A topological model for partial equivariance in deep learning and data analysis
This work addresses the challenge of partial equivariance in deep learning and data analysis, which is incremental as it builds upon existing group-equivariant methods.
The authors tackled the problem of encoding partial equivariance in neural networks by proposing a topological model and introducing P-GENEO operators, which respect transformations in a non-expansive way, and they demonstrated that the resulting spaces have convenient approximation and convexity properties.
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.