  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.
