Graph Signal Processing (GSP) is an emerging research field that extends the concepts of digital signal processing to graphs. GSP has numerous applications in different areas such as sensor networks, machine learning, and image processing. The sampling and reconstruction of static graph signals have played a central role in GSP. However, many real-world graph signals are inherently time-varying and the smoothness of the temporal differences of such graph signals may be used as a prior assumption. In the current work, we assume that the temporal differences of graph signals are smooth, and we introduce a novel algorithm based on the extension of a Sobolev smoothness function for the reconstruction of time-varying graph signals from discrete samples. We explore some theoretical aspects of the convergence rate of our Time-varying Graph signal Reconstruction via Sobolev Smoothness (GraphTRSS) algorithm by studying the condition number of the Hessian associated with our optimization problem. Our algorithm has the advantage of converging faster than other methods that are based on Laplacian operators without requiring expensive eigenvalue decomposition or matrix inversions. The proposed GraphTRSS is evaluated on several datasets including two COVID-19 datasets and it has outperformed many existing state-of-the-art methods for time-varying graph signal reconstruction. GraphTRSS has also shown excellent performance on two environmental datasets for the recovery of particulate matter and sea surface temperature signals.

中文翻译：

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通过 Sobolev 平滑度重建时变图信号


图信号处理（GSP）是一个新兴的研究领域，它将数字信号处理的概念扩展到图。 GSP 在传感器网络、机器学习和图像处理等不同领域有广泛的应用。静态图信号的采样和重建在GSP中发挥了核心作用。然而，许多现实世界的图信号本质上是随时间变化的，并且此类图信号的时间差异的平滑度可以用作先验假设。在当前的工作中，我们假设图信号的时间差异是平滑的，并且我们引入了一种基于 Sobolev 平滑函数扩展的新算法，用于从离散样本重建时变图信号。我们通过研究与优化问题相关的 Hessian 矩阵的条件数，探讨了通过 Sobolev 平滑度 (GraphTRSS) 算法进行时变图信号重建的收敛速度的一些理论方面。我们的算法的优点是比基于拉普拉斯算子的其他方法收敛得更快，而不需要昂贵的特征值分解或矩阵求逆。所提出的 GraphTRSS 在多个数据集（包括两个 COVID-19 数据集）上进行了评估，它的性能优于许多现有的时变图信号重建的最先进方法。 GraphTRSS 还在颗粒物和海面温度信号恢复的两个环境数据集上表现出了出色的性能。