Signal Representations on Graphs: Tools and Applications

TL;DR

This paper introduces a comprehensive framework for representing and modeling various types of signals on graphs, providing theoretical foundations, algorithms, and real-world applications for smooth, piecewise-constant, and piecewise-smooth graph signals.

Contribution

It develops explicit definitions and dictionaries for different classes of graph signals, and analyzes their effectiveness in approximation and sampling tasks with practical case studies.

Findings

Effective graph dictionaries for different signal classes

Theoretical analysis of approximation and sampling methods

Successful real-world applications demonstrating the framework

Abstract

We present a framework for representing and modeling data on graphs. Based on this framework, we study three typical classes of graph signals: smooth graph signals, piecewise-constant graph signals, and piecewise-smooth graph signals. For each class, we provide an explicit definition of the graph signals and construct a corresponding graph dictionary with desirable properties. We then study how such graph dictionary works in two standard tasks: approximation and sampling followed with recovery, both from theoretical as well as algorithmic perspectives. Finally, for each class, we present a case study of a real-world problem by using the proposed methodology.