基于高阶光滑性和自适应低秩性的时变图信号恢复
Time-Varying Graph Signal Recovery Using High-Order Smoothness and Adaptive Low-rankness
AI总结：
针对时变图信号恢复问题，提出结合高阶Sobolev光滑性与误差函数加权核范数正则化的方法，设计两种高效算法并证明其一收敛性，实验验证其效果优于现有最优方法。
摘要：
时变图信号恢复已广泛应用于气候变化、环境灾害监测和流行病学研究等诸多场景。选择合适的正则化项来描述底层信号的特征至关重要，例如信号在图域上的光滑性，以及以矩阵形式建模的时空信号的低秩结构。作为最常用的选择之一，图拉普拉斯算子（graph Laplacian）通常被用于设计图正则化项，以从部分观测数据中重建定义在图上的信号。本研究提出了一种时变图信号恢复方法，该方法基于高阶Sobolev光滑性，并采用误差函数加权核范数正则化来实现低秩约束。我们提出了两种基于交替方向乘子法（alternating direction method of multipliers）和迭代重加权的高效算法，并详细证明了其中一种算法的收敛性。我们在合成数据集和真实数据集上开展了多种数值实验，结果表明，与图信号恢复领域的现有最优方法相比，所提方法具有更优的有效性。
英文摘要：
Time-varying graph signal recovery has been widely used in many applications, including climate change, environmental hazard monitoring, and epidemic studies. It is crucial to choose appropriate regularizations to describe the characteristics of the underlying signals, such as the smoothness of the signal over the graph domain and the low-rank structure of the spatial-temporal signal modeled in a matrix form. As one of the most popular options, the graph Laplacian is commonly adopted in designing graph regularizations for reconstructing signals defined on a graph from partially observed data. In this work, we propose a time-varying graph signal recovery method based on the high-order Sobolev smoothness and an error-function weighted nuclear norm regularization to enforce the low-rankness. Two efficient algorithms based on the alternating direction method of multipliers and iterative reweighting are proposed, and convergence of one algorithm is shown in detail. We conduct various numerical experiments on synthetic and real-world data sets to demonstrate the proposed method's effectiveness compared to the state-of-the-art in graph signal recovery.
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