The paper presents theoretical results on the expressive power of Euclidean graph neural networks (GNNs) for point cloud analysis. It focuses on the ability of these models to separate (or distinguish) point clouds that are not related by permutations and rigid motions.
The key insights are:
Two iterations of the 1-EWL (Euclidean Weisfeiler-Leman) test can separate almost all point clouds in any dimension, except for a set of measure zero.
A single iteration of the vanilla 3-EWL test is complete for 3D point clouds, meaning it can distinguish any pair of non-isomorphic point clouds.
The 2-SEWL (Euclidean Weisfeiler-Leman with Special Euclidean information) test, which incorporates additional geometric information, is also complete for 3D point clouds with a single iteration.
The authors show how to construct differentiable architectures for point clouds, such as 2-SEWLnet, that have the same separation power as the Euclidean k-WL tests, while keeping the complexity reasonable.
The paper provides a theoretical foundation for understanding the expressive power of Euclidean GNNs and offers practical guidelines for designing complete and efficient point cloud architectures.
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by Snir Hordan,... ที่ arxiv.org 04-01-2024
https://arxiv.org/pdf/2301.13821.pdfสอบถามเพิ่มเติม