On Graph Uncertainty Principle and Eigenvector Delocalization

TL;DR

This paper explores the uncertainty principle in graph signal processing, linking it to eigenvector delocalization, and derives bounds for random regular graphs and general graphs.

Contribution

It introduces new connections between graph uncertainty bounds and eigenvector delocalization, and provides computational methods for bounds on arbitrary graphs.

Findings

Uncertainty bounds are established for random d-regular graphs.

Numerically efficient approximations for uncertainty bounds are proposed.

Theoretical links between uncertainty principles and eigenvector properties are demonstrated.

Abstract

Uncertainty principles present an important theoretical tool in signal processing, as they provide limits on the time-frequency concentration of a signal. In many real-world applications the signal domain has a complicated irregular structure that can be described by a graph. In this paper, we focus on the global uncertainty principle on graphs and propose new connections between the uncertainty bound for graph signals and graph eigenvectors delocalization. We also derive uncertainty bounds for random $d$-regular graphs and provide numerically efficient upper and lower approximations for the uncertainty bound on an arbitrary graph.