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.

本文提出了一种拓扑模型来编码神经网络中的部分等变性。为此，我们引入了一类名为P-GENEOs的操作符，它以一种非扩展的方式改变通过测量表达的数据，尊重某些变换集合的作用。如果作用的变换集合是一个群，我们得到了所谓的GENEOs。我们研究了测量空间，其定义域受到某些自映射的作用，以及连接这些空间之间的P-GENEOs空间。我们定义了伪度量，并展示了所得空间的一些性质。特别地，我们展示了这些空间具有便利的逼近和凸性质。

深度学习与数据分析中的部分等变性拓扑模型