Revisiting Generalized p-Laplacian Regularized Framelet GCNs: Convergence, Energy Dynamic and as Non-Linear Diffusion

Dai Shi, Zhiqi Shao, Yi Guo, Qibin Zhao, Junbin Gao

Abstract: This paper presents a comprehensive theoretical analysis of the graph p-Laplacian regularized framelet network (pL-UFG) to establish a solid understanding of its properties. We conduct a convergence analysis on pL-UFG, addressing the gap in the understanding of its asymptotic behaviors. Further by investigating the generalized Dirichlet energy of pL-UFG, we demonstrate that the Dirichlet energy remains non-zero throughout convergence, ensuring the avoidance of over-smoothing issues. Additionally, we elucidate the energy dynamic perspective, highlighting the synergistic relationship between the implicit layer in pL-UFG and graph framelets. This synergy enhances the model's adaptability to both homophilic and heterophilic data. Notably, we reveal that pL-UFG can be interpreted as a generalized non-linear diffusion process, thereby bridging the gap between pL-UFG and differential equations on the graph. Importantly, these multifaceted analyses lead to unified conclusions that offer novel insights for understanding and implementing pL-UFG, as well as other graph neural network (GNN) models. Finally, based on our dynamic analysis, we propose two novel pL-UFG models with manually controlled energy dynamics. We demonstrate empirically and theoretically that our proposed models not only inherit the advantages of pL-UFG but also significantly reduce computational costs for training on large-scale graph datasets.

Submission Length: Long submission (more than 12 pages of main content)

Previous TMLR Submission Url: https://openreview.net/forum?id=VCXDpyxkEj

Changes Since Last Submission:

Dear Action Editor and Reviewers:

Thank you so much for your valuable comments on our submission. In this camera-ready manuscript, we have carefully addressed the grammatical errors, inappropriate citation formats, and unclear statements in our previous manuscript. We hope our changes can further enhance the readability of our manuscript. Please let us know if any further changes are required.

Once again, we thank all reviewers and editors for your efforts in our submission. Your patience is greatly appreciated.

Kind Regards,

Authors

Assigned Action Editor: ~Giannis_Nikolentzos1

Submission Number: 1778