Total Variation Denoising for Graph Signals Enhanced with Moreau Envelope

Graph signal denoising plays a crucial role in processing complex network data. However, traditional Total variation (TV) denoising methods often suffer from excessive smoothing and high computational cost. In this paper, we propose an Moreau-enhanced TV (MTV) denoising method for graph signals, based on the Moreau envelope. By introducing a non-convex regularization term, this method improves the conventional \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _{1}$$\end{document}-norm model, effectively reducing edge information loss caused by over-smoothing and preserving the structural features of graph signals. Compared to the traditional TV approach, the MTV method strikes a better balance between noise suppression and signal feature preservation. It excels in capturing signal jumps and mitigating error accumulation, particularly for periodic cyclic and complex random graph signals. Additionally, the proposed method demonstrates strong robustness to changes in graph density, maintaining stable denoising performance even as edge probabilities increase, thus overcoming the instability commonly observed in traditional methods on complex topologies. Experimental results validate that the MTV method outperforms existing techniques across various graph topologies and signal distributions, especially in high-dimensional and complex signal scenarios, highlighting its theoretical and practical significance in graph signal processing.