Introduction to Graph Signal Processing

The necessity to process signals living in non-Euclidean domains, such as signals defined on the top of a graph, has led to the extension of signal processing techniques to the graph setting. Among different approaches, graph signal processing distinguishes itself by providing a Fourier analysis of these signals. Analogously to the Fourier transform for time and image signals, the graph Fourier transform decomposes the graph signals decomposes in terms of the harmonics provided by the underlying topology. For instance, a graph signal characterized by a slow variation between adjacent nodes has a low frequency content. Along with the graph Fourier transform, graph filters are the key tool to alter the graph frequency content of a graph signal. This thesis focuses on graph filters that are performed distributively in the node domain–that is, each node needs to exchange information only within its neighbor to perform a given filtering operation. Similarly to the classical filters, we propose ways to design and implement distributed finite impulse response and infinite impulse response graph filters. One of the key contributions of this thesis is to bring the temporal dimension to graph signal processing and build upon a graph-time signal processing framework. This is done in different ways. First, we analyze the effects that the temporal variations on the graph signal and graph topology have on the filtering output. Second, we introduce the notion of joint graph-time filtering. Third, we presentpr a statistical analysis of the distributed graph filtering when the graph signal and the graph topology change randomly in time. Finally, we extend the sampling framework from the reconstruction of graph signals to the observation and tracking of time-varying graph processes. We characterize the behavior of the distributed autoregressivemoving average (ARMA) graph filters when the graph signal and the graph topology are time-varying. The latter analysis is exploited in two ways: i ) to quantify the limitations of graph filters in a dynamic environment, such as a moving sensors processing a time-varying signal in a sensor network; and i i ) to provide ways for filtering with low computation and communication complexity time-varying graph signals. We develop the notion of distributed graph-time filtering, which is an operation that jointly processes the graph frequencies of a time-varying graph signal on one hand and its temporal frequencies on the other hand. We propose distributed finite impulse response and infinite impulse response recursions to implement a two-dimensional graphtime filtering operation. Finally, we propose design strategies to find the filter coefficients that approximate a desired two-dimensional frequency response. We extend the analysis of graph filters to a stochastic environment, i.e., when the graph topology and the graph signal change randomly over time. By characterizing the first and second order moments of the filter output, we quantify the impact of the graph signal and the graph topology randomness into the distributed filtering operation. The latter allows us to develop the notion of graph filtering in the mean, which is also used to ease the computational burden of classical graph filters. Finally, we propose a sampling framework for time-varying graph signals. Particularly, when the graph signal changes over time following a state-space model, we extend the graph signal sampling theory to the tasks of observing and tracking the time-varying graph signal froma few relevant nodes. The latter theory considers the graph signal sampling as a particular case and shows that tools from sparse sensing and sensor selection can be used for sampling.

Graph is a great model that can efficiently deal with structural data and has excellent research value and application value in our life. Graph theory and signal processing have been perfectly combined to form graph signal processing theory, which provides a theory for studying high-dimensional and structural signals. The wireless sensor network (WSN) is a typical data transmission network based on a graph structure. Because of redundant sensor nodes in WSN, it results in large energy consumption and slow transmission rate. To solve this problem, we propose a new algorithm to optimize the number of nodes by using graph signals processing and compressed sensing (CS) theory. The new node optimization algorithm can not only reduce energy consumption but also increase the transmission rate.

Generalized sampling of graph signals with the prior information based on graph fractional Fourier transform

Sampling is a fundamental topic in graph signal processing with applications in estimation, clustering, and video compression. In contrast to traditional signal processing, however, the irregularity of the signal domain makes the selection of the sampling points non-trivial and hard to analyze. Indeed, although graph signal reconstruction is well-understood in the noiseless case, performance bounds for the interpolation of noisy samples exist mainly for randomized sampling schemes. This paper addresses this issue by deriving a lower bound on the mean-square interpolation error for graph signals. This bound is universal in the sense that it is not restricted to a specific sampling method and holds for all sampling sets. Simulations illustrate the tightness of the bound, which is then used to evaluate the performance of greedy sampling. Finally, a solution to the complexity issues of kernel principal component analysis is proposed using graph signal sampling.

This paper investigates the relation between graph signal processing and consensus of multi-agent systems. The graph signal processing is a technique to process graph signals, that is, signals whose values are on the vertices of graphs and whose structures are specified by the edges. By considering the combination of the states of agents and the graph describing the network structure between them as a graph signal, we show that the multi-agent consensus corresponds to low-pass filtering of the graph signal. This reveals a connection between the two distinct areas, i.e., the graph signal processing and the control of multi-agent systems. In addition, we provide a design method of consensus controllers based on the graph signal processing. In the proposed method, the controllers of agents are designed so that the spatial frequency of the states of the agents becomes a desired one. This enables us to construct controllers of multi-agent systems in the spatial frequency domain.

State-of-the-art data analysis tools have to deal with high-dimensional data. Fortunately, the inherent dimensionality of data is often much smaller, as it has an internal structure limiting its degrees of freedom. In most cases, this structure can be approximated using a graph, i.e, a set of nodes connected by edges. Based on this idea, graphs have been largely used in semi-supervised and unsupervised learning. The canonical assumption when incorporating a graph prior is to assume that the signal is smooth with respect to the graph, i.e, that connected pieces of data have similar values. Nevertheless, this hypothesis is insufficient to characterize more complex relations between the data and its structure, the graph. To this end, the field of Graph Signal Processing (GSP) extends the graph smoothness hypothesis to a more general assumption. Its key idea is to think of the signal as a sum of harmonic modes, which are obtained from the eigen-decomposition of the Laplacian, an operator representing the second order derivative. Thus, the essence of GSP resides in the Fourier transform defined by the projection on the Laplacian eigenvectors basis, which allows us to filter graph signals. However, GSP suffers from a counterintuitive irregularity. Contrarily to the classical Fourier basis, the graph Laplacian eigenvectors are not spreading uniformly on the vertex set and can be highly concentrated in a region of the graph. This non-uniformity results in a lack of any intuitive translation operator and consequently poses challenges to generalize some of the classical signal processing theory, e.g., stationarity, uncertainty principles, or sampling. In this thesis, we answer these difficulties by accepting that each node is unique and should be treated specially according to its surrounding structure. This is achieved using an operator called localization that shifts a kernel defined in the spectral domain around a specific node while adapting to the local graph structure. In the first part of the thesis, we focus on harmonic analysis in GSP. We start by making use of the norm of localized kernels to capture the spectral local features of the graph vertices. To illustrate the relevance of this analysis we then introduce structural clustering, an algorithm that groups nodes according to the role they play in the graph. Then, we bound the vertex-frequency concentration of graph filter banks atoms. Because of the irregular structure of graphs, we construct a local uncertainty principle that is, again, driven by the norm of a localized kernel. In the second part of the thesis, we tackle GSP problems from a machine learning perspective. We use the localization operator to extend the notion of stationarity to graph signals. It consists of a graph signal probabilistic framework allowing us to optimally solve inverse problems. Finally, we show how the graph total variation can be used in semi-supervised learning and prove its consistency with respect to the underlying graph manifold.

The study of sampling signals on graphs, with the goal of building an analog of sampling for standard signals in the time and spatial domains, has attracted considerable attention recently. Beyond adding to the growing theory on graph signal processing (GSP), sampling on graphs has various promising applications. In this article, we review current progress on sampling over graphs focusing on theory and potential applications. Although most methodologies used in graph signal sampling are designed to parallel those used in sampling for standard signals, sampling theory for graph signals significantly differs from the theory of Shannon--Nyquist and shift-invariant sampling. This is due in part to the fact that the definitions of several important properties, such as shift invariance and bandlimitedness, are different in GSP systems. Throughout this review, we discuss similarities and differences between standard and graph signal sampling and highlight open problems and challenges.

The emerging field of graph signal processing (GSP) allows one to transpose classical signal processing operations (e.g., filtering) to signals on graphs. The GSP framework is generally built upon the graph Laplacian, which plays a crucial role in studying graph properties and measuring graph signal smoothness. Here, instead, we propose the graph modularity matrix as the centerpiece of GSP to incorporate knowledge about graph community structure when processing signals on the graph but without the need for community detection. We study this approach in several generic settings, such as filtering, optimal sampling and reconstruction, surrogate data generation, and denoising. Feasibility is illustrated by a small-scale example and a transportation network data set as well as one application in human neuroimaging where community-aware GSP reveals relationships between behavior and brain features that are not shown by Laplacian-based GSP. This work demonstrates how concepts from network science can lead to new, meaningful operations on graph signals.

The Graph Signal Processing (GSP) is a mathematical framework that extends the Discrete Signal Processing (DSP) tools such as filtering and signals decomposition to graph data structures. In this paper, we explore the application of the GSP framework to distributed wireless sensor networks to reduce the measured noise and/or estimate the signal of missing sensors. The context is that of distributed monitoring applications, such as the monitoring of large buildings, such as hospitals, public areas and farmlands. Using simulation tools, we analysed the ability of GSP in reducing the noise and in estimating sensor's data in different WSN scenarios. We modelled the sensor networks as Graph structures and apply Graph Shift and Graph Laplacian operations on such graph signals. The analysis of obtained results shows that GSP may represent a valuable tool in the considered scenarios with outstanding performance.

This paper initiates a discussion on the application of the graph signal processing to exploration of complex and heterogeneous data and systems, and especially for the case of environmental monitoring in smart habitat of city, country, and continent. This emerging approach relates to the objects, which can be represented by a networked structure, but enables also the reconstruction of network-like associations from data when this kind of structured organization is not apparent. In this paper, the sensor network is perceived as the fundamental layer of the smart habitat, and inference is provided not only by direct operation on acquired signals, but also network model is identified for recorded ozone (O <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> ) data in 100 measurement points deployed in Poland. It means that a graph as a mathematical representation of the complex network is generated, which links system features and behaviors coded in measured data sets. Results of multiscale projections are commented for ozone data sets. Furthermore, in opposite to the classical signal processing, the spectral analysis for graph signals is demonstrated, including reconstruction of graph Laplacian and Fourier transform calculation for signals spanned on graph vertices. Finally, local (related to the location of sensor in network) properties and behaviors are clustered based on spectral maps generated for graph signals.

In response to the problem of insufficient adaptability of parameterized learning in local observation graph signal scenes in multi-task learning scenarios, a graph signal processing technique for multi-task scenarios is studied and introduced. In this technology, the research is based on the improved diffusion least mean square algorithm for clustering sampling, using different graph filters to train parameters and a clustering strategy to complete sampling. In addition, a multi-task power grid voltage graph signal detection technique with improved high pass filtering is proposed, and power grid data security detection is achieved through threshold setting and projection detection. In the analysis of graph signal processing, the research model has better steady-state learning performance. For example, in the transient mean square deviation test of multitasking scenarios, its convergence value is −31.25 dB, which is better than similar models. In the single-task transient mean square deviation test, the research model is closer to the reference standard, with a convergence value of −34.25 dB and better learning ability. In power application analysis, the research model performs better in graph signal detection, such as in voltage amplitude scenarios where the maximum detection probability is 100%. In the voltage angle scenario, when the voltage angle is higher than ±−2°, the detection probability of the research model is higher than 92.25%. It can be seen that the technology proposed by the research has excellent application effects. This study will provide technical support for the improvement of graph signal technology and data security.

In graph signal processing (GSP), prior information on the dependencies in the signal is collected in a graph which is then used when processing or analyzing the signal. Blind source separation (BSS) techniques have been developed and analyzed in different domains, but for graph signals the research on BSS is still in its infancy. In this paper, this gap is filled with two contributions. First, a nonparametric BSS method, which is relevant to the GSP framework, is refined, the Cramér-Rao bound (CRB) for mixing and unmixing matrix estimators in the case of Gaussian moving average graph signals is derived, and for studying the achievability of the CRB, a new parametric method for BSS of Gaussian moving average graph signals is introduced. Second, we also consider BSS of non-Gaussian graph signals and two methods are proposed. Identifiability conditions show that utilizing both graph structure and non-Gaussianity provides a more robust approach than methods which are based on only either graph dependencies or non-Gaussianity. It is also demonstrated by numerical study that the proposed methods are more efficient in separating non-Gaussian graph signals.

Chapter 8 - Graph Signal Processing

This paper introduces a design method for densergraph-frequency graph Fourier frames (DGFFs) to enhance graph signal processing and analysis. The graph Fourier transform (GFT) enables us to analyze graph signals in the graph spectral domain and facilitates various graph signal processing tasks, such as filtering, sampling and reconstruction, denoising, and so on. However, the conventional GFT faces two significant limitations. First, unlike the discrete Fourier transform and its variants (such as discrete cosine transforms), the graph frequencies of the derived graph Fourier basis (GFB) from a given graph tend to be unevenly distributed or localized, which leads to biased spectral analysis. Second, the GFB used in GFT does not provide an efficient sparse representation of graph signals compared to overcomplete systems like frames. To overcome these challenges, we propose adding oscillating vectors with intermediate graph frequencies between the original vectors in the GFB for both undirected and directed graphs, constructing GFFs with densergraph frequencies. The resulting DGFFs are expected to enable more accurate graph signal analysis. Furthermore, we propose a graph filtering method based on the DGFFs. In experiments, we apply the DGFFs to practical applications such as graph signal recovery, demonstrating superior performance compared to existing GFBs.

Sampling is a fundamental topic in graph signal processing, having found applications in estimation, clustering, and video compression. In contrast to traditional signal processing, the irregularity of the signal domain makes selecting a sampling set non-trivial and hard to analyze. Indeed, though conditions for graph signal interpolation from noiseless samples exist, they do not lead to a unique sampling set. The presence of noise makes choosing among these sampling sets a hard combinatorial problem. Although greedy sampling schemes are commonly used in practice, they have no performance guarantee. This work takes a twofold approach to address this issue. First, universal performance bounds are derived for the Bayesian estimation of graph signals from noisy samples. In contrast to currently available bounds, they are not restricted to specific sampling schemes and hold for any sampling sets. Second, this paper provides near-optimal guarantees for greedy sampling by introducing the concept of approximate submodularity and updating the classical greedy bound. It then provides explicit bounds on the approximate supermodularity of the interpolation mean-square error showing that it can be optimized with worst-case guarantees using greedy search even though it is not supermodular. Simulations illustrate the derived bound for different graph models and show an application of graph signal sampling to reduce the complexity of kernel principal component analysis.