G-invariant Persistent Homology

Frosini, Patrizio (2013) G-invariant Persistent Homology. [Preprint]

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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 greatly 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 the group 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.

Abstract
Document type
Preprint
Creators
CreatorsAffiliationORCID
Frosini, Patrizio
Keywords
Natural pseudo-distance, filtering function, group action, lower bound, stability, shape comparison
Subjects
DOI
Deposit date
15 Mar 2013 15:17
Last modified
15 Mar 2013 15:17
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