Geometry-Aware Simplicial Message Passing
For researchers in geometric deep learning, this work provides a theoretical framework to characterize the expressivity of geometry-aware message passing on simplicial complexes, addressing a known limitation of combinatorial tests.
The paper introduces the Geometric Simplicial Weisfeiler-Lehman (GSWL) test, which incorporates vertex coordinates into color refinement to distinguish geometric simplicial complexes that are indistinguishable by combinatorial tests. It proves that geometry-aware simplicial message passing schemes are bounded by GSWL and can match its discriminating power, validated on synthetic and mesh datasets.
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.