AUTHOR=Ferrari Lucia , Frosini Patrizio , Quercioli Nicola , Tombari Francesca TITLE=A topological model for partial equivariance in deep learning and data analysis JOURNAL=Frontiers in Artificial Intelligence VOLUME=6 YEAR=2023 URL=https://www.frontiersin.org/journals/artificial-intelligence/articles/10.3389/frai.2023.1272619 DOI=10.3389/frai.2023.1272619 ISSN=2624-8212 ABSTRACT=

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, we obtain the so-called GENEOs. We then study the spaces of measurements, whose domains are subjected 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.