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.

本文介绍了一种使用代数拓扑学中的表示稳定现象定义等变神经网络的方法，使得网络在固定维度上训练后能够接受任意维度的输入，并提供了简单的开源实现和初步的数值实验。

任意维度等变神经网络

Any-dimensional equivariant neural networks

Filters in a Convolutional Neural Network (CNN) contain model parameters
learned from enormous amounts of data. In this paper, we suggest to decompose
convolutional filters in CNN as a truncated expansion with pre-fixed bases,
namely the Decomposed Convolutional Filters network (DCFNet), where the
expansion coefficients remain learned from data. Such a structure not only
reduces the number of trainable parameters and computation, but also imposes
filter regularity by bases truncation. Through extensive experiments, we
consistently observe that DCFNet maintains accuracy for image classification
tasks with a significant reduction of model parameters, particularly with
Fourier-Bessel (FB) bases, and even with random bases. Theoretically, we
analyze the representation stability of DCFNet with respect to input
variations, and prove representation stability under generic assumptions on the
expansion coefficients. The analysis is consistent with the empirical
observations.

本文提出了基于截断展开的分解卷积滤波器（DCFNet）神经网络，使用预设基进行卷积滤波器的分解，通过扩展系数的学习来减少可以训练的参数和计算量，并通过基截断来加强滤波器的规律性。实验证明，DCFNet 可以在减少模型参数的同时，保持图像分类任务的准确性，尤其是在使用 Fourier-Bessel 基时效果显著，理论分析表明，DCFNet 对于输入的变化有较好的表示稳定性，并且经验观察与理论分析是一致的。