Harmonic Persistent Homology

Abstract

.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 as well as the subspaces associated to the bars of the barcodes 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 maximize the "relative essential content" among all representatives of the bar, where the relative essential content is the weight a particular cycle puts on the set of essential simplices.Keywordsharmonic homologypersistent homologybarcodesessential simplicesstabilityMSC codes55N31

Locations

• SIAM Journal on Applied Algebra and Geometry - View
• arXiv (Cornell University) - View - PDF