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. We first rigorously bound 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 show that GeoNGNN, the geometric counterpart of one of the simplest subgraph graph neural networks (subgraph GNNs), 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 theoretical power of invariant models, contributing to a rigorous and comprehensive understanding of their capabilities. We also empirically evaluated GeoNGNN, the simplest model within the large E(3)-complete family we established, which achieves competitive results to models relying on high-order invariant/equivariant representations on molecule-relevant tasks.

翻译：不变模型作为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来生成有意义的几何表示。这类模型以结构简洁、实验效果良好和计算效率高为特点。然而，其理论表达能力尚不明确，限制了对这类模型潜力的深入理解。在本工作中，我们专注于刻画广泛不变模型的理论表达能力。我们首先严格界定了最经典的不变模型——融合距离信息的消息传递神经网络（DisGNN）的表达能力上限，将其无法区分的案例限制在仅具有高度对称性的点云。随后，我们证明GeoNGNN（一种最简单的子图图神经网络在几何领域的对应模型）能够有效打破这些极端案例的对称性，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）传统图学习中发展的大多数子图GNN可以无缝扩展到几何场景并保持E(3)完备性；2）DimeNet、GemNet和SphereNet这三种成熟的不变模型同样能够实现E(3)完备性。我们的理论结果填补了不变模型理论能力方面的空白，有助于对其能力形成严谨而全面的理解。我们还对GeoNGNN（我们所建立的大型E(3)完备模型族中最简单的模型）进行了实证评估，其在分子相关任务上取得了与依赖高阶不变/等变表示的模型相竞争的结果。