Topological data analysis and topological deep learning beyond persistent homology: a review

Area of Science:

• Applied Mathematics
• Data Science
• Topology

Background:

• Topological Data Analysis (TDA) uses topology for data insights.
• Persistent homology is a key TDA technique, often used with topological deep learning (TDL).
• Limitations of persistent homology include abstraction and data type restrictions.

Purpose of the Study:

• To provide a comprehensive review of TDA and TDL beyond persistent homology.
• To analyze advanced topological methods for complex data analysis.
• To guide the selection of appropriate topological tools for various data types.

Main Methods:

• Review of persistent topological Laplacians and Dirac operators for spectral representations.
• Exploration of sheaf theory, Mayer topology, and interaction topology formulations.
• Discussion of differential topology techniques for manifold data, including persistent de Rham cohomology and Hodge decomposition.
• Analysis of geometric topology approaches for 1D curves, such as persistent Jones polynomials and Khovanov homology.

Main Results:

• Advanced TDA and TDL methods capture topological invariants and homotopic evolution.
• Techniques are presented for diverse data, including point clouds, sequential data, manifolds, and curves.
• Guidance on selecting appropriate topological tools based on data characteristics is provided.
• A review of topological representations, software, and machine learning vectorizations is included.

Conclusions:

• This review expands the scope of TDA and TDL beyond persistent homology.
• Advanced topological methods offer powerful tools for analyzing complex, diverse datasets.
• The findings facilitate more effective application of TDA and TDL across various scientific and industrial domains.