This work presents a comprehensive theoretical analysis of graph p-Laplacian based framelet network (pL-UFG) to establish a solid understanding of its properties. We begin by conducting a convergence analysis of the p-Laplacian based implicit layer integrated after the framelet convolution, providing insights into the asymptotic behavior of pL-UFG. By exploring the generalized Dirichlet energy of pL-UFG, we demonstrate that the Dirichlet energy remains non-zero, ensuring the avoidance of over-smoothing issues in pL-UFG as it approaches convergence. Furthermore, we elucidate the dynamic energy perspective through which the implicit layer in pL-UFG synergizes with graph framelets, enhancing the model's adaptability to both homophilic and heterophilic data. Remarkably, we establish that the implicit layer can be interpreted as a generalized non-linear diffusion process, enabling training using diverse schemes. These multifaceted analyses lead to unified conclusions that provide novel insights for understanding and implementing pL-UFG, contributing to advancements in the field of graph-based deep learning.

本文针对基于图p-拉普拉斯的框架网络（pL-UFG）进行了全面的理论分析，着重探讨了其性质，包括收敛分析、广义狄利克雷能量、隐式层动态能量等。研究表明，隐式层可以解释为广义非线性扩散过程，有利于使用多种方案进行训练。研究结果提供了实现pL-UFG的新思路，并促进了基于图的深度学习领域的发展。

重新审视广义p-Laplacian正则化Framelet GCNs: 收敛性， 能量动态和使用非线性扩散进行训练

重新审视广义p-Laplacian正则化Framelet GCNs: 收敛性，能量动态和使用非线性扩散进行训练