Generalized Permutants and Graph GENEOs

This paper is part of a line of research devoted to developing a compositional and geometric theory of Group Equivariant Non-Expansive Operators (GENEOs) for Geometric Deep Learning. It has two objectives. The first objective is to generalize the notions of permutants and permutant measures, originally defined for the identity of a single “perception pair”, to a map between two such pairs. The second and main objective is to extend the application domain of the whole theory, which arose in the set-theoretical and topological environments, to graphs. This is performed using classical methods of mathematical definitions and arguments. The theoretical outcome is that, both in the case of vertex-weighted and edge-weighted graphs, a coherent theory is developed. Several simple examples show what may be hoped from GENEOs and permutants in graph theory and how they can be built. Rather than being a competitor to other methods in Geometric Deep Learning, this theory is proposed as an approach that can be integrated with such methods