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：Weisfeiler-Lehman（WL）检验是判断图同构的基础迭代算法。已有研究指出，该检验构成了多种图神经网络架构的设计基础，而这类网络的性能与能力可通过该检验的表达力来理解。受三维物体相关数据集在机器学习应用中最新发展的启发，我们研究了WL检验对完全距离图表示的欧几里得点云何时具备完备性——即能否区分任意点云的等距构型。主要结论表明：对于任意$d\ge 2$的$d$维欧氏空间中的点云，$(d-1)$维WL检验具有完备性，且仅需三次迭代即可实现。该结果在$d=2,3$时具有紧致性。我们还发现，$d$维WL检验仅需一次迭代即可达到完备性。