We introduce a general method for achieving robust group-invariance in group-equivariant convolutional neural networks ($G$-CNNs), which we call the $G$-triple-correlation ($G$-TC) layer. The approach leverages the theory of the triple-correlation on groups, which is the unique, lowest-degree polynomial invariant map that is also complete. Many commonly used invariant maps--such as the max--are incomplete: they remove both group and signal structure. A complete invariant, by contrast, removes only the variation due to the actions of the group, while preserving all information about the structure of the signal. The completeness of the triple correlation endows the $G$-TC layer with strong robustness, which can be observed in its resistance to invariance-based adversarial attacks. In addition, we observe that it yields measurable improvements in classification accuracy over standard Max $G$-Pooling in $G$-CNN architectures. We provide a general and efficient implementation of the method for any discretized group, which requires only a table defining the group's product structure. We demonstrate the benefits of this method for $G$-CNNs defined on both commutative and non-commutative groups--$SO(2)$, $O(2)$, $SO(3)$, and $O(3)$ (discretized as the cyclic $C8$, dihedral $D16$, chiral octahedral $O$ and full octahedral $O_h$ groups)--acting on $\mathbb{R}^2$ and $\mathbb{R}^3$ on both $G$-MNIST and $G$-ModelNet10 datasets.

翻译：我们提出一种在群等变卷积神经网络（$G$-CNNs）中实现鲁棒群不变性的通用方法，称为$G$-三重相关（$G$-TC）层。该方法利用群上的三重相关理论——该理论是唯一且最低次的多项式不变映射，同时也是完备的。许多常用的不变映射（如最大值映射）是不完备的：它们会同时去除群结构和信号结构。相比之下，完备不变映射仅移除由群作用引起的变量变化，同时保留信号结构的所有信息。三重相关的完备性赋予了$G$-TC层强鲁棒性，这体现在其对抗基于不变性的对抗攻击的能力上。此外，我们观察到，在$G$-CNN架构中，该方法相较于标准Max $G$-Pooling在分类准确率上具有可测量的提升。我们为任意离散群提供了通用且高效的实现方法，该方法仅需一个定义群乘积结构的表即可。我们通过作用于$\mathbb{R}^2$和$\mathbb{R}^3$上的交换群与非交换群——$SO(2)$、$O(2)$、$SO(3)$和$O(3)$（离散化为循环群$C8$、二面体群$D16$、手性八面体群$O$和全八面体群$O_h$）——在$G$-MNIST和$G$-ModelNet10数据集上验证了该方法的优势。