On $C^0$-persistent homology and trees

Daniel Perez (1, 2, 3)

1 LMO - Laboratoire de Mathématiques d'Orsay
2 DMA - Département de Mathématiques et Applications - ENS Paris
3 DATASHAPE - Understanding the Shape of Data

The study of the topology of the superlevel sets of stochastic processes on [0, 1] in probability led to the introduction of R-trees which characterize the connected components of the superlevel-sets. We provide a generalization of this construction to more general deterministic continuous functions on some path-connected, compact topological set X and reconcile the probabilistic approach with the traditional methods of persistent homology. We provide an algorithm which functorially links the tree T f associated to a function f and study some invariants of these trees, which in 1D turn out to be linked to the regularity of f .

Domaines: Mathématiques [math]; Topologie algébrique [math.AT]; Informatique [cs]; Géométrie algorithmique [cs.CG]

Daniel Perez. On $C^0$-persistent homology and trees. 2020. ⟨hal-03040819v2⟩