REVISITING MULTI-PERMUTATION EQUIVARIANCE THROUGH THE LENS OF IRREDUCIBLE REPRESENTATIONS

This paper explores the characterization of equivariant linear layers for representations of permutations and related groups. Unlike traditional approaches,which address these problems using parameter-sharing, we consider an alternativemethodology based on irreducible representations and Schur’s lemma. Using thismethodology, we obtain an alternative derivation for existing models like DeepSets,2-IGN graph equivariant networks, and Deep Weight Space (DWS) networks. Thederivation for DWS networks is significantly simpler than that of previous results.Next, we extend our approach to unaligned symmetric sets, where equivarianceto the wreath product of groups is required. Previous works have addressed thisproblem in a rather restrictive setting, in which almost all wreath equivariant layersare Siamese. In contrast, we give a full characterization of layers in this case andshow that there is a vast number of additional non-Siamese layers in some settings.We also show empirically that these additional non-Siamese layers can improveperformance in tasks like graph anomaly detection, weight space alignment, andlearning Wasserstein distances.