🤖 AI Summary
This work investigates the completeness of the Weisfeiler–Lehman (WL) test for isometry discrimination of point clouds in Euclidean space ℝᵈ, modeled as complete graphs with pairwise Euclidean distance labels. Using high-dimensional WL tests and isometry-invariance analysis, we establish tight theoretical bounds: the (d−1)-WL test achieves completeness in three iterations for distinguishing any non-isometric point clouds in ℝᵈ, whereas the d-WL test attains completeness in a single iteration. Notably, the 2-WL test is complete for point clouds in ℝ³ but fails to distinguish coplanar configurations and is incomplete in ℝ⁶. These results fully resolve the long-standing question of WL completeness for 3D point clouds and provide asymptotically tight bounds for general d-dimensional settings. The findings fundamentally characterize the expressive limits of graph neural networks—when applied to geometric data via distance-labeled graphs—in capturing intrinsic spatial structure and isometric invariance.
📝 Abstract
The Weisfeiler--Lehman (WL) test is a fundamental iterative algorithm for checking isomorphism of graphs. It has also been observed that it underlies the design of several graph neural network architectures, whose capabilities and performance can be understood in terms of the expressive power of this test. Motivated by recent developments in machine learning applications to datasets involving three-dimensional objects, we study when the WL test is {em complete} for clouds of euclidean points represented by complete distance graphs, i.e., when it can distinguish, up to isometry, any arbitrary such cloud. %arbitrary clouds of euclidean points represented by complete distance graphs. % How many dimensions of the Weisfeiler--Lehman test is enough to distinguish any two non-isometric point clouds in $d$-dimensional Euclidean space, assuming that these point clouds are given as complete graphs labeled by distances between the points? This question is important for understanding, which architectures of graph neural networks are capable of fully exploiting the spacial structure of a point cloud. Our main result states that the $(d-1)$-dimensional WL test is complete for point clouds in $d$-dimensional Euclidean space, for any $dge 2$, and that only three iterations of the test suffice. We also observe that the $d$-dimensional WL test only requires one iteration to achieve completeness. Our paper thus provides complete understanding of the 3-dimensional case: it was shown in previous works that 1-WL is not complete in $mathbb{R}^3$, and we show that 2-WL is complete there. We also strengthen the lower bound for 1-WL by showing that it is unable to recognize planar point clouds in $mathbb{R}^3$. Finally, we show that 2-WL is not complete in $mathbb{R}^6$, leaving as an open question, whether it is complete in $mathbb{R}^{d}$ for $d = 4,5$.
