# A Topological Framework for Deep Learning

arXiv: https://arxiv.org/abs/2008.13697 · HTML: https://mlsanity.com/paper/2008.13697/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Aug 2020.

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

- [Topological Expressivity of ReLU Neural Networks](https://mlsanity.com/paper/2310.11130/) (2023)
- [Topological Deep Learning: Classification Neural Networks](https://mlsanity.com/paper/2102.08354/) (2021)
- [Singular value decomposition based matrix surgery](https://mlsanity.com/paper/2302.11446/) (2023)
- [Defining Neural Network Architecture through Polytope Structures of Dataset](https://mlsanity.com/paper/2402.02407/) (2024)
- [Data Topology-Dependent Upper Bounds of Neural Network Widths](https://mlsanity.com/paper/2305.16375/) (2023)

## Abstract (arXiv)

We utilize classical facts from topology to show that the classification problem in machine learning is always solvable under very mild conditions. Furthermore, we show that a softmax classification network acts on an input topological space by a finite sequence of topological moves to achieve the classification task. Moreover, given a training dataset, we show how topological formalism can be used to suggest the appropriate architectural choices for neural networks designed to be trained as classifiers on the data. Finally, we show how the architecture of a neural network cannot be chosen independently from the shape of the underlying data. To demonstrate these results, we provide example datasets and show how they are acted upon by neural nets from this topological perspective.