# Topological Learning in Multi-Class Data Sets

arXiv: https://arxiv.org/abs/2301.09734 · HTML: https://mlsanity.com/paper/2301.09734/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 60 references that are in the map.
- First posted on arXiv 23 Jan 2023.

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

- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)
- [Approximating Continuous Functions by ReLU Nets of Minimal Width](https://mlsanity.com/paper/1710.11278/) (2017)
- [Discovering physical concepts with neural networks](https://mlsanity.com/paper/1807.10300/) (2018)
- [Geometric Deep Learning on Molecular Representations](https://mlsanity.com/paper/2107.12375/) (2021)
- [Topology of deep neural networks](https://mlsanity.com/paper/2004.06093/) (2020)
- [Deep Neural Network Approximation Theory](https://mlsanity.com/paper/1901.02220/) (2019)
- [Real-time gravitational-wave science with neural posterior estimation](https://mlsanity.com/paper/2106.12594/) (2021)
- [AI Poincar'e: Machine Learning Conservation Laws from Trajectories](https://mlsanity.com/paper/2011.04698/) (2020)
- [On transversality of bent hyperplane arrangements and the topological expressiveness of ReLU neural networks](https://mlsanity.com/paper/2008.09052/) (2020)
- [Deep Networks and the Multiple Manifold Problem](https://mlsanity.com/paper/2008.11245/) (2020)
- [Parameterized Machine Learning for High-Energy Physics](https://mlsanity.com/paper/1601.07913/) (2016)
- [Reformulation of the No-Free-Lunch Theorem for Entangled Data Sets](https://mlsanity.com/paper/2007.04900/) (2020)
- [Deep Learning Topological Invariants of Band Insulators](https://mlsanity.com/paper/1805.10503/) (2018)
- [Deep Reinforcement Learning Control of Quantum Cartpoles](https://mlsanity.com/paper/1910.09200/) (2019)

## Abstract (arXiv)

We specialize techniques from topological data analysis to the problem of characterizing the topological complexity (as defined in the body of the paper) of a multi-class data set. As a by-product, a topological classifier is defined that uses an open sub-covering of the data set. This sub-covering can be used to construct a simplicial complex whose topological features (e.g., Betti numbers) provide information about the classification problem. We use these topological constructs to study the impact of topological complexity on learning in feedforward deep neural networks (DNNs). We hypothesize that topological complexity is negatively correlated with the ability of a fully connected feedforward deep neural network to learn to classify data correctly. We evaluate our topological classification algorithm on multiple constructed and open source data sets. We also validate our hypothesis regarding the relationship between topological complexity and learning in DNN's on multiple data sets.