Topological Deep Learning: A Review of an Emerging Paradigm

TL;DR

This paper reviews the emerging field of topological deep learning, highlighting how topological data analysis enhances deep learning models with stable, shape-aware representations and discussing future research directions.

Contribution

It provides a comprehensive overview of TDA concepts, explores their integration into deep learning, and discusses the challenges and future prospects of topological deep learning.

Findings

TDA offers stable, shape-aware data summaries for deep learning.

Integration of TDA techniques into deep models is increasing.

Topological analytics can improve understanding of deep model behavior.

Abstract

Topological data analysis (TDA) provides insight into data shape. The summaries obtained by these methods are principled global descriptions of multi-dimensional data whilst exhibiting stable properties such as robustness to deformation and noise. Such properties are desirable in deep learning pipelines but they are typically obtained using non-TDA strategies. This is partly caused by the difficulty of combining TDA constructs (e.g. barcode and persistence diagrams) with current deep learning algorithms. Fortunately, we are now witnessing a growth of deep learning applications embracing topologically-guided components. In this survey, we review the nascent field of topological deep learning by first revisiting the core concepts of TDA. We then explore how the use of TDA techniques has evolved over time to support deep learning frameworks, and how they can be integrated into different aspects of deep learning. Furthermore, we touch on TDA usage for analyzing existing deep models; deep topological analytics. Finally, we discuss the challenges and future prospects of topological deep learning.