A General Framework for Robust G-Invariance in G-Equivariant Networks

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 \textit{complete}. Many commonly used invariant maps\textemdash such as the \texttt{max}\textemdash 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.