mlsanityData to 14 Oct 2026

Geometry-Aware Simplicial Message Passing

arXiv 2605.06061 ↗First posted 7 May 2026cs.LG cs.CG math.ATcited by 0

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Abstract (arXiv)

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.

Key facts. No citing papers in the map yet (data through 14 Oct 2026). Builds on 24 of its 36 references that are in the map. First posted on arXiv 7 May 2026.

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24 of 36 references.

References · 24

  1. 2016Semi-Supervised Classification with Graph Convolutional Networkscited by 11,361
  2. 2017PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Spacecited by 4,401
  3. 2018How Powerful are Graph Neural Networks?cited by 4,393
  4. 2018Dynamic Graph CNN for Learning on Point Cloudscited by 2,223
  5. 2017Deep Setscited by 1,421
  6. 2018Weisfeiler and Leman Go Neural: Higher-order Graph Neural Networkscited by 881
  7. 2019Provably Powerful Graph Networkscited by 389
  8. 2021Geometric and Physical Quantities Improve E(3) Equivariant Message Passingcited by 228
  9. 2021Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networkscited by 221
  10. 2021Weisfeiler and Lehman Go Cellular: CW Networkscited by 218
  11. 2021Scalars are universal: Equivariant machine learning, structured like classical physicscited by 127
  12. 2021Weisfeiler and Leman go Machine Learning: The Story so farcited by 122
12 more
  1. 2023On the Expressive Power of Geometric Graph Neural Networkscited by 90
  2. 2024Position: Topological Deep Learning is the New Frontier for Relational Learningcited by 66
  3. 2022Geometrically Equivariant Graph Neural Networks: A Surveycited by 56
  4. 2023E(n) Equivariant Message Passing Simplicial Networkscited by 23
  5. 2023Differentiable Euler Characteristic Transforms for Shape Classificationcited by 18
  6. 2024MANTRA: The Manifold Triangulations Assemblagecited by 12
  7. 2023An Invitation to the Euler Characteristic Transformcited by 9
  8. 2024Diss-l-ECT: Dissecting Graph Data with Local Euler Characteristic Transformscited by 6
  9. 2024Topology meets Machine Learning: An Introduction using the Euler Characteristic Transformcited by 4
  10. 2025LEAP: Local ECT-Based Learnable Positional Encodings for Graphscited by 2
  11. 2026The Logical Expressiveness of Topological Neural Networkscited by 2
  12. 2025On the Stability of the Euler Characteristic Transform for a Perturbed Embeddingcited by 1

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