Persistent Homology: A Pedagogical Introduction with Biological Applications
Aurelie Jodelle Kemme, Collins Amburo Agyingi
TL;DR
The paper provides a pedagogy-friendly introduction to persistent homology, positioning PH within Topological Data Analysis to uncover multi-scale data shape and robustness to noise. It formalizes the pipeline from point clouds through filtrations to homology, Betti numbers, and persistence visualizations, and demonstrates the method with a 3-1m-Supercoiled DNA case study. By detailing concrete steps, practical tools (e.g., Ripser, GUDHI, Eirene), and interpretation of $\beta_k$ and persistence diagrams, the work aims to bridge theory and application for novices. The study highlights PH’s potential to augment bio-data analysis and suggests avenues for integrating topological summaries with neural and graph-based models, advancing scalable, interpretable data analysis in biology and beyond.
Abstract
Persistent Homology (PH) is a fundamental tool in computational topology, designed to uncover the intrinsic geometric and topological features of data across multiple scales. Originating within the broader framework of Topological Data Analysis (TDA), PH has found diverse applications ranging from protein structure and knot analysis to financial domains such as Bitcoin behaviour and stock market dynamics. Despite its growing relevance, there remains a lack of accessible resources that bridge the gap between theoretical foundations and practical implementation for beginners. This paper offers a clear and comprehensive introduction to persistent homology, guiding readers from core concepts to real-world applications. Specifically, we illustrate the methodology through the analysis of a 3-1 supercoiled DNA structure. The paper is tailored for readers without prior exposure to algebraic topology, aiming to demystify persistent homology and foster its broader adoption in data analysis tasks.