用克里福代数提升图神经网络的几何对称性表达能力
A Clifford Algebraic Approach to E(n)-Equivariant High-order Graph Neural Networks
- 基于克里福代数构建高阶消息传递机制，实现E(n)等变性
- 在多体、运动捕捉和MD17数据集上超越现有方法
- 适合需要几何对称性的分子与物理系统建模
设计能处理数据对称性的神经网络架构至关重要，尤其对于在欧几里得变换下保持等变性的几何图。当前基于消息传递的等变图神经网络（EGNN）表达能力受限，而高阶图神经网络虽能克服此局限，却缺乏等变性质，限制了其在化学与物理科学中的应用。本文提出克里福群等变图神经网络（CG-EGNN），通过引入克里福代数框架，将高阶局部结构融入消息传递过程。利用克里福代数特性，模型可从位置特征中学习等变函数。高阶消息传递机制使模型获取更丰富的邻域信息，提升性能。我们证明了k跳消息传递框架的普遍性，表明加入k跳机制可增强表达能力。实验验证，CG-EGNN在多体、CMU运动捕捉及MD17等基准任务上均优于此前方法，展现了其在几何深度学习中的有效性。
原文摘要 · Abstract (English)
Designing neural network architectures that can handle data symmetry is crucial. This is especially important for geometric graphs whose properties are equivariance under Euclidean transformations. Current equivariant graph neural networks (EGNNs), particularly those using message passing, have a limitation in expressive power. Recent high-order graph neural networks can overcome this limitation, yet they lack equivariance properties, representing a notable drawback in certain applications in chemistry and physical sciences. In this paper, we introduce the Clifford Group Equivariant Graph Neural Networks (CG-EGNNs), a novel EGNN that enhances high-order message passing by integrating high-order local structures in the context of Clifford algebras. As a key benefit of using Clifford algebras, CG-EGNN can learn functions that capture equivariance from positional features. By adopting the high-order message passing mechanism, CG-EGNN gains richer information from neighbors, thus improving model performance. Furthermore, we establish the universality property of the $k$-hop message passing framework, showcasing greater expressive power of CG-EGNNs with additional $k$-hop message passing mechanism. We empirically validate that CG-EGNNs outperform previous methods on various benchmarks including n-body, CMU motion capture, and MD17, highlighting their effectiveness in geometric deep learning.
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