Graph Signal Processing over a Probability Space of Shift Operators

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 maybe 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, convolution, and band-pass filters in this framework, which subsumes traditional GSP theory as the special case where the probability space consists of a single shift operator. We show that a convolution filter under this framework is the expectation of random convolution filters in traditional GSP, while the notion of bandlimitedness requires additional wriggle 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. Numerical results on both synthetic and real datasets verify the superiority of performing GSP over a probability space of shift operators versus restricting to a single shift operator.