Building a Vietoris–Rips Filtration for Persistent Homology
Summary
This article explains how to turn a point cloud and its pairwise distances into a Vietoris–Rips filtration for topological data analysis. Vertices enter at zero, edges enter at their endpoint distance, and triangles enter at the longest of their three edges. The implementation described enumerates simplices through dimension two, subject to a maximum distance cutoff, then sorts them by filtration value with lower-dimensional faces first when values tie.
The article also covers lookup tables for mapping vertices and edges to global simplex indices, and sparse boundary columns over Z/2 with entries ordered so pivots are easy to retrieve during reduction. It reports a sanity check using 78 points: combinatorial simplex counts and total boundary entries matched their expected formulas. The method can reveal persistent connected components and loops, but the article stops before persistence reduction and diagram interpretation. Its main practical limit is cubic growth in the number of triangles, which makes the approach costly as point clouds grow.
Key ideas
- A Vietoris–Rips edge enters when its endpoint distance is within the chosen scale.
- A triangle enters at the largest of its three edge distances, ensuring all faces are already present.
- Sorting ties by dimension keeps faces before cofaces in the filtration.
- Vertex and edge lookup tables trade additional memory for fast boundary construction.
- Triangle counts grow cubically with point count, motivating a low dimension cap.
Tags
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.