  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 reconstruction guarantees rely on
constructing embedded complexes which become difficult in high dimensions. We
show that the persistence diagrams used for estimating local homology, can be
approximated using families of Vietoris-Rips complexes, whose simple
constructions are robust in any dimension. To the best of our knowledge, our
results, for the first time, make applications based on local homology, such as
stratification learning, feasible in high dimensions.
