On the Expressive Power of Spectral Invariant Graph Neural Networks

Bohang Zhang, Lingxiao Zhao, Haggai Maron
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:60496-60526, 2024.

Abstract

Incorporating spectral information to enhance Graph Neural Networks (GNNs) has shown promising results but raises a fundamental challenge due to the inherent ambiguity of eigenvectors. Various architectures have been proposed to address this ambiguity, referred to as spectral invariant architectures. Notable examples include GNNs and Graph Transformers that use spectral distances, spectral projection matrices, or other invariant spectral features. However, the potential expressive power of these spectral invariant architectures remains largely unclear. The goal of this work is to gain a deep theoretical understanding of the expressive power obtainable when using spectral features. We first introduce a novel message-passing framework for designing spectral invariant GNNs, called Eigenspace Projection GNN (EPNN). Our comprehensive analysis shows that EPNN essentially unifies all prior spectral invariant architectures, in that they are either strictly less expressive or equivalent to EPNN. A fine-grained expressiveness hierarchy among different architectures is also established. On the other hand, we present a surprising result that EPNN itself is bounded by a recently proposed class of Subgraph GNNs, implying that all these spectral invariant architectures are strictly less expressive than 3-WL. Finally, we demonstrate that these spectral features offer no additional advantage when combined with more expressive GNNs.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-zhang24ck, title = {On the Expressive Power of Spectral Invariant Graph Neural Networks}, author = {Zhang, Bohang and Zhao, Lingxiao and Maron, Haggai}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {60496--60526}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/zhang24ck/zhang24ck.pdf}, url = {https://proceedings.mlr.press/v235/zhang24ck.html}, abstract = {Incorporating spectral information to enhance Graph Neural Networks (GNNs) has shown promising results but raises a fundamental challenge due to the inherent ambiguity of eigenvectors. Various architectures have been proposed to address this ambiguity, referred to as spectral invariant architectures. Notable examples include GNNs and Graph Transformers that use spectral distances, spectral projection matrices, or other invariant spectral features. However, the potential expressive power of these spectral invariant architectures remains largely unclear. The goal of this work is to gain a deep theoretical understanding of the expressive power obtainable when using spectral features. We first introduce a novel message-passing framework for designing spectral invariant GNNs, called Eigenspace Projection GNN (EPNN). Our comprehensive analysis shows that EPNN essentially unifies all prior spectral invariant architectures, in that they are either strictly less expressive or equivalent to EPNN. A fine-grained expressiveness hierarchy among different architectures is also established. On the other hand, we present a surprising result that EPNN itself is bounded by a recently proposed class of Subgraph GNNs, implying that all these spectral invariant architectures are strictly less expressive than 3-WL. Finally, we demonstrate that these spectral features offer no additional advantage when combined with more expressive GNNs.} }
Endnote
%0 Conference Paper %T On the Expressive Power of Spectral Invariant Graph Neural Networks %A Bohang Zhang %A Lingxiao Zhao %A Haggai Maron %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-zhang24ck %I PMLR %P 60496--60526 %U https://proceedings.mlr.press/v235/zhang24ck.html %V 235 %X Incorporating spectral information to enhance Graph Neural Networks (GNNs) has shown promising results but raises a fundamental challenge due to the inherent ambiguity of eigenvectors. Various architectures have been proposed to address this ambiguity, referred to as spectral invariant architectures. Notable examples include GNNs and Graph Transformers that use spectral distances, spectral projection matrices, or other invariant spectral features. However, the potential expressive power of these spectral invariant architectures remains largely unclear. The goal of this work is to gain a deep theoretical understanding of the expressive power obtainable when using spectral features. We first introduce a novel message-passing framework for designing spectral invariant GNNs, called Eigenspace Projection GNN (EPNN). Our comprehensive analysis shows that EPNN essentially unifies all prior spectral invariant architectures, in that they are either strictly less expressive or equivalent to EPNN. A fine-grained expressiveness hierarchy among different architectures is also established. On the other hand, we present a surprising result that EPNN itself is bounded by a recently proposed class of Subgraph GNNs, implying that all these spectral invariant architectures are strictly less expressive than 3-WL. Finally, we demonstrate that these spectral features offer no additional advantage when combined with more expressive GNNs.
APA
Zhang, B., Zhao, L. & Maron, H.. (2024). On the Expressive Power of Spectral Invariant Graph Neural Networks. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:60496-60526 Available from https://proceedings.mlr.press/v235/zhang24ck.html.

Related Material