关于不变几何深度学习模型的完备性
On the Completeness of Invariant Geometric Deep Learning Models
AI总结：
该研究刻画全连接条件下不变几何深度学习模型的理论表达能力，证明多种经典及子图类不变几何模型可实现E(3)-完备性，填补了该领域的表达能力理论空白。
摘要：
不变模型是几何深度学习模型的重要类别，能够通过利用点云中的信息性几何特征生成有意义的几何表示。这类模型具备简洁性、良好的实验效果与计算效率的特点，但其理论表达能力仍不明确，限制了对这类模型潜力的深入理解。本研究聚焦于刻画全连接条件下各类不变模型的理论表达能力。我们首先严格刻画了最经典的不变模型——融合距离的消息传递神经网络（DisGNN）的表达能力，将其不可识别的情形限定为仅高度对称的点云。随后我们证明，GeoNGNN（最简单的子图图神经网络之一的几何对应模型）能够有效打破这些极端情形的对称性，从而实现E(3)-完备性。通过将GeoNGNN作为理论工具，我们进一步证明：1）传统图学习中开发的大多数子图GNN可无缝扩展到几何场景并具备E(3)-完备性；2）DimeNet、GemNet和SphereNet这三种成熟的不变模型也都能实现E(3)-完备性。我们的理论成果填补了不变模型表达能力的研究空白，有助于严谨全面地理解其性能边界。
英文摘要：
Invariant models, one important class of geometric deep learning models, are capable of generating meaningful geometric representations by leveraging informative geometric features in point clouds. These models are characterized by their simplicity, good experimental results and computational efficiency. However, their theoretical expressive power still remains unclear, restricting a deeper understanding of the potential of such models. In this work, we concentrate on characterizing the theoretical expressiveness of a wide range of invariant models under fully-connected conditions. We first rigorously characterize the expressiveness of the most classic invariant model, message-passing neural networks incorporating distance (DisGNN), restricting its unidentifiable cases to be only highly symmetric point clouds. We then prove that GeoNGNN, the geometric counterpart of one of the simplest subgraph graph neural networks, can effectively break these corner cases' symmetry and thus achieve E(3)-completeness. By leveraging GeoNGNN as a theoretical tool, we further prove that: 1) most subgraph GNNs developed in traditional graph learning can be seamlessly extended to geometric scenarios with E(3)-completeness; 2) DimeNet, GemNet and SphereNet, three well-established invariant models, are also all capable of achieving E(3)-completeness. Our theoretical results fill the gap in the expressive power of invariant models, contributing to a rigorous and comprehensive understanding of their capabilities.
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