提出解决等变网络对称性增强问题的新方法,提升模型表达能力。
Reducing Symmetry Increase in Equivariant Neural Networks

- 从特征空间结构出发,证明对称性增加存在下确界。
- 设计可计算算法求解下确界,并给出避免有害对称性增强的特征设计指南。
- 在合成数据和QM9数据集上验证理论有效性,适用于科学计算场景。
等变神经网络(ENNs)在科学领域广泛应用,但处理对称输入时表达能力下降:输出表示对超出输入对称性的变换保持不变。其数学本质是输入经等变映射后对称性增加。现有研究仅关注特定案例,缺乏对成因的深入理解与通用缓解策略。本文首次系统刻画对称性增加现象,提出一个严谨框架:(i) 对任意特征空间与输入对称群,证明对称性增加存在由特征空间结构决定的下确界;(ii) 基于此构建可计算算法求解该下确界,并提出实用的特征设计准则以防止有害对称性增强;(iii) 在标准正则性假设下,证明多数等变映射中该准则有效降低对称性增加。通过合成数据与真实世界QM9数据集的可视化与实验验证了理论预测的正确性。
原文摘要 · Abstract (English)
Equivariant Neural Networks (ENNs) have empowered numerous applications in scientific fields. Despite their remarkable capacity for representing geometric structures, ENNs suffer from degraded expressivity when processing symmetric inputs: the output representations are invariant to transformations that extend beyond the input's symmetries. The mathematical essence of this phenomenon is that a symmetric input, after being processed by an equivariant map, experiences an increase in symmetry. While prior research has documented symmetry increase in specific cases, a rigorous understanding of its underlying causes and general reduction strategies remains lacking. In this paper, we provide a detailed and in-depth characterization of symmetry increase together with a principled framework for its reduction: (i) For any given feature space and input symmetry group, we prove that the increased symmetry admits an infimum determined by the structure of the feature space; (ii) Building on this foundation, we develop a computable algorithm to derive this infimum, and propose practical guidelines for feature design to prevent harmful symmetry increases. (iii) Under standard regularity assumptions, we demonstrate that for most equivariant maps, our guidelines effectively reduce symmetry increase. To complement our theoretical findings, we provide visualizations and experiments on both synthetic datasets and the real-world QM9 dataset. The results validate our theoretical predictions.
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