Steerable convolutional neural networks (CNNs) provide a general framework
for building neural networks equivariant to translations and other
transformations belonging to an origin-preserving group $G$, such as
reflections and rotations. They rely on standard convolutions with
$G$-steerable kernels obtained by analytically solving the group-specific
equivariance constraint imposed onto the kernel space. As the solution is
tailored to a particular group $G$, the implementation of a kernel basis does
not generalize to other symmetry transformations, which complicates the
development of group equivariant models. We propose using implicit neural
representation via multi-layer perceptrons (MLPs) to parameterize $G$-steerable
kernels. The resulting framework offers a simple and flexible way to implement
Steerable CNNs and generalizes to any group $G$ for which a $G$-equivariant MLP
can be built. We apply our method to point cloud (ModelNet-40) and molecular
data (QM9) and demonstrate a significant improvement in performance compared to
standard Steerable CNNs.

使用 MLP 来参数化 G - 可操作的卷积核，提出了一种简单灵活的 Steerable CNNs 框架，可推广到任何可建立 G - 等变 MLP 的群组 G。在点云（ModelNet-40）和分子数据（QM9）上应用我们的方法，与标准的 Steerable CNNs 相比，性能有显著提高。

可控卷积神经网络的隐式神经卷积核

Implicit Neural Convolutional Kernels for Steerable CNNs

The big empirical success of group equivariant networks has led in recent
years to the sprouting of a great variety of equivariant network architectures.
A particular focus has thereby been on rotation and reflection equivariant CNNs
for planar images. Here we give a general description of $E(2)$-equivariant
convolutions in the framework of Steerable CNNs. The theory of Steerable CNNs
thereby yields constraints on the convolution kernels which depend on group
representations describing the transformation laws of feature spaces. We show
that these constraints for arbitrary group representations can be reduced to
constraints under irreducible representations. A general solution of the kernel
space constraint is given for arbitrary representations of the Euclidean group
$E(2)$ and its subgroups. We implement a wide range of previously proposed and
entirely new equivariant network architectures and extensively compare their
performances. $E(2)$-steerable convolutions are further shown to yield
remarkable gains on CIFAR-10, CIFAR-100 and STL-10 when used as a drop-in
replacement for non-equivariant convolutions.

该研究描述了在可旋转卷积神经网络框架中的 $E (2)$- 等变卷积，提出了转换特性表示描述特征空间变换法则的群表示。研究人员证明了这些约束可以通过使用不可约表示约简为任意群表示的约束，并通过实现一系列先前提出的和全新的等变网络架构进行了广泛比较，表明当用作非等变卷积的替代品时，在 CIFAR-10、CIFAR-100 和 STL-10 上使用 $E (2)$- 可旋转卷积可以取得显著的改进。