We consider statistical graph signal processing (GSP) in a generalized framework where each vertex of a graph is associated with an element from a Hilbert space. This general model encompasses various signals such as the traditional scalar-valued graph signal, multichannel graph signal, and discrete- and continuous-time graph signals, allowing us to build a unified theory of graph random processes. We introduce the notion of joint wide-sense stationarity in this generalized GSP framework, which allows us to characterize a graph random process as a combination of uncorrelated oscillation modes across both the vertex and Hilbert space domains. We elucidate the relationship between the notions of wide-sense stationarity in different domains, and derive the Wiener filters for denoising and signal completion under this framework. Numerical experiments on both real and synthetic datasets demonstrate the utility of our generalized approach in achieving better estimation performance compared to traditional GSP or the time-vertex framework.

中文翻译：

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广义图信号处理中的广义平稳性

我们在广义框架中考虑统计图信号处理（GSP），其中图的每个顶点都与来自希尔伯特空间的元素相关联。这个通用模型包含了各种信号，例如传统的标量值图信号、多通道图信号以及离散和连续时间图信号，使我们能够建立一个统一的图随机过程理论。我们在这个广义 GSP 框架中引入了联合广义平稳性的概念，它允许我们将图随机过程描述为顶点和希尔伯特空间域中不相关振荡模式的组合。我们阐明了不同领域广义平稳性概念之间的关系，并在该框架下推导出用于去噪和信号补全的维纳滤波器。