Classical persistent homology is a powerful mathematical tool for shape comparison. Unfortunately, it is not tailored to study the action of transformation groups that are different from the group Homeo(X) of all self-homeomorphisms of a topological space X. This fact restricts its use in applications. In order to obtain better lower bounds for the natural pseudo-distance d_G associated with a subgroup G of Homeo(X), we need to adapt persistent homology and consider G-invariant persistent homology. Roughly speaking, the main idea consists in defining persistent homology by means of a set of chains that is invariant under the action of G. In this paper, we formalize this idea and prove the stability of the persistent Betti number functions in G-invariant persistent homology with respect to the natural pseudo-distance d_G. We also show how G-invariant persistent homology could be used in applications concerning shape comparison, when the invariance group is a proper subgroup of the group of all self-homeomorphisms of a topological space. In this paper, we will assume that the space X is triangulable, in order to guarantee that the persistent Betti number functions are finite without using any tameness assumption.
Patrizio Frosini (2015). G-invariant persistent homology. MATHEMATICAL METHODS IN THE APPLIED SCIENCES, 38(6), 1190-1199 [10.1002/mma.3139].
G-invariant persistent homology
FROSINI, PATRIZIO
2015
Abstract
Classical persistent homology is a powerful mathematical tool for shape comparison. Unfortunately, it is not tailored to study the action of transformation groups that are different from the group Homeo(X) of all self-homeomorphisms of a topological space X. This fact restricts its use in applications. In order to obtain better lower bounds for the natural pseudo-distance d_G associated with a subgroup G of Homeo(X), we need to adapt persistent homology and consider G-invariant persistent homology. Roughly speaking, the main idea consists in defining persistent homology by means of a set of chains that is invariant under the action of G. In this paper, we formalize this idea and prove the stability of the persistent Betti number functions in G-invariant persistent homology with respect to the natural pseudo-distance d_G. We also show how G-invariant persistent homology could be used in applications concerning shape comparison, when the invariance group is a proper subgroup of the group of all self-homeomorphisms of a topological space. In this paper, we will assume that the space X is triangulable, in order to guarantee that the persistent Betti number functions are finite without using any tameness assumption.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.