用不可约表示重新解析排列等变层，简化经典模型推导并发现新类型非对称结构。
Revisiting Multi-Permutation Equivariance through the Lens of Irreducible Representations
- 基于不可约表示与舒尔引理，重构等变层设计方法。
- 在非对齐对称集合上发现大量非对称（非Siamese）等变层，数量显著增加。
- 实验证明新结构提升图异常检测、权重空间对齐等任务性能。
本文从不可约表示视角重新分析排列及相关群的等变线性层。不同于传统参数共享方法，本工作基于不可约表示与舒尔引理，给出了DeepSets、2-IGN图等变网络及Deep Weight Space（DWS）网络的更简洁推导，其中DWS的推导复杂度显著降低。进一步将方法扩展至非对齐对称集合场景，此时需满足对群的褶积积（wreath product）等变性。先前工作仅在高度受限条件下处理该问题，且几乎所有层均为对称（Siamese）结构。本文给出完整刻画，证明在某些设置下存在大量额外的非对称层。实验表明，这些新层在图异常检测、权重空间对齐和学习Wasserstein距离等任务中可提升性能。代码已开源于GitHub。
原文摘要 · Abstract (English)
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}.
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