Generalized Permutants and Graph GENEOs

TL;DR

This paper introduces a novel framework connecting Topological Data Analysis and Geometric Deep Learning by extending GENEOs to graph structures, using generalized permutants to analyze graph data.

Contribution

It develops the concepts of generalized permutants and measures, enabling the construction of GENEOs for graphs, bridging topology and deep learning.

Findings

Demonstrates how GENEOs can transform and analyze graph structures

Introduces new mathematical tools for graph-based data analysis

Provides experimental evidence of the operators' effectiveness

Abstract

In this paper we establish a bridge between Topological Data Analysis and Geometric Deep Learning, adapting the topological theory of group equivariant non-expansive operators (GENEOs) to act on the space of all graphs weighted on vertices or edges. This is done by showing how the general concept of GENEO can be used to transform graphs and to give information about their structure. This requires the introduction of the new concepts of generalized permutant and generalized permutant measure and the mathematical proof that these concepts allow us to build GENEOs between graphs. An experimental section concludes the paper, illustrating the possible use of our operators to extract information from graphs. This paper is part of a line of research devoted to developing a compositional and geometric theory of GENEOs for Geometric Deep Learning.