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

Lucia Ferrari; Patrizio Frosini; Nicola Quercioli; Francesca Tombari

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arXiv · 2308.13357

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

Lucia Ferrari·Patrizio Frosini·Nicola Quercioli·Francesca Tombari

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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.

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Lucia Ferrari, Patrizio Frosini, Nicola Quercioli, Francesca Tombari. 2023-08-25. A topological model for partial equivariance in deep learning and data analysis. https://arxiv.org/abs/2308.13357

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