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The natural occurrence of singular spaces in applications has led to recentinvestigations on performing topological data analysis (TDA) in a stratifiedframework. In many applications, there is no a priori information on whatpoints should be regarded as singular or regular. For this purpose we describea fully implementable process that provably approximates the stratification fora large class of two-strata Whitney stratified spaces from sufficiently closenon-stratified samples. Additionally, in this work, we establish a notion ofpersistent stratified homotopy type obtained from a sample with two strata. Inanalogy to the non-stratified applications in TDA which rely on a series ofconvenient properties of (persistent) homotopy types of sufficiently regularspaces, we show that our persistent stratified homotopy type behaves much likeits non-stratified counterpart and exhibits many properties (such as stability,and inference results) necessary for an application in TDA. In total, ourresults combine to a sampling theorem guaranteeing the (approximate) inferenceof (persistent) stratified homotopy types of sufficiently regular two-strataWhitney stratified spaces.

From Samples to Persistent Stratified Homotopy Types