Topological Deep Learning: Classification Neural Networks

TL;DR

This paper introduces a topological framework for understanding classification problems in deep learning, providing insights into when neural networks can or cannot solve these problems.

Contribution

It formalizes classification in deep learning using topology, revealing conditions for solvability and offering new perspectives beyond traditional methods.

Findings

Identifies topological conditions for neural network classification success

Highlights limitations of neural networks in certain topological settings

Provides a novel formalism linking topology and deep learning

Abstract

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.