A U-match Algorithm for Persistent Relative Homology

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.