Graph Signal Processing over a Probability Space of Shift Operators

TL;DR

This paper extends graph signal processing to handle a probability space of shift operators, enabling analysis when graph topology is uncertain or random, and introduces new Fourier and filtering concepts for this setting.

Contribution

It develops a GSP framework over a probability space of shift operators, generalizing classical GSP to uncertain or random graph topologies.

Findings

MFC filters are expectations of classical random convolution filters.

Bandlimitedness requires more than fixed points of band-pass filters.

Framework applicable to synthetic and real datasets.

Abstract

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.