Topological Machine Learning with Unreduced Persistence Diagrams

TL;DR

This paper introduces methods to generate topological features from unreduced boundary matrices and demonstrates that using unreduced persistence diagrams can match or outperform reduced ones in machine learning tasks, with potential computational benefits.

Contribution

The paper presents novel techniques for creating feature vectors from unreduced boundary matrices and compares their effectiveness to reduced diagrams in machine learning pipelines.

Findings

Unreduced diagrams can perform as well as or better than reduced diagrams in some tasks.

Using unreduced boundary matrices can reduce computational costs.

Topological features from unreduced diagrams retain significant information for machine learning.

Abstract

Supervised machine learning pipelines trained on features derived from persistent homology have been experimentally observed to ignore much of the information contained in a persistence diagram. Computing persistence diagrams is often the most computationally demanding step in such a pipeline, however. To explore this, we introduce several methods to generate topological feature vectors from unreduced boundary matrices. We compared the performance of pipelines trained on vectorizations of unreduced PDs to vectorizations of fully-reduced PDs across several data and task types. Our results indicate that models trained on PDs built from unreduced diagrams can perform on par and even outperform those trained on fully-reduced diagrams on some tasks. This observation suggests that machine learning pipelines which incorporate topology-based features may benefit in terms of computational cost and performance by utilizing information contained in unreduced boundary matrices.