A U-match Algorithm for Persistent Relative Homology

TL;DR

This work extends topological data analysis to persistent relative homology (PRH) by introducing the U-match decomposition framework, enabling a two-step matrix reduction to compute the PRH barcode and relative cycle representatives in time $O(m^3)$ for a dataset with $m$ cells. It develops a rigorous algebraic foundation for PRH, defines two compatible U-match factorizations, and shows how to extract birth/death data via explicit sublevel functions $f_{\ker}$ and $f_{im}$. The paper also identifies a practical optimization for lag filtrations, where $G_{\bullet}X$ lags behind $F_{\bullet}X$, yielding performance comparable to standard $R=DV$ methods, and provides an implementation integrated with Open Applied Topology. Overall, the methods deliver both the PRH barcode and concrete representative cycles, broadening applicability to arbitrarily filtrated pairs and enhancing interpretability of relative topological features. These contributions offer a transparent, algebraically grounded pathway to analyze relative topological structure in noisy, high-dimensional data and facilitate broader adoption in TDA workflows.

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.

Figures (3)

• Figure 1: Two levels of a filtered chain complex, over a filtered topological space ${F_{\bullet}X}$, whose chain maps are of the form $f_p: C_p(F_{a}X) \hookrightarrow C_p(F_bX)$ for some $a \leq b$.
• Figure 2: A point cloud consisting of nodes on a $3 \times 3$ grid where adjacent nodes $a,b$ on the grid satisfy $\|a-b\|_2 = 1/2$ and diagonally adjacent nodes satisfy $\|a-b\|_2 = \sqrt{2}/2$. Included are examples of relative (purple) and absolute (orange) $1$-cycles (left) together with the standard persistent homology barcode (center) and persistent relative homology barcode (right) corresponding to a filtration with lag $l = 1/4$. Notice on the right figure that no class can have lifetime larger than $l$, including essential $0$-homology.
• Figure 3: A point cloud and (overlaid) Vietoris-Rips complex for fixed filtration value $\varepsilon$(top row, left) together with a sequence of three persistent relative homology barcodes computed with lag parameters $\delta_1 = 0.05$(top row, right), $\delta_2 = 0.5$(bottom row, left) and $\delta_3 = 5$(bottom row, right).

Theorems & Definitions (20)

• Definition 1
• Definition 2
• Lemma 3
• Definition 4
• Theorem 5
• proof
• Theorem 6
• proof
• Theorem 7
• proof