Stratified Formal Deformations and Intersection Homology of Data Point Clouds

TL;DR

This paper introduces stratified formal deformations and spines to simplify complexes for intersection homology computation, demonstrating their effectiveness on Vietoris-Rips complexes from sampled data.

Contribution

It develops new combinatorial transformations that preserve intersection homology, enabling efficient analysis of complex data structures.

Findings

Transformations preserve intersection homology during complex reduction

Algorithms successfully compute intersection homology of stratified spines

Applications to data sampled near singular spaces demonstrate practical utility

Abstract

Intersection homology is a topological invariant which detects finer information in a space than ordinary homology. Using ideas from classical simple homotopy theory, we construct local combinatorial transformations on simplicial complexes under which intersection homology remains invariant. In particular, we obtain the notions of stratified formal deformations and stratified spines of a complex, leading to reductions of complexes prior to computation of intersection homology. We implemented the algorithmic execution of such transformations, as well as the calculation of intersection homology, and apply these algorithms to investigate the intersection homology of stratified spines in Vietoris-Rips type complexes associated to point sets sampled near given, possibly singular, spaces.