Geometry-Aware Simplicial Message Passing

The paper introduces the Geometric Simplicial Weisfeiler–Lehman (GSWL) test, which extends the classic WL and SWL tests by incorporating vertex coordinates into color refinement for geometric simplicial complexes. It demonstrates that geometry‑aware simplicial message passing schemes are bounded by GSWL in expressivity and can match GSWL’s discriminating power on any fixed finite family of complexes. By combining GSWL with the Euler Characteristic Transform, the authors provide a complete invariant and an approximation framework, validated through experiments that reveal a clear hierarchy from combinatorial to geometry‑aware models.

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