Wide-Sense Stationarity in Generalized Graph Signal Processing

TL;DR

This paper extends graph signal processing to a generalized framework where signals are elements of a Hilbert space, introducing joint wide-sense stationarity and deriving Wiener filters for improved denoising and completion.

Contribution

It introduces a unified theory of graph random processes with joint wide-sense stationarity, encompassing various signal types and deriving new filtering methods.

Findings

Better estimation performance on real and synthetic datasets

Unified theory linking different notions of stationarity

Derivation of Wiener filters for denoising and completion

Abstract

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.