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

• DOI: 10.48550/arxiv.2310.18564
• Publication Date: 2023-01-01
• Authors: Sophia Sanborn; Nina Miolane
• Keywords: Equivariant map; Robustness

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