A topological model for (partial) equivariance in deep learning

A group equivariant operator transforms measurements of data into other measurements while respecting certain group actions on them. If such operators are also non-expansive, then we talk about GENEOs. GENEOs have been introduced to model neural networks and inject geometrical knowledge about the data, encoded by the group action, to be preserved by the machine. However, data rarely satisfy rigid symmetries due to noise, incompleteness or symmetry-breaking features, and hence the group axioms may not be satisfied. Thus, we propose a topological model to encode partial equivariance in neural networks. To this end, we introduce a class of operators, called P-GENEOs, that change data expressed by measurements, respecting the action of certain sets of transformations, in a non-expansive way. We then study the spaces of measurements, whose domains are subject to the action of certain self-maps, and the space of P-GENEOs between these spaces.