Challenges and Applications of Graph Signal Processing

A mis primeros maestros, mis abuelitos Jaime y Norma. To my beloved family: Jaime (mi abuelito), Norma (mi abuelita), Tatiana (mi mami), Neg (mi sis), and Ross (mi hermanita). I also want to thank my dad Jaime.

ArXiv, 2019

Graph signal processing deals with signals which are observed on an irregular graph domain. While many approaches have been developed in classical graph theory to cluster vertices and segment large graphs in a signal independent way, signal localization based approaches to the analysis of data on graph represent a new research direction which is also a key to big data analytics on graphs. To this end, after an overview of the basic definitions in graphs and graph signals, we present and discuss a localized form of the graph Fourier transform. To establish an analogy with classical signal processing, spectral- and vertex-domain definitions of the localization window are given next. The spectral and vertex localization kernels are then related to the wavelet transform, followed by a study of filtering and inversion of the localized graph Fourier transform. For rigour, the analysis of energy representation and frames in the localized graph Fourier transform is extended to the energy fo...

IEEE Signal Processing Magazine

International Journal of Circuits, Systems and Signal Processing, 2022

In this paper, we review the development of the traditional graph signal processing methodology, and the recent research areas that are applying graph neural networks on graph data. For the popular topics on processing the graph data with neural networks, the main models/frameworks, dataset and applications are discussed in details. Some challenges and open problems are provided, which serve as the guidance for future research directions.

arXiv (Cornell University), 2022

Graph signal processing (GSP) is a framework to analyze and process graph-structured data. Many research works focus on developing tools such as Graph Fourier transforms (GFT), filters, and neural network models to handle graph signals. Such approaches have successfully taken care of "signal processing" in many circumstances. In this paper, we want to put emphasis on "graph signals" themselves. Although there are characterizations of graph signals using the notion of bandwidth derived from GFT, we want to argue here that graph signals may contain hidden geometric information of the network, independent of (graph) Fourier theories. We shall provide a framework to understand such information, and demonstrate how new knowledge on "graph signals" can help with "signal processing".

Foundations and Trends® in Machine Learning, 2020

IEEE Signal Processing Magazine, 2000

In applications such as social, energy, transportation, sensor, and neuronal networks, high-dimensional data naturally reside on the vertices of weighted graphs. The emerging field of signal processing on graphs merges algebraic and spectral graph theoretic concepts with computational harmonic analysis to process such signals on graphs. In this tutorial overview, we outline the main challenges of the area, discuss different ways to define graph spectral domains, which are the analogues to the classical frequency domain, and highlight the importance of incorporating the irregular structures of graph data domains when processing signals on graphs. We then review methods to generalize fundamental operations such as filtering, translation, modulation, dilation, and downsampling to the graph setting, and survey the localized, multiscale transforms that have been proposed to efficiently extract information from high-dimensional data on graphs. We conclude with a brief discussion of open issues and possible extensions.

2020

Graph signal processing allows us to analyze a graph signal by transforming it into the frequency domain. In general, the procedure requires one to know the signal value at every node. In real situations, it is possible that only partial observations are made, called subgraph signals. In this paper, we introduce a subgraph signal processing framework. It allows us to define Fourier transform and the notion of frequency domain with signals available on a subset of nodes. As a consequence, we are able to give meaningful frequency interpretation of subgraph signals and perform standard signal processing tasks.

arXiv (Cornell University), 2019

The area of Data Analytics on graphs promises a paradigm shift as we approach information processing of classes of data, which are typically acquired on irregular but structured domains (social networks, various ad-hoc sensor networks). Yet, despite its long history, current approaches mostly focus on the optimization of graphs themselves, rather than on directly inferring learning strategies, such as detection, estimation, statistical and probabilistic inference, clustering and separation from signals and data acquired on graphs. To fill this void, we first revisit graph topologies from a Data Analytics point of view, and establish a taxonomy of graph networks through a linear algebraic formalism of graph topology (vertices, connections, directivity). This serves as a basis for spectral analysis of graphs, whereby the eigenvalues and eigenvectors of graph Laplacian and adjacency matrices are shown to convey physical meaning related to both graph topology and higher-order graph properties, such as cuts, walks, paths, and neighborhoods. Through a number of carefully chosen examples, we demonstrate that the isomorphic nature of graphs enables the basic properties and descriptors to be preserved throughout the data analytics process, even in the case of reordering of graph vertices, where classical approaches fail. Next, to illustrate estimation strategies performed on graph signals, spectral analysis of graphs is introduced through eigenanalysis of mathematical descriptors of graphs and in a generic way. Finally, a framework for vertex clustering and graph segmentation is established based on graph spectral representation (eigenanalysis) which illustrates the power of graphs in various data association tasks. The supporting examples demonstrate the promise of Graph Data Analytics in modeling structural and functional/semantic inferences. At the same time, Part I serves as a basis for Part II and Part III which deal with theory, methods and applications of processing Data on Graphs and Graph Topology Learning from data. Contents 1 Introduction 2 2 Graph Definitions and Properties 3 2.1 Basic Definitions. .. .. .. .. .. .. .. 3 2.2 Some Frequently Used Graph Topologies. . 5 2.3 Properties of Graphs and Associated Matrices 7 3 Spectral Decomposition of Graph Matrices 10 3.

