Revisiting Multi-Permutation Equivariance through the Lens of Irreducible Representations (2410.06665v1)

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}.

• The paper presents alternative derivations of equivariant layers, notably simplifying the analysis for DWS networks.
• It extends the method to unaligned symmetric sets, revealing numerous non-Siamese layers that boost task performance.
• Empirical results validate that these approaches enhance predictions in graph anomaly detection and weight space alignment tasks.

Revisiting Multi-Permutation Equivariance through the Lens of Irreducible Representations

Overview

This paper investigates the characterization of equivariant linear layers within the framework of permutation representations, utilizing irreducible representations and Schur's Lemma as methodological tools. The traditional approach to equivariance, often reliant on parameter-sharing, is supplemented by an exploration of representation theory, particularly irreducible representations. The work provides alternative derivations for established models, such as DeepSets, 2-IGN graph equivariant networks, and Deep Weight Space (DWS) networks, with the DWS derivation being notably simpler compared to previous analyses.

Key Contributions

1. Alternative Derivations: The paper offers an alternative route to deriving the equivariant layer characterizations for DeepSets, 2-IGN, and DWS. The DWS networks, in particular, benefit from a more streamlined derivation process absent in previous literature.
2. Extension to Unaligned Symmetric Sets: The authors extend their approach to datasets composed of unaligned symmetric elements, where equivariance to the wreath product of groups is required. Unlike previous works that constrained almost all wreath equivariant layers to be Siamese, this paper provides a comprehensive layer characterization. It confirms the existence of numerous non-Siamese layers that enhance performance in tasks related to graph anomaly detection, weight space alignment, and Wasserstein distances.
3. Empirical Validation: The additional non-Siamese layers are empirically shown to improve task performance, further validating the theoretical contributions of the paper.

Numerical and Theoretical Implications

The introduction of irreducible representations simplifies the derivation of layer characterizations, potentially impacting the design of neural networks by reducing the complexity associated with parameter-sharing schemes. The findings suggest that considering non-Siamese, wreath-equivariant layers can significantly enhance predictive capabilities in specific machine learning tasks, implying a broader scope for application beyond those traditionally considered.

Future Prospects

The implications of utilizing irreducible representations extend into designing more efficient neural architectures that respect multi-permutation equivariance without the overhead of exhaustive parameter-sharing. There is potential for further research into how these representations might be leveraged in more complex neural architectures, extending beyond permutation-based symmetries. Additionally, exploring connections to other symmetry-related problems in machine learning could yield new insights and methodologies.

Conclusion

The paper provides a significant theoretical advancement in understanding multi-permutation equivariance through the lens of irreducible representations. By offering simpler, more generalized derivations and expanding the scope of equivariance with empirical validation, it opens new avenues for research and application in AI and machine learning, improving the interpretability and efficiency of complex neural networks.