  We study the persistent homology of both functional data on compact
topological spaces and structural data presented as compact metric measure
spaces. One of our goals is to define persistent homology so as to capture
primarily properties of the shape of a signal, eliminating otherwise highly
persistent homology classes that may exist simply because of the nature of the
domain on which the signal is defined. We investigate the stability of these
invariants using metrics that downplay regions where signals are weak. The
distance between two signals is small if they exhibit high similarity in
regions where they are strong, regardless of the nature of their full domains,
in particular allowing different homotopy types. Consistency and estimation of
persistent homology of metric measure spaces from data are studied within this
framework. We also apply the methodology to the construction of multi-scale
topological descriptors for data on compact Riemannian manifolds via metric
relaxations derived from the heat kernel.
