A topological model for partial equivariance in deep learning and data
  analysis
Lucia Ferrari Patrizio Frosini Nicola Quercioli Francesca Tombari
In this article, 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. If the set of transformations acting is a group, then we obtain the
so-called GENEOs. 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. We define pseudo-metrics on them and show some properties of the
resulting spaces. In particular, we show how such spaces have convenient
approximation and convexity properties.