On the topological theory of Group Equivariant Non-Expansive Operators

In this thesis we aim to provide a general topological and geometrical framework for group equivariance in the machine learning context. A crucial part of this framework is a synergy between persistent homology and the theory of group actions. In our approach, instead of focusing on data, we focus on suitable operators defined on the functions that represent the data. In particular, we define group equivariant non-expansive operators (GENEOs), which are maps between function spaces endowed with the actions of groups of transformations. We investigate the topological, geometric and metric properties of the space of GENEOs. We begin by defining suitable pseudo-metrics for the function spaces, the equivariance groups, and the set of GENEOs and proving some results about our model. Basing on these pseudo-metrics, we prove that the space of GENEOs is compact and convex, under the assumption that the function spaces are compact and convex. These results provide fundamental guarantees in a machine learning perspective. We show some new methods to build different classes of GENEOs in order to populate and approximate the space of GENEOs. Moreover, we define a suitable Riemannian structure on manifolds of GENEOs making available the use of gradient descent methods.