2015

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.

International Journal of Mathematics Trends and Technology, 2020

The Signal Processing on Graph (SPG) is an emerging field of research aiming to develop accurate methods for big data analysis by combining graph theory and classical signal processing methods. One key method in signal processing on graph is the so-called Graph Fourier Transform (GFT) which is a generalization of the Classical Fourier Transform (defined for data lying on regular domains :1D for times series or 2D for images) to data lying on networks. Those network data are viewed like a set of interrelated data points lying on a graph whose graph vertices map the data points and graph links encode the relationship between data. In the classical framework, the Fourier transform is a linear operator that performs the mapping of a vector from its initial representation domain to the frequency domain through the Fourier matrix which is an orthonormal basis formed by complex exponential vectors constructed from powers of the complex number. Those vectors are of a key importance in the properties of the transform and its applications. However, for each graph Fourier transform proposed in the literature, although its graph Fourier matrix is orthonormal, its vectors are not complex as in the classical framework, limiting the extension and the use of some useful properties of the classical Fourier transform to the graph signals framework. In this work, we present a method to define a complex orthonormal basis for the graph Fourier transform that allows to perform spectral analysis for graph signals in the frequency domain. The graph Fourier basis we defined is identical to the Fourier basis when applied to graph signals defined on a regular domain. We applied the proposed method successfully to signal detection on an irregularly sampled sensor network.

Using projections on the (generalized) eigenvectors associated to matrices that characterize the topological structure, several authors have constructed generalizations of the Fourier transform on graphs. By exploring mappings of the spectrum of these matrices we show how to construct more general transforms, in particular wavelet-like transforms on graphs. For time-series, tomograms, a generalization of the Radon transforms to arbitrary pairs of non-commuting operators, are positive bilinear transforms with a rigorous probabilistic interpretation which provide a full characterization of the signals and are robust in the presence of noise. Here the notion of tomogram transform is also extended to signals on arbitrary graphs

Cooperative and Graph Signal Processing, 2018

The aim of this chapter is to give an overview of the recent advances related to sampling and recovery of signals defined over graphs. First, we illustrate the conditions for perfect recovery of bandlimited graph signals from samples collected over a selected set of vertexes. Then, we describe some sampling design criteria proposed in the literature to mitigate the effect of noise and model mismatching when performing graph signal recovery. Finally, we illustrate algorithms and optimal sampling strategies for adaptive recovery and tracking of dynamic graph signals, where both sampling set and signal values are allowed to vary with time. Numerical simulations carried out over both synthetic and real data illustrate the potential advantages of graph signal processing methods for sampling, interpolation, and tracking of signals observed over irregular domains such as, e.g., technological or biological networks.

IEEE Signal Processing Magazine, 2019

Comptes Rendus Physique

The legacy of Joseph Fourier in science is vast, especially thanks to the essential tool that is the Fourier Transform. The flexibility of this analysis, its computational efficiency and the physical interpretation it offers makes it a cornerstone in many scientific domains. With the explosion of digital data, both in quantity and diversity, the generalization of the tools based on Fourier Transform is mandatory. In data science, new problems arose for the processing of irregular data such as social networks, biological networks or other data on networks. Graph Signal Processing is a promising approach to deal with those. The present text is an overview of the state-of-the-art in Graph Signal Processing, focusing on how to define a Fourier Transform for data on graphs, how to interpret it and how to use it to process such data. It closes showing some examples of use. Along the way, the review reveals how Fourier's work remains modern and universal, and how his concepts, coming from physics and blended with mathematics, computer science and signal processing, play a key role to answer the modern challenges in data science.