Some new methods to build group equivariant non-expansive operators in TDA

TL;DR

This paper introduces new methods for constructing group equivariant non-expansive operators (GENEOs) in topological data analysis, enhancing the ability to incorporate symmetry and invariance in data representations.

Contribution

The paper presents novel techniques for building G-equivariant non-expansive operators from function spaces, advancing the theoretical framework for symmetry-aware data analysis.

Findings

New methods for constructing GENEOs are proposed.

The approach enhances invariance properties in topological data analysis.

The methods are applicable to data spaces with group symmetries.

Abstract

Group equivariant operators are playing a more and more relevant role in machine learning and topological data analysis. In this paper we present some new results concerning the construction of $G$-equivariant non-expansive operators (GENEOs) from a space $\varPhi$ of real-valued bounded continuous functions on a topological space $X$ to $\varPhi$ itself. The space $\varPhi$ represents our set of data, while $G$ is a subgroup of the group of all self-homeomorphisms of $X$, representing the invariance we are interested in.