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

• URL: http://arxiv.org/abs/2305.15639v4
• Date: Tue, 19 Sep 2023 03:57:06 GMT
• Title: Revisiting Generalized p-Laplacian Regularized Framelet GCNs: Convergence, Energy Dynamic and Training with Non-Linear Diffusion
• Authors: Dai Shi, Zhiqi Shao, Yi Guo, Qibin Zhao, Junbin Gao
• Abstract summary: This paper presents a theoretical analysis of the graph p-Laplacian regularized framelet network (pL-UFG) We conduct a convergence analysis on pL-UFG, addressing the gap in the understanding of its behaviors. We propose two novel pL-UFG models with manually controlled energy dynamics.
• Score: 44.4195350090039
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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