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Topological Deep Learning: A Review of an Emerging Paradigm

Ali Zia, Abdelwahed Khamis, James Nichols, Zeeshan Hayder, Vivien Rolland, Lars Petersson

TL;DR

Topological Deep Learning targets integrating topological data analysis with deep models to obtain global, deformation-tolerant representations. The paper surveys core TDA concepts (simplicial complexes, persistent homology, and diagram representations) and categorizes methods to embed, augment, or regularize neural networks with topological information, including post-training analytics. It presents a unified taxonomy of existing approaches, discusses practical implementations and libraries, and highlights open challenges. Overall, topology-guided learning offers robustness, data efficiency, and interpretability benefits across domains, while computational and theoretical hurdles remain.

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.

Topological Deep Learning: A Review of an Emerging Paradigm

TL;DR

Topological Deep Learning targets integrating topological data analysis with deep models to obtain global, deformation-tolerant representations. The paper surveys core TDA concepts (simplicial complexes, persistent homology, and diagram representations) and categorizes methods to embed, augment, or regularize neural networks with topological information, including post-training analytics. It presents a unified taxonomy of existing approaches, discusses practical implementations and libraries, and highlights open challenges. Overall, topology-guided learning offers robustness, data efficiency, and interpretability benefits across domains, while computational and theoretical hurdles remain.

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.
Paper Structure (15 sections, 3 equations, 1 figure)

This paper contains 15 sections, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview of TDA
  3. Simplicial complexes and persistent homology
  4. Representations of persistent homology
  5. Homological feature vectorizations
  6. Topological Deep Learning (TDL)
  7. Learning Topological Features Embedding
  8. Integration of Topological Representations
  9. Topological Loss
  10. Deep Topological Analytics
  11. Discussion
  12. Challenges
  13. Successes and Future Directions:
  14. Implementations
  15. Conclusion

Figures (1)

  • Figure 1: Topological Deep Learning introduces TDA methods to deep models leading to topological neural architectures that can potentially address deep learning limitations. This is done by plugging topological components for (a) learning features Embedding (Section \ref{['sec:tda_embedding']}), (b) enhancing the learned Representations (Section \ref{['sec:tda_integration']}), and/or (c) regularizing the model using a topological Loss (Section \ref{['sec:top_loss']}). Beyond that, (d) TDA can be used post-training to reveal insights of trained models (interpretability) (Section \ref{['sec:tda-measure']}).