reviews: 
Comments: This work introduces an innovative approach for generating sparse yet topologically connected geometric graphs by projecting points onto a spherical surface, constructing a corresponding polyhedron, and deriving a graph from its edge connectivity. The method addresses computational limitations inherent in earlier approaches that relied on power graphs. The paper establishes theoretical foundations by demonstrating that highly expressive message-passing neural networks with carefully selected node and edge feature initializations can differentiate among arbitrary graphs. Notably, the authors establish that through a sparse graph structure with maximum degree 3, first-order Weisfeiler-Leman expressivity can distinguish all generic point cloud configurations without requiring fully connected representations. Experimental validation confirms the practical utility of the approach and the role of polyhedron face angles in achieving universality on sparse settings, with consistent improvements demonstrated across synthetic problems, classification benchmarks, and regression tasks.
Strengths: 1. The proposed technique leverages a computationally efficient O(n*log(n)) algorithm and demonstrates negligible computational cost in practice, making it scalable to larger datasets
2. Introduces a fresh perspective to an underexplored domain: constructing geometric graphs through distance threshold-based approaches. Both theoretical analysis and experimental evidence support the validity of this methodology.
3. Offers a framework-independent solution applicable across diverse message-passing neural network variants and similar architectures
Weaknesses: 1. The theoretical contributions rely substantially on existing work, and the underlying algorithm for graph construction is also established, suggesting an element of incremental innovation. However, the integration of these concepts represents a meaningful contribution to the machine learning landscape.
2. Empirical findings are reliable and reproducible but do not establish new state-of-the-art benchmarks; instead, they provide performance gains primarily for less expressive models. Additional experiments employing SCHull graphs with standard initializations on more sophisticated architectures (such as MACE or SEGNN) would strengthen the evaluation. While achieving comparable performance to these methods on distance-constrained graphs with improved computational efficiency is valuable, this comparison would be informative. A more comprehensive empirical analysis along these lines could justify a higher assessment.
3. The review of geometric deep learning literature appears incomplete. Notably absent is discussion of expressivity considerations for non-generic point clouds and comparative analysis with architectures designed for universality on arbitrary point cloud inputs. The following references address these gaps: Hordan, S., Amir, T. and Dym, N., 2024. Weisfeiler Leman for Euclidean Equivariant Machine Learning. ICML 2024; Li, Zian, et al. "Is distance matrix enough for geometric deep learning?" Advances in Neural Information Processing Systems 36 (2024). Resolving these concerns would likely strengthen the contribution.
Recommendation: 10
Reviewer confidence: 5
Soundness/correctness: 4
Clarity: 3