We introduce a unified framework for group equivariant networks on
homogeneous spaces derived from a Fourier perspective. We consider
tensor-valued feature fields, before and after a convolutional layer. We
present a unified derivation of kernels via the Fourier domain by leveraging
the sparsity of Fourier coefficients of the lifted feature fields. The sparsity
emerges when the stabilizer subgroup of the homogeneous space is a compact Lie
group. We further introduce a nonlinear activation, via an elementwise
nonlinearity on the regular representation after lifting and projecting back to
the field through an equivariant convolution. We show that other methods
treating features as the Fourier coefficients in the stabilizer subgroup are
special cases of our activation. Experiments on $SO(3)$ and $SE(3)$ show
state-of-the-art performance in spherical vector field regression, point cloud
classification, and molecular completion.

从 Fourier 角度出发，我们在同质空间上引入了一种统一的群等变网络框架，其中考虑了张量值特征场及其卷积层前后的特征，通过利用提升特征场的 Fourier 系数的稀疏性，通过 Fourier 域统一推导卷积核并实现非线性激活，该方法在 spherical vector field regression、point cloud classification 和 molecular completion 等任务中达到了最先进的性能水平。

基于傅里叶变换的同变网络同构空间中的核与非线性设计

Unified Fourier-based Kernel and Nonlinearity Design for Equivariant  Networks on Homogeneous Spaces

We study how group symmetry helps improve data efficiency and generalization
for end-to-end differentiable planning algorithms when symmetry appears in
decision-making tasks. Motivated by equivariant convolution networks, we treat
the path planning problem as \textit{signals} over grids. We show that value
iteration in this case is a linear equivariant operator, which is a (steerable)
convolution. This extends Value Iteration Networks (VINs) on using
convolutional networks for path planning with additional rotation and
reflection symmetry. Our implementation is based on VINs and uses steerable
convolution networks to incorporate symmetry. The experiments are performed on
four tasks: 2D navigation, visual navigation, and 2 degrees of freedom (2DOFs)
configuration space and workspace manipulation. Our symmetric planning
algorithms improve training efficiency and generalization by large margins
compared to non-equivariant counterparts, VIN and GPPN.

该论文研究了如何使用群对称性改善端到端可微规划算法的数据效率和泛化能力，当对称性出现在决策任务中时。 通过等变卷积网络的启发，将路径规划问题视为网格上的信号，并表明在此情况下，价值迭代是一个线性的等变算子，即一个（可转向）卷积。 基于 VIN 的实现，使用等变卷积网络来纳入对称性。 在三个路径规划任务和一个工作空间操纵任务中的实验表明，我们的算法较 VIN 和 GPPN 的非等变协方差算法提高了大量的训练效率和泛化能力。

使用可转向卷积将对称性纳入可微规划中

Integrating Symmetry into Differentiable Planning with Steerable Convolutions

Although group convolutional networks are able to learn powerful
representations based on symmetry patterns, they lack explicit means to learn
meaningful relationships among them (e.g., relative positions and poses). In
this paper, we present attentive group equivariant convolutions, a
generalization of the group convolution, in which attention is applied during
the course of convolution to accentuate meaningful symmetry combinations and
suppress non-plausible, misleading ones. We indicate that prior work on visual
attention can be described as special cases of our proposed framework and show
empirically that our attentive group equivariant convolutional networks
consistently outperform conventional group convolutional networks on benchmark
image datasets. Simultaneously, we provide interpretability to the learned
concepts through the visualization of equivariant attention maps.

提出了一种基于视觉注意力机制的等变卷积方法 AGECN，解决了对称模式下位置和姿态关系不明确的问题，实现了对图像数据集的高效表征。