A Survey of Vectorization Methods in Topological Data Analysis.

Area of Science:

• Computational topology
• Machine learning
• Data science

Background:

• Topological data analysis (TDA) offers powerful tools for understanding complex data structures.
• Persistent homology barcodes are a key output of TDA, capturing shape information.
• Effective vectorization of these barcodes is crucial for their integration into supervised learning.

Purpose of the Study:

• To systematically evaluate and compare existing methods for vectorizing persistent homology barcodes.
• To identify the most effective vectorization techniques for supervised classification tasks.
• To provide a framework for understanding and experimenting with these methods.

Main Methods:

• Reviewed and categorized thirteen distinct vectorization techniques for persistent homology barcodes.
• Developed an organizational framework to classify these vectorization approaches.
• Conducted a comprehensive benchmarking study using three established supervised classification datasets.

Main Results:

• Identified significant performance variations among the thirteen vectorization methods.
• Discovered that a straightforward vectorization approach, relying on elementary summary statistics, achieved superior performance.
• Demonstrated the effectiveness of simple methods over more complex topological feature extraction techniques in these tasks.

Conclusions:

• The choice of vectorization method significantly impacts the success of integrating topological features into machine learning models.
• Simple summary statistic-based vectorizations can be highly competitive and efficient for classification.
• A web application is provided to aid researchers in exploring and selecting appropriate vectorization strategies.