On the Degrees of Freedom of Signals on Graphs

TL;DR

This paper explores the conditions under which signals on graphs can be perfectly localized and reconstructed, providing theoretical insights and practical algorithms for sampling and reconstruction of band-limited graph signals.

Contribution

It establishes the conditions for perfect localization and reconstruction of graph signals, and introduces an algorithm for sampling and reconstructing band-limited signals.

Findings

Conditions for perfect localization of signals on graphs.

Algorithm for reconstructing band-limited graph signals.

Method to select bandwidth minimizing reconstruction error.

Abstract

Continuous-time signals are well known for not being perfectly localized in both time and frequency domains. Conversely, a signal defined over the vertices of a graph can be perfectly localized in both vertex and frequency domains. We derive the conditions ensuring the validity of this property and then, building on this theory, we provide the conditions for perfect reconstruction of a graph signal from its samples. Next, we provide a finite step algorithm for the reconstruction of a band-limited signal from its samples and then we show the effect of sampling a non perfectly band-limited signal and show how to select the bandwidth that minimizes the mean square reconstruction error.