Geometry-Aware Simplicial Message Passing
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.
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BibTeX
@misc{wang2026geometryaware,
title = {{Geometry-Aware Simplicial Message Passing}},
author = {Elena Xinyi Wang and Bastian Rieck},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.06061},
eprint = {2605.06061},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.06061},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Geometry-Aware Simplicial Message Passing
AU - Elena Xinyi Wang
AU - Bastian Rieck
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2605.06061
UR - https://arxiv.org/abs/2605.06061
AB - 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.
ER -