发表机构
Gaoling School of Artificial Intelligence, Renmin University of China（中国人民大学高瓴人工智能学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对等变神经网络处理对称输入时的对称性增加问题，通过理论刻画提出降低策略，经合成及QM9数据集实验验证了方法的有效性。
AI 中文摘要
等变神经网络（Equivariant Neural Networks, ENNs）已为多个科学领域的应用提供了支撑。尽管其在表示几何结构方面表现出色，但处理对称输入时存在表达能力下降的问题：输出表示对超出输入对称性的变换具有不变性。该现象的数学本质是，对称输入经等变映射处理后会出现对称性增加。现有研究虽已记录特定场景下的对称性增加，但对其根本原因及通用降低策略仍缺乏严谨理解。本文对对称性增加进行了详细深入的刻画，并提出了一套用于降低对称性增加的原则性框架：（i）对于给定的特征空间和输入对称群，证明增加的对称性存在由特征空间结构决定的下确界；（ii）基于该结论，开发了可计算该下确界的算法，并提出了防止有害对称性增加的特征设计实用准则；（iii）在标准正则性假设下，证明对于大多数等变映射，所提准则可有效降低对称性增加。为补充理论发现，在合成数据集和真实世界QM9数据集上开展了可视化与实验，结果验证了理论预测。
英文摘要
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.
