Some remarks on the algebraic properties of group invariant operators in persistent homology

Abstract

Topological data analysis is a new approach to processing digital data, focusing on the fact that topological properties are quite important for efficient data comparison. In particular, persistent topology and homology are relevant mathematical tools in TDA, and their study is attracting more and more researchers. As a matter of fact, in many applications data can be represented by continuous real-valued functions defined on a topological space X, and persistent homology can be efficiently used to compare these data by describing the homological changes of the sub-level sets of those functions. However, persistent homology is invariant under the action of the group (formula presented) of all self-homeomorphisms of X, while in many cases an invariance with respect to a proper subgroup G of (formula presented) is preferable. Interestingly, it has been recently proved that this restricted invariance can be obtained by applying G-invariant non-expansive operators to the considered functions. As a consequence, in order to proceed along this line of research we need methods to build G-invariant non-expansive operators. According to this perspective, in this paper we prove some new results about the algebra of GINOs.