Approximating local homology from samples

Abstract. Recently, multi-scale notions of local homology (a variant of persistent homology) have been used to study the local structure of spaces around a given point from a point cloud sample. Current recon-struction guarantees rely on constructing embedded complexes which become difficult in high dimensions. We show that the persistence dia-grams used for estimating local homology, can be approximated using families of Vietoris-Rips complexes, whose simple constructions are ro-bust in any dimension. To the best of our knowledge, our results, for the first time, make applications based on local homology, such as stratifica-tion learning, feasible in high dimensions.