  In this paper, we investigate a sheaf-theoretic interpretation of
stratification learning from geometric and topological perspectives. Our main
result is the construction of stratification learning algorithms framed in
terms of a sheaf on a partially ordered set with the Alexandroff topology. We
prove that the resulting decomposition is the unique minimal stratification for
which the strata are homogeneous and the given sheaf is constructible. In
particular, when we choose to work with the local homology sheaf, our algorithm
gives an alternative to the local homology transfer algorithm given in Bendich
et al. (2012), and the cohomology stratification algorithm given in Nanda
(2017). Additionally, we give examples of stratifications based on the
geometric techniques of Breiding et al. (2018), illustrating how the
sheaf-theoretic approach can be used to study stratifications from both
topological and geometric perspectives. This approach also points toward future
applications of sheaf theory in the study of topological data analysis by
illustrating the utility of the language of sheaf theory in generalizing
existing algorithms.
