Signals on Graphs: Uncertainty Principle and Sampling

TL;DR

This paper extends signal processing concepts to graph signals, establishing an uncertainty principle, analyzing sampling strategies, and proposing algorithms for signal recovery and noise handling on graphs.

Contribution

It introduces a new uncertainty principle for graph signals, links sampling strategies to signal recovery, and generalizes reconstruction methods to frame-based approaches.

Findings

Maximally concentrated graph signals characterized.

Uncertainty principle for graph signals derived.

Sampling location significantly impacts recovery performance.

Abstract

In many applications, the observations can be represented as a signal defined over the vertices of a graph. The analysis of such signals requires the extension of standard signal processing tools. In this work, first, we provide a class of graph signals that are maximally concentrated on the graph domain and on its dual. Then, building on this framework, we derive an uncertainty principle for graph signals and illustrate the conditions for the recovery of band-limited signals from a subset of samples. We show an interesting link between uncertainty principle and sampling and propose alternative signal recovery algorithms, including a generalization to frame-based reconstruction methods. After showing that the performance of signal recovery algorithms is significantly affected by the location of samples, we suggest and compare a few alternative sampling strategies. Finally, we provide the conditions for perfect recovery of a useful signal corrupted by sparse noise, showing that this problem is also intrinsically related to vertex-frequency localization properties.