Geometry-Aware Simplicial Message Passing

2026-05-07

AI Summary

Traditional message-passing networks and their simplicial complex extensions struggle to distinguish mesh structures with identical connectivity but differing geometric embeddings, due to a lack of geometric awareness. This work proposes the Geometric Simplicial Weisfeiler–Lehman (GSWL) test, which for the first time incorporates vertex coordinates into the color refinement process, thereby establishing a geometric-aware simplicial message-passing framework. By integrating an Euler characteristic transform, the framework’s expressive power is fully characterized. We prove that GSWL is equivalent to geometric-aware message passing over families of finite geometric simplicial complexes, thus constructing a hierarchical theory bridging combinatorial and geometric expressivity. Experiments on both synthetic and real-world mesh data demonstrate that the proposed approach substantially enhances expressive power, clearly revealing the modeling advantages conferred by explicit geometric information.

Abstract

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.

Problem

• Weisfeiler–Lehman test
• simplicial complexes
• geometric expressivity
• message passing
• mesh representation

Innovation

• Geometric Simplicial Weisfeiler–Lehman
• geometry-aware message passing
• simplicial complexes
• expressivity characterization
• Euler Characteristic Transform