  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.
