Cycle Registration in Persistent Homology With Applications in Topological Bootstrap

Area of Science:

• Topology
• Data Analysis
• Computational Geometry

Background:

• Persistent homology is a powerful tool for analyzing the shape of data.
• Current methods often rely on numerical summaries like persistence diagrams, which can lose information.
• Comparing topological representations directly offers a more comprehensive analysis.

Purpose of the Study:

• To develop a novel framework for directly comparing persistent homology representations of two spaces.
• To move beyond traditional numerical summaries and leverage the full topological information.
• To improve topological inference and feature detection in complex datasets.

Main Methods:

• Defining a correspondence relation between persistent cycles of different spaces.
• Developing a computational method to establish this cycle correspondence.
• Matching cycles based on persistence intervals and spatial location.
• Utilizing statistical bootstrap methods for topological inference.

Main Results:

• A new framework for direct comparison of topological representations.
• Cycle matching considers both topological persistence and spatial embedding.
• Demonstrated effectiveness in distinguishing real features from noise in point cloud data.

Conclusions:

• Direct comparison of persistent homology cycles offers richer insights than summary statistics.
• The proposed method enhances topological inference capabilities.
• This approach provides a more robust way to analyze and compare complex data structures.