Signal Processing on Directed Graphs

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".

ArXiv, 2022

Graph filtering is the cornerstone operation in graph signal processing (GSP). Thus, understanding it is key in developing potent GSP methods. Graph filters are local and distributed linear operations, whose output depends only on the local neighborhood of each node. Moreover, a graph filter’s output can be computed separately at each node by carrying out repeated exchanges with immediate neighbors. Graph filters can be compactly written as polynomials of a graph shift operator (typically, a sparse matrix description of the graph). This has led to relating the properties of the filters with the spectral properties of the corresponding matrix – which encodes global structure of the graph. In this work, we propose a framework that relies solely on the local distribution of the neighborhoods of a graph. The crux of this approach is to describe graphs and graph signals in terms of a measurable space of rooted balls. Leveraging this, we are able to seamlessly compare graphs of different ...

ArXiv, 2020

Many modern data analytics applications on graphs operate on domains where graph topology is not known a priori, and hence its determination becomes part of the problem definition, rather than serving as prior knowledge which aids the problem solution. Part III of this monograph starts by addressing ways to learn graph topology, from the case where the physics of the problem already suggest a possible topology, through to most general cases where the graph topology is learned from the data. A particular emphasis is on graph topology definition based on the correlation and precision matrices of the observed data, combined with additional prior knowledge and structural conditions, such as the smoothness or sparsity of graph connections. For learning sparse graphs (with small number of edges), the least absolute shrinkage and selection operator, known as LASSO is employed, along with its graph specific variant, graphical LASSO. For completeness, both variants of LASSO are derived in an...

2019

The focus of Part I of this monograph has been on both the fundamental properties, graph topologies, and spectral representations of graphs. Part II embarks on these concepts to address the algorithmic and practical issues centered round data/signal processing on graphs, that is, the focus is on the analysis and estimation of both deterministic and random data on graphs. The fundamental ideas related to graph signals are introduced through a simple and intuitive, yet illustrative and general enough case study of multisensor temperature field estimation. The concept of systems on graph is defined using graph signal shift operators, which generalize the corresponding principles from traditional learning systems. At the core of the spectral domain representation of graph signals and systems is the Graph Discrete Fourier Transform (GDFT). The spectral domain representations are then used as the basis to introduce graph signal filtering concepts and address their design, including Chebys...

2015 49th Asilomar Conference on Signals, Systems and Computers, 2015

Joint filtering of signals indexed on a graph consists in filtering not only the signal, but also the graph by an appropriate downsampling. Existing methods for filtering and downsampling graph signals approximate graphs as sums of bipartite graphs or use nodal domains of the Laplacian. Here, a different method is introduced, and is based on the partitioning in meaningful subgraphs of the graph itself, e.g. network's communities; this partition may be interpreted as a coarsening of the graph and may also be tailored to be aware of the signal structure. A method is proposed to create filterbanks that compute, for graph signals, an approximation and several details using the partition to downsample the graph. This means that we jointly filter the graph and the graph signal; it leads to the design of a new subgraphbased filterbank for graph signals. This design is tested on simple examples for compression and denoising.

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.

IEEE Transactions on Signal Processing, 2019

Graph signal processing (GSP) has become an important tool in many areas such as image processing, networking learning and analysis of social network data. In this paper, we propose a broader framework that not only encompasses traditional GSP as a special case, but also includes a hybrid framework of graph and classical signal processing over a continuous domain. Our framework relies extensively on concepts and tools from functional analysis to generalize traditional GSP to graph signals in a separable Hilbert space with infinite dimensions. We develop a concept analogous to Fourier transform for generalized GSP and the theory of filtering and sampling such signals.

Graph signal processing(GSP) is a representation of data in graphical format with directed or undirected vertices. In many applications such as big data networks, economic and social networks analysis signals with graph is relevant. Harmonic analysis for processing the signals with spectral and algebric graphical thereotical concepts are merged and analyzed with respect to signal processing schemes on graphs. In this work, main challenges of GSP are discussed with Graph Spectral Domains (GSD) and when processing the signals on graph. The information is extracted efficiently from the highdimensional data by using operators of signals on graph and transformation of graph on signal are highlighted in this work. Finally, a brief discussion of open issues of GSP are reviewed.

