Stationary signal processing on graphs

TL;DR

This paper extends the concept of stationarity to signals on arbitrary graphs, enabling efficient spectral analysis and improved denoising and regression methods for graph-structured data.

Contribution

It introduces a generalized notion of stationarity on graphs, characterizes it via a Power Spectral Density, and develops Wiener filtering techniques for graph signals.

Findings

Stationary graph signals are characterized by a well-defined Power Spectral Density.

Efficient estimation of the Power Spectral Density is possible even for large graphs.

The proposed methods improve denoising and regression performance on graph signals.

Abstract

Graphs are a central tool in machine learning and information processing as they allow to conveniently capture the structure of complex datasets. In this context, it is of high importance to develop flexible models of signals defined over graphs or networks. In this paper, we generalize the traditional concept of wide sense stationarity to signals defined over the vertices of arbitrary weighted undirected graphs. We show that stationarity is expressed through the graph localization operator reminiscent of translation. We prove that stationary graph signals are characterized by a well-defined Power Spectral Density that can be efficiently estimated even for large graphs. We leverage this new concept to derive Wiener-type estimation procedures of noisy and partially observed signals and illustrate the performance of this new model for denoising and regression.