Towards a characterization of the uncertainty curve for graphs

TL;DR

This paper explores the uncertainty principle on graphs by characterizing signals that achieve the minimal spectral-graph localization trade-off, extending previous work to a broader class of graphs.

Contribution

It provides a new characterization of signals that attain the uncertainty curve for a wider range of graphs, advancing understanding of signal localization on graphs.

Findings

Characterization of signals on graphs achieving the uncertainty curve

Extension of previous results to more general graph classes

Deeper insight into the localization trade-off in graph signal processing

Abstract

Signal processing on graphs is a recent research domain that aims at generalizing classical tools in signal processing, in order to analyze signals evolving on complex domains. Such domains are represented by graphs, for which one can compute a particular matrix, called the normalized Laplacian. It was shown that the eigenvalues of this Laplacian correspond to the frequencies of the Fourier domain in classical signal processing. Therefore, the frequency domain is not the same for every support graph. A consequence of this is that there is no non-trivial generalization of Heisenberg's uncertainty principle, that states that a signal cannot be fully localized both in the time domain and in the frequency domain. A way to generalize this principle, introduced by Agaskar and Lu, consists in determining a curve that represents a lower bound on the compromise between precision in the graph domain and precision in the spectral domain. The aim of this paper is to propose a characterization of the signals achieving this curve, for a larger class of graphs than the one studied by Agaskar and Lu.