A topological model for partial equivariance in deep learning and data analysis

TL;DR

This paper introduces a topological framework for modeling partial equivariance in neural networks using P-GENEO operators, which respect transformation actions and have favorable approximation properties.

Contribution

It presents a novel topological model and operators for encoding partial equivariance, extending existing concepts to more general transformation sets.

Findings

Defined P-GENEOs and GENEOs for data transformations

Established pseudo-metrics and properties of measurement spaces

Showed approximation and convexity properties of these spaces

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, 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.