Another look at recovering local homology from samples of stratified sets

Abstract

Recovering homological features of spaces from samples has become one of the central themes of topological data analysis, leading to many successful applications. Many of the results in this area focus on global homological features of a subset of a Euclidean space. In this case, homology recovery predicates on imposing well understood geometric conditions on the underlying set. Typically, these conditions guarantee that small enough neighborhoods of the set have the same homology as the set itself. Existing work on recovering local homological features of a space from samples employs similar conditions locally. However, such local geometric conditions may vary from point to point and can potentially degenerate. For instance, the size of local homology preserving neighborhoods across all points of interest may not be bounded away from zero. In this paper, we introduce more general and robust conditions for local homology recovery and show that tame homology stratified sets, including Whitney stratified sets, satisfy these conditions away from strata boundaries, thus obtaining control over the regions where local homology recovery may not be feasible. Moreover, we show that true local homology of such sets can be computed from good enough samples using Vietoris–Rips complexes.