Uncertainty Principle and Sampling of Signals Defined on Graphs

TL;DR

This paper develops an uncertainty principle for signals on graphs, linking it to sampling strategies and signal recovery, with implications for various network-based data applications.

Contribution

It introduces an uncertainty principle for graph signals, establishes a sampling theorem, and explores how sample placement affects recovery performance.

Findings

Uncertainty principle for signals on graphs established.

Sampling location significantly impacts signal recovery.

Proposed alternative sampling strategies outperform existing methods.

Abstract

In many applications, from sensor to social networks, gene regulatory networks or big data, observations can be represented as a signal defined over the vertices of a graph. Building on the recently introduced Graph Fourier Transform, the first contribution of this paper is to provide an uncertainty principle for signals on graph. As a by-product of this theory, we show how to build a dictionary of maximally concentrated signals on vertex/frequency domains. Then, we establish a direct relation between uncertainty principle and sampling, which forms the basis for a sampling theorem of signals defined on graph. Based on this theory, we show that, besides sampling rate, the samples' location plays a key role in the performance of signal recovery algorithms. Hence, we suggest a few alternative sampling strategies and compare them with recently proposed methods.