Revisiting Multi-Permutation Equivariance through the Lens of Irreducible Representations

TL;DR

This paper introduces a novel approach using irreducible representations to characterize permutation-equivariant layers, simplifying existing models and extending to unaligned symmetric sets, with empirical improvements in various tasks.

Contribution

It provides a new methodology based on irreducible representations for permutation equivariance, simplifying derivations and characterizing non-Siamese layers for unaligned sets.

Findings

Simplified derivation of DeepSets, 2-IGN, and DWS models.

Full characterization of layers for unaligned symmetric sets.

Empirical performance gains in graph anomaly detection and Wasserstein distance learning.

Abstract

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 alternative methodology based on irreducible representations and Schur's lemma. Using this methodology, we obtain an alternative derivation for existing models like DeepSets, 2-IGN graph equivariant networks, and Deep Weight Space (DWS) networks. The derivation for DWS networks is significantly simpler than that of previous results. Next, we extend our approach to unaligned symmetric sets, where equivariance to the wreath product of groups is required. Previous works have addressed this problem in a rather restrictive setting, in which almost all wreath equivariant layers are Siamese. In contrast, we give a full characterization of layers in this case and show 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 improve performance in tasks like graph anomaly detection, weight space alignment, and learning Wasserstein distances. Our code is available at \href{https://github.com/yonatansverdlov/Irreducible-Representations-of-Deep-Weight-Spaces}{GitHub}.