A U-match Algorithm for Persistent Relative Homology

TL;DR

This paper introduces a new algorithm for computing persistent relative homology, extending topological data analysis to better understand the structure of data relative to subspaces, with efficient computation and correctness guarantees.

Contribution

It presents a simple, transparent, and general two-step matrix reduction algorithm for persistent relative homology with comparable complexity to standard persistent homology.

Findings

Algorithm is correct and efficient

Implementation optimized for specific cases

Computational complexity comparable to ordinary persistent homology

Abstract

A central problem in data-driven scientific inquiry is how to interpret structure in noisy, high-dimensional data. Topological data analysis (TDA) provides a solution via persistent homology, which encodes features of interest as topological holes within a filtration of data. The present work extends this framework to a related invariant which uncovers topological structure of a space relative to a subspace: persistent relative homology (PRH). We show that this invariant can be computed in a simple, highly transparent and general manner, using a two-step matrix reduction technique with worst-case time complexity comparable to ordinary persistent homology. We provide proofs demonstrating the correctness and computational complexity of this approach in addition to a performance-optimized implementation for a special case.