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We study the persistent homology of both functional data on compacttopological spaces and structural data presented as compact metric measurespaces. One of our goals is to define persistent homology so as to captureprimarily properties of the shape of a signal, eliminating otherwise highlypersistent homology classes that may exist simply because of the nature of thedomain on which the signal is defined. We investigate the stability of theseinvariants using metrics that downplay regions where signals are weak. Thedistance between two signals is small if they exhibit high similarity inregions where they are strong, regardless of the nature of their full domains,in particular allowing different homotopy types. Consistency and estimation ofpersistent homology of metric measure spaces from data are studied within thisframework. We also apply the methodology to the construction of multi-scaletopological descriptors for data on compact Riemannian manifolds via metricrelaxations derived from the heat kernel.

A Topological Study of Functional Data and Fr\'echet Functions of Metric Measure Spaces