G-invariant Persistent Homology
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.
Cite
@article{arxiv.1212.0655,
title = {G-invariant Persistent Homology},
author = {Patrizio Frosini},
journal= {arXiv preprint arXiv:1212.0655},
year = {2013}
}
Comments
14 pages, 4 figures. Remark 4.2 has been expanded to become Subsection 4.2, including a new example (Example 4.3). The section "Discussion and further research" and some references have been added. Small changes in the text