Convolutional neural networks revolutionized computer vision and natrual
language processing. Their efficiency, as compared to fully connected neural
networks, has its origin in the architecture, where convolutions reflect the
translation invariance in space and time in pattern or speech recognition
tasks. Recently, Cohen and Welling have put this in the broader perspective of
invariance under symmetry groups, which leads to the concept of group
equivaiant neural networks and more generally steerable neural networks. In
this article, we review the architecture of such networks including equivariant
layers and filter banks, activation with capsules and group pooling. We apply
this formalism to the symmetric group, for which we work out a number of
details on representations and capsules that are not found in the literature.

本研究论文探讨卷积神经网络在对称群中的应用，提出了群等变神经网络的概念和架构，以及使用多种层和滤波器的方法，为对称群的表示和胶囊的细节做出了数学分析。

等变可操控神经网络：对对称群的特别强调综述

Equivariant and Steerable Neural Networks: A review with special emphasis on the symmetric group

Recent work in equivariant deep learning bears strong similarities to
physics. Fields over a base space are fundamental entities in both subjects, as
are equivariant maps between these fields. In deep learning, however, these
maps are usually defined by convolutions with a kernel, whereas they are
partial differential operators (PDOs) in physics. Developing the theory of
equivariant PDOs in the context of deep learning could bring these subjects
even closer together and lead to a stronger flow of ideas. In this work, we
derive a $G$-steerability constraint that completely characterizes when a PDO
between feature vector fields is equivariant, for arbitrary symmetry groups
$G$. We then fully solve this constraint for several important groups. We use
our solutions as equivariant drop-in replacements for convolutional layers and
benchmark them in that role. Finally, we develop a framework for equivariant
maps based on Schwartz distributions that unifies classical convolutions and
differential operators and gives insight about the relation between the two.

研究在相等变深度学习和物理领域中共享一致的数学概念，并通过极化微分算子和基于调和分析的方法提出了一个新的框架，使我们能够更好地将基于显式对称性应变的捕捉无论是在深度学习还是物理学中都可以更精确地进行表达。

可变等变神经网络的可定向偏微分算子

Steerable Partial Differential Operators for Equivariant Neural Networks

We provide a general self-attention formulation to impose group equivariance
to arbitrary symmetry groups. This is achieved by defining positional encodings
that are invariant to the action of the group considered. Since the group acts
on the positional encoding directly, group equivariant self-attention networks
(GSA-Nets) are steerable by nature. Our experiments on vision benchmarks
demonstrate consistent improvements of GSA-Nets over non-equivariant
self-attention networks.

提出了一种通用的自注意力公式来对任意对称群实现群等变性，通过定义对群考虑不变的位置编码来实现。GSA-Nets 可以通过群作用于位置编码的直接操作进行指导，实验证明 GSA-Nets 比非等变自注意网络具有一致的改进。