Leveraging Persistent Homology for Topological Feature Extraction in Machine Learning: The PH-AML Pipeline

Abstract from my paper

Conventional Machine learning models often rely on purely statistical data while overlooking the rich global structure of complex datasets. In this paper, we propose PHAML (Persistent Homology-Augmented Machine Learning), a pipeline designed to address this shortcoming by effectively extracting topological descriptors from high-dimensional data and inserting them into standard supervised models. Specifically, we construct simplicial complexes through the Vietoris–Rips filtrations over the input data, compute persistent homology to obtain persistent landscapes and barcodes and convert this data into quantitative feature vectors to feed to ML models such as Support Vector Machine and Logistic Regression. Those topological features are integrated with traditional inputs, yielding an augmented representation of the data featuring global shape and general variations. PHAML is assessed on synthetic and real-world datasets, demonstrating improvements in classification performance, accuracy, and noise robustness and leveraging topological invariants' stability under perturbations. We also compare a baseline model with conventional features with a PHAML model to quantify the performance improvements attributable to persistent homology. Our results underscore the benefit of combining topological descriptors with classical features, highlighting the promise of persistent homology-based machine learning models in which global structure is preserved.

Project Update – 28/02/2025

• The SVM classifier currently achieves 100% accuracy on the synthetic sphere vs. torus dataset. This is likely due to the clear topological separation captured by persistent homology (e.g., presence/absence of 1D holes)

Project Update – 14/04/2025

• Ongoing validation of PHAML using dimensionality reduction (e.g., PCA) to visualize the decision boundary in the augmented feature space (topological + traditional features).
• If the projection shows clear class separation, it supports the hypothesis that persistent homology captures meaningful global structure.
• Next steps include robustness tests and comparing results across alternative models (e.g., logistic regression).

Installation

Specific versions must be used to prevent conflicts from dependencies requiring different package versions.

Requirements

• Python 3.7+
• Julia 1.6
• ripser
• persim
• scikit-learn
• NumPy
• Matplotlib