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Title: A topological study of functional data and Frechet functions of metric measure spaces
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.  more » « less
Award ID(s):
1722995
PAR ID:
10174714
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Journal of applied and computational topology
Volume:
3
ISSN:
2367-1726
Page Range / eLocation ID:
359-380
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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