  The natural occurrence of singular spaces in applications has led to recent
investigations on performing topological data analysis (TDA) in a stratified
framework. In many applications, there is no a priori information on what
points should be regarded as singular or regular. For this purpose we describe
a fully implementable process that provably approximates the stratification for
a large class of two-strata Whitney stratified spaces from sufficiently close
non-stratified samples. Additionally, in this work, we establish a notion of
persistent stratified homotopy type obtained from a sample with two strata. In
analogy to the non-stratified applications in TDA which rely on a series of
convenient properties of (persistent) homotopy types of sufficiently regular
spaces, we show that our persistent stratified homotopy type behaves much like
its non-stratified counterpart and exhibits many properties (such as stability,
and inference results) necessary for an application in TDA. In total, our
results combine to a sampling theorem guaranteeing the (approximate) inference
of (persistent) stratified homotopy types of sufficiently regular two-strata
Whitney stratified spaces.
