Shapes of Uncertainty in Spectral Graph Theory

TL;DR

This paper introduces a versatile framework for uncertainty principles in spectral graph theory, integrating various existing relations and analyzing the shapes of uncertainty regions for graph signals.

Contribution

It develops a unified approach to uncertainty principles in spectral graph theory, incorporating general filter functions and characterizing uncertainty shapes using numerical range analysis.

Findings

Unifies multiple uncertainty relations in spectral graph theory.

Characterizes the shapes of uncertainty regions for graph signals.

Provides computational tools for analyzing space-frequency localization.

Abstract

We present a flexible framework for uncertainty principles in spectral graph theory. In this framework, general filter functions modeling the spatial and spectral localization of a graph signal can be incorporated. It merges several existing uncertainty relations on graphs, among others the Landau-Pollak principle describing the joint admissibility region of two projection operators, and uncertainty relations based on spectral and spatial spreads. Using theoretical and computational aspects of the numerical range of matrices, we are able to characterize and illustrate the shapes of the uncertainty curves and to study the space-frequency localization of signals inside the admissibility regions.