移位算子概率空间上的图信号处理
Graph Signal Processing over a Probability Space of Shift Operators
摘要：
图形信号处理（GSP）使用移位算子来定义图形信号集合的傅立叶基。通常选择移位运算符来捕获图的拓扑。然而，在许多应用中，图的拓扑结构可能是未知的先验，其结构不确定，或从预定义的集合中随机生成的每个观察。每一个图的拓扑结构产生一个不同的移位算子。在本文中，我们开发了一个GSP框架上的概率空间的移位算子。我们开发了相应的概念，傅里叶变换，MFC滤波器，带通滤波器，其中包含经典的GSP理论作为特殊情况下，概率空间由一个单一的移位运算符。我们表明，在这个框架下的MFC滤波器是在经典的GSP随机卷积滤波器的期望，而bandlimitedness的概念需要额外的回旋余地，从简单的带通滤波器的一个固定点。我们开发了一种机制，便于从一个空间的移位运算符映射到另一个，这使得我们的框架被应用到一组丰富的场景。我们演示了如何使用合成和真实的数据集的理论可以应用。
英文摘要：
Graph signal processing (GSP) uses a shift operator to define a Fourier basis for the set of graph signals. The shift operator is often chosen to capture the graph topology. However, in many applications, the graph topology may be unknown a priori, its structure uncertain, or generated randomly from a predefined set for each observation. Each graph topology gives rise to a different shift operator. In this paper, we develop a GSP framework over a probability space of shift operators. We develop the corresponding notions of Fourier transform, MFC filters, and band-pass filters, which subsumes classical GSP theory as the special case where the probability space consists of a single shift operator. We show that an MFC filter under this framework is the expectation of random convolution filters in classical GSP, while the notion of bandlimitedness requires additional wiggle room from being simply a fixed point of a band-pass filter. We develop a mechanism that facilitates mapping from one space of shift operators to another, which allows our framework to be applied to a rich set of scenarios. We demonstrate how the theory can be applied by using both synthetic and real datasets.
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