A Spectral Graph Uncertainty Principle

TL;DR

This paper establishes a spectral uncertainty principle for graphs, quantifying the fundamental tradeoff between a signal's localization on a graph and in its spectral domain, with theoretical and computational insights.

Contribution

It introduces a spectral graph uncertainty principle, characterizes the uncertainty region, and provides efficient algorithms and analytical expressions for specific graph classes.

Findings

Uncertainty curve achieved by eigenvectors of Laplacian eigenvalues

Convexity of the uncertainty region enables fast approximation

Analytical bounds for Erdős-Rényi random graphs

Abstract

The spectral theory of graphs provides a bridge between classical signal processing and the nascent field of graph signal processing. In this paper, a spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed. Just as the classical result provides a tradeoff between signal localization in time and frequency, this result provides a fundamental tradeoff between a signal's localization on a graph and in its spectral domain. Using the eigenvectors of the graph Laplacian as a surrogate Fourier basis, quantitative definitions of graph and spectral "spreads" are given, and a complete characterization of the feasibility region of these two quantities is developed. In particular, the lower boundary of the region, referred to as the uncertainty curve, is shown to be achieved by eigenvectors associated with the smallest eigenvalues of an affine family of matrices. The convexity of the uncertainty curve allows it to be found to within $\varepsilon$ by a fast approximation algorithm requiring $O(\varepsilon^{-1/2})$ typically sparse eigenvalue evaluations. Closed-form expressions for the uncertainty curves for some special classes of graphs are derived, and an accurate analytical approximation for the expected uncertainty curve of Erd\H{o}s-R\'enyi random graphs is developed. These theoretical results are validated by numerical experiments, which also reveal an intriguing connection between diffusion processes on graphs and the uncertainty bounds.