Traditional supervised learning aims to learn an unknown mapping by fitting a function to a set of input-output pairs with a fixed dimension. The fitted function is then defined on inputs of the same dimension. However, in many settings, the unknown mapping takes inputs in any dimension; examples include graph parameters defined on graphs of any size and physics quantities defined on an arbitrary number of particles. We leverage a newly-discovered phenomenon in algebraic topology, called representation stability, to define equivariant neural networks that can be trained with data in a fixed dimension and then extended to accept inputs in any dimension. Our approach is user-friendly, requiring only the network architecture and the groups for equivariance, and can be combined with any training procedure. We provide a simple open-source implementation of our methods and offer preliminary numerical experiments.

翻译：传统监督学习旨在通过将函数拟合到固定维度的输入-输出对上来学习未知映射，且拟合后的函数仅定义于同维度的输入。然而在许多场景中，未知映射接受任意维度的输入，例如定义于任意规模图的图参数，以及定义于任意数量粒子的物理量。我们利用代数拓扑中一个名为"表示稳定性"的新发现现象，构建了可先在固定维度数据上训练，继而扩展至接受任意维度输入的等变神经网络。该方法具有用户友好特性，仅需指定网络架构与等变性群组，并可与任何训练流程兼容。我们提供基于该方法的简洁开源实现，并展示了初步数值实验结果。