Adaptive Partitioning for Template Functions on Persistence Diagrams

TL;DR

This paper introduces an adaptive partitioning method for persistence diagrams in topological data analysis, enhancing feature extraction for machine learning by improving stability and classification performance.

Contribution

It proposes a novel adaptive partitioning approach and parameter selection framework for template functions on persistence diagrams, advancing feature representation techniques.

Findings

Improved classification accuracy with adaptive partitioning.

Enhanced stability of persistence diagram features.

Competitive performance compared to existing methods.

Abstract

As the field of Topological Data Analysis continues to show success in theory and in applications, there has been increasing interest in using tools from this field with methods for machine learning. Using persistent homology, specifically persistence diagrams, as inputs to machine learning techniques requires some mathematical creativity. The space of persistence diagrams does not have the desirable properties for machine learning, thus methods such as kernel methods and vectorization methods have been developed. One such featurization of persistence diagrams by Perea, Munch and Khasawneh uses continuous, compactly supported functions, referred to as "template functions," which results in a stable vector representation of the persistence diagram. In this paper, we provide a method of adaptively partitioning persistence diagrams to improve these featurizations based on localized information in the diagrams. Additionally, we provide a framework to adaptively select parameters required for the template functions in order to best utilize the partitioning method. We present results for application to example data sets comparing classification results between template function featurizations with and without partitioning, in addition to other methods from the literature.