From group equivariant to partial group equivariant non-expansive operators

Incorporating symmetries as an inductive bias into neural network architectures, Geometric Deep Learning (GDL) and Topological Data Analysis (TDA) have given an improvement in the development of deep learning models. In recent years, a line of research has emerged forming a bridge between GDL and TDA: a topological-geometrical theory of Group Equivariant Non-Expansive Operators (GENEOs). In the theory of GENEOs the collection of all symmetries is represented by a group, but in some applications, the group axioms are not maintained since real-world data rarely follows strict mathematical symmetries due to noisy or incomplete data or to symmetry breaking features. The main aim of this thesis is to give a generalization of the results obtained for GENEOs to a new mathematical framework where the property of equivariance is maintained only for some transformations, encoding a partial equivariance with respect to the action of the group of all transformations. To this end, we introduce the concept of Partial Group Equivariant Non-Expansive Operator (P-GENEO), extending the results obtained for GENEOs to a more general set-up, where the sets of transformations are represented by subsets with a weaker structure than the one of group.