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. Our main result states that the $(d-1)$-dimensional WL test is complete for point clouds in $d$-dimensional Euclidean space, for any $d\ge 2$, and that only three iterations of the test suffice. Our result is tight for $d = 2, 3$. We also observe that the $d$-dimensional WL test only requires one iteration to achieve completeness.

翻译：摘要：Weisfeiler-Lehman（WL）检验是用于图同构性判定的基础迭代算法。已有研究表明，该算法构成了多种图神经网络架构的设计基础，其能力与性能可通过这一检验的表达能力来理解。受机器学习在三维物体数据集应用中最新进展的启发，我们研究了WL检验对于由完全距离图表示的欧氏点云是否具有完备性——即能否区分任意此类点云（在等距变换意义下）。主要结果表明：对于任意$d\ge 2$，$(d-1)$维WL检验在$d$维欧氏空间点云上具有完备性，且仅需三次迭代即可完成。该结果在$d=2,3$情形下具有紧致性。我们还观察到$d$维WL检验仅需一次迭代即可达到完备性。