Group-convolutional neural networks (GCNNs) are among the most important methods for introducing symmetry as an inductive bias in deep learning: In each linear layer, GCNNs sample a transformation group $G$ densely and correlate data and filters in different poses (with suitable anti-aliasing for steerable GCNNs) to maintain equivariance with respect to $G$. Unfortunately, applying filters to many data items resulting from this sampling is expensive (even for translations alone, i.e., in ordinary CNNs), and costs grow exponentially with increasing degrees of freedom (such as translations and rotations in 3D), which often hinders practical applications. In this paper, we propose sampling in feature space, i.e., replacing geometrically dense samples with representative samples selected by feature similarity. This decouples geometric resolution from memory and processing costs during training and inference, providing a novel way to trade off computational effort and accuracy. Our main empirical finding is that a coarse feature-space sampling already preserves classification accuracy remarkably well, which permits precomputation based on geometric similarity, accelerating the training of equivariant 3D classifiers substantially.

翻译：群卷积神经网络（GCNN）是将对称性作为归纳偏置引入深度学习的最重要方法之一：在每个线性层中，GCNN密集采样变换群$G$，并将不同姿态下的数据与滤波器相关联（对于可导向GCNN需配合适当的抗混叠处理），以保持关于$G$的等变性。然而，将滤波器应用于此采样产生的众多数据项成本高昂（即使在仅考虑平移的普通CNN中也是如此），且计算开销随自由度（如三维空间中的平移与旋转）增加而呈指数增长，这往往阻碍实际应用。本文提出在特征空间中进行采样，即用基于特征相似性选取的代表性样本替代几何密集采样。这使训练与推理过程中的几何分辨率与内存及计算开销解耦，为权衡计算效率与精度提供了新途径。主要实验发现是：粗粒度的特征空间采样已能出色保持分类精度，这使得基于几何相似性的预计算成为可能，从而显著加速等变三维分类器的训练。