  A central challenge in topological data analysis is the interpretation of
barcodes. The classical algebraic-topological approach to interpreting homology
classes is to build maps to spaces whose homology carries semantics we
understand and then to appeal to functoriality. However, we often lack such
maps in real data; instead, we must rely on a cross-dissimilarity measure
between our observations of a system and a reference. In this paper, we develop
a pair of computational homological algebra approaches for relating persistent
homology classes and barcodes: persistent extension, which enumerates potential
relations between cycles from two complexes built on the same vertex set, and
the method of analogous bars, which utilizes persistent extension and the
witness complex built from a cross-dissimilarity measure to provide relations
across systems. We provide an implementation of these methods and demonstrate
their use in comparing cycles between two samples from the same metric space
and determining whether topology is maintained or destroyed under clustering
and dimensionality reduction.
