Neural networks for point clouds, which respect their natural invariance to permutation and rigid motion, have enjoyed recent success in modeling geometric phenomena, from molecular dynamics to recommender systems. Yet, to date, no model with polynomial complexity is known to be complete, that is, able to distinguish between any pair of non-isomorphic point clouds. We fill this theoretical gap by showing that point clouds can be completely determined, up to permutation and rigid motion, by applying the 3-WL graph isomorphism test to the point cloud's centralized Gram matrix. Moreover, we formulate an Euclidean variant of the 2-WL test and show that it is also sufficient to achieve completeness. We then show how our complete Euclidean WL tests can be simulated by an Euclidean graph neural network of moderate size and demonstrate their separation capability on highly symmetrical point clouds.

翻译：针对点云的神经网络能够自然地保持其排列和刚体运动的对称性，近年来在分子动力学、推荐系统等几何现象建模领域取得了显著成功。然而，迄今为止，尚不存在已知的多项式复杂度模型能够做到完全区分任意一对非同构点云。我们通过理论填补了这一空白：证明了只需对点云的中心化格拉姆矩阵应用3-WL图同构测试，即可在排列和刚体运动意义下完全确定点云。此外，我们提出了2-WL测试的欧几里得变体，并证明该变体同样足以实现完整性。随后我们展示了如何通过中等规模的欧几里得图神经网络模拟完整的欧几里得WL测试，并在高度对称的点云上验证了其分离能力。