem in machine learning in a topological setting. Using this topological framework, we show when the classification problem is possible or not possible in the context of neural networks. Finally, we demonstrate how our topological setting immediately illuminates aspects of this problem that are not as readily apparent using traditional tools.