Signal Processing with a Distribution of Graph 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. Numerical results on real datasets verify the superiority of performing GSP over a probability space of shift operators versus restricting to a single shift operator.