IEEE Transactions on Signal Processing, 2017

A number of applications in engineering, social sciences, physics, and biology involve inference over networks. In this context, graph signals are widely encountered as descriptors of vertex attributes or features in graph-structured data. Estimating such signals in all vertices given noisy observations of their values on a subset of vertices has been extensively analyzed in the literature of signal processing on graphs (SPoG). This paper advocates kernel regression as a framework generalizing popular SPoG modeling and reconstruction and expanding their capabilities. Formulating signal reconstruction as a regression task on reproducing kernel Hilbert spaces of graph signals permeates benefits from statistical learning, offers fresh insights, and allows for estimators to leverage richer forms of prior information than existing alternatives. A number of SPoG notions such as bandlimitedness, graph filters, and the graph Fourier transform are naturally accommodated in the kernel framework. Additionally, this paper capitalizes on the so-called representer theorem to devise simpler versions of existing Tikhonov regularized estimators, and offers a novel probabilistic interpretation of kernel methods on graphs based on graphical models. Motivated by the challenges of selecting the bandwidth parameter in SPoG estimators or the kernel map in kernel-based methods, the present paper further proposes two multi-kernel approaches with complementary strengths. Whereas the first enables estimation of the unknown bandwidth of bandlimited signals, the second allows for efficient graph filter selection. Numerical tests with synthetic as well as real data demonstrate the merits of the proposed methods relative to state-of-the-art alternatives.

2021 IEEE Statistical Signal Processing Workshop (SSP)

In this paper, we develop a signal processing framework of a network without explicit knowledge of the network topology. Instead, we make use of knowledge on the distribution of operators on the network. This makes the framework flexible and useful when accurate knowledge of graph topology is unavailable. Moreover, the usual graph signal processing is a special case of our framework by using the delta distribution. The main elements of the theory include Fourier transform, theory of filtering and sampling.

arXiv (Cornell University), 2015

The structure of networks describing interactions between entities gives significant insights about how these systems work. Recently, an approach has been proposed to transform a graph into a collection of signals, using a multidimensional scaling technique on a distance matrix representing relations between vertices of the graph as points in a Euclidean space: coordinates are interpreted as components, or signals, indexed by the vertices. In this article, we propose several extensions to this approach: We first extend the current methodology, enabling us to highlight connections between properties of the collection of signals and graph structures, such as communities, regularity or randomness, as well as combinations of those. A robust inverse transformation method is next described, taking into account possible changes in the signals compared to original ones. This technique uses, in addition to the relationships between the points in the Euclidean space, the energy of each signal, coding the different scales of the graph structure. These contributions open up new perspectives by enabling processing of graphs through the processing of the corresponding collection of signals. A technique of denoising of a graph by filtering of the corresponding signals is then described, suggesting considerable potential of the approach.

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.

2020

With the explosive growth of information and communication, data is being generated at an unprecedented rate from various sources, including multimedia, sensor networks, biological systems, social networks, and physical infrastructure. Research in graph signal processing aims to develop tools for processing such data by providing a framework for the analysis of high-dimensional datadefined on irregular graph domains. Graph signal processing extends fundamental signal processingconcepts to data supported on graphs that we refer to as graph signals. In this work, we study two fraternal problems: (1) sampling and (2) reconstruction of signals on graphs. Both of these problems are eminent topics in the field of signal processing over the past decades and have meaningful implications for many real-world problems including semi-supervised learning and activelearning on graphs. Sampling is the task of choosing or measuring some representative subset of the signal such that we can interpola...

IEEE Transactions on Signal Processing

We consider statistical graph signal processing (GSP) in a generalized framework where each vertex of a graph is associated with an element from a Hilbert space. This general model encompasses various signals such as the traditional scalar-valued graph signal, multichannel graph signal, and discrete-and continuous-time graph signals, allowing us to build a unified theory of graph random processes. We introduce the notion of joint wide-sense stationarity in this generalized GSP framework, which allows us to characterize a graph random process as a combination of uncorrelated oscillation modes across both the vertex and Hilbert space domains. We elucidate the relationship between the notions of wide-sense stationarity in different domains, and derive the Wiener filters for denoising and signal completion under this framework. Numerical experiments on both real and synthetic datasets demonstrate the utility of our generalized approach in achieving better estimation performance compared to traditional GSP or the time-vertex framework.