The Weisfeiler--Lehman (WL) test and its simplicial extension (SWL) characterize the combinatorial expressivity of message passing networks, but they are blind to geometry, i.e., meshes with identical connectivity but different embeddings are indistinguishable. We introduce the Geometric Simplicial Weisfeiler--Lehman (GSWL) test, which incorporates vertex coordinates into color refinement for geometric simplicial complexes. In addition, we show that (i) the expressivity of geometry-aware simplicial message passing schemes is bounded above by GSWL, and (ii) that there exist parameters such that the discriminating power of GSWL is matched by these schemes on any fixed finite family of geometric simplicial complexes. Combined with the Euler Characteristic Transform (ECT), a complete invariant for geometric simplicial complexes, this yields a geometric expressivity characterization together with an approximation framework. Experiments on synthetic and mesh datasets serve to validate our theory, showing a clear hierarchy from combinatorial to geometry-aware models.

翻译：Weisfeiler–Lehman（WL）检验及其单纯形扩展（SWL）刻画了消息传递网络的组合表达能力，但它们对几何结构不敏感——即具有相同连接性但不同嵌入的网格无法被区分。我们引入了几何单纯形Weisfeiler–Lehman（GSWL）检验，该方法将顶点坐标纳入颜色细化过程，以处理几何单纯形复形。此外，我们证明：（i）几何感知单纯形消息传递方案的表达能力上限由GSWL界定；（ii）存在参数使得GSWL的判别能力可由这些方案在任何固定有限几何单纯形复形族上匹配。结合作为几何单纯形复形完全不变量的欧拉特征变换（ECT），这给出了几何表达能力刻画及近似框架。在合成数据集和网格数据集上的实验验证了我们的理论，揭示了从组合模型到几何感知模型的清晰层次结构。