3 research outputs found
Harmonic Persistent Homology
We introduce harmonic persistent homology spaces for filtrations of finite
simplicial complexes. As a result we can associate concrete subspaces of cycles
to each bar of the barcode of the filtration. We prove stability of the
harmonic persistent homology subspaces under small perturbations of functions
defining them. We relate the notion of "essential simplices" introduced in an
earlier work to identify simplices which play a significant role in the birth
of a bar, with that of harmonic persistent homology. We prove that the harmonic
representatives of simple bars maximizes the "relative essential content"
amongst all representatives of the bar, where the relative essential content is
the weight a particular cycle puts on the set of essential simplices.Comment: 35 pages, 5 figures. Comments welcom