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

TL;DR

This paper provides a comprehensive theoretical analysis of the pL-UFG graph neural network, exploring its convergence, energy dynamics, and relation to non-linear diffusion, leading to improved models with reduced computational costs.

Contribution

It offers the first convergence and energy dynamic analysis of pL-UFG, revealing its relation to non-linear diffusion and proposing new models with controlled energy dynamics and lower computational costs.

Findings

pL-UFG converges with non-zero Dirichlet energy avoiding over-smoothing

Energy dynamics analysis shows synergy between implicit layers and framelets

Proposed models reduce training costs while maintaining advantages

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.