Stability of 2Parameter Persistent Homology
Abstract
The Čech and Rips constructions of persistent homology are stable with respect to perturbations of the input data. However, neither is robust to outliers, and both can be insensitive to topological structure of highdensity regions of the data. A natural solution is to consider 2parameter persistence. This paper studies the stability of 2parameter persistent homology: We show that several related densitysensitive constructions of bifiltrations from data satisfy stability properties accommodating the addition and removal of outliers. Specifically, we consider the multicover bifiltration, Sheehy's subdivision bifiltrations, and the degree bifiltrations. For the multicover and subdivision bifiltrations, we get 1Lipschitz stability results closely analogous to the standard stability results for 1parameter persistent homology. Our results for the degree bifiltrations are weaker, but they are tight, in a sense. As an application of our theory, we prove a law of large numbers for subdivision bifiltrations of random data.
 Publication:

arXiv eprints
 Pub Date:
 October 2020
 arXiv:
 arXiv:2010.09628
 Bibcode:
 2020arXiv201009628B
 Keywords:

 Mathematics  Algebraic Topology;
 Computer Science  Computational Geometry
 EPrint:
 45 pages