Graph-time signal processing : Filtering and sampling strategies

In many applications involving wireless sensor networks (WSNs), the observed data can be modeled as signals defined over graphs. As a consequence, an increasing interest has been witnessed to develop new methods to analyze graph signals, leading to the emergence of the field of Graph Signal Processing. One of the most important processing tools in this field is graph filters, which can be easily implemented distributedly over networks by means of cooperation among the nodes. Most of previous works related to graph filters assume the same connection probability in both link directions when transmitting an information between two neighboring nodes. This assumption is not realistic in practice due to the typical random link asymmetry in WSNs caused by interferences and background noise. This paper proposes solutions to cope with the problem of graph filtering of noisy time-varying input graph signals over random time-varying asymmetric WSNs. We design an extension to node-variant graph filters that can provide a trade-off between the expected error and variance, by optimizing the filter coefficients adaptively, resulting in an accurate graph filtering. Numerical experiments carried out over different random WSNs illustrate the efficiency of the proposed solutions.

The rapid development of signal processing on graphs provides a new perspective for processing large-scale data associated with irregular domains. In many practical applications, it is necessary to handle massive data sets through complex networks, in which most nodes have limited computing power. Designing efficient distributed algorithms is critical for this task. This paper focuses on the distributed reconstruction of a time-varying bandlimited graph signal based on observations sampled at a subset of selected nodes. A distributed least square reconstruction (DLSR) algorithm is proposed to recover the unknown signal iteratively, by allowing neighboring nodes to communicate with one another and make fast updates. DLSR uses a decay scheme to annihilate the out-of-band energy occurring in the reconstruction process, which is inevitably caused by the transmission delay in distributed systems. Proof of convergence and error bounds for DLSR are provided in this paper, suggesting that the algorithm is able to track time-varying graph signals and perfectly reconstruct time-invariant signals. The DLSR algorithm is numerically experimented with synthetic data and real-world sensor network data, which verifies its ability in tracking slowly time-varying graph signals.

We consider the problem of recovering random graph signals from nonlinear measurements. For this setting, closed-form Bayesian estimators are usually intractable and even numerical evaluation may be difficult to compute for large networks. In this paper, we propose a graph signal processing (GSP) framework for random graph signal recovery that utilizes information on the structure behind the data. First, we develop the GSP-linear minimum mean-squared-error (GSP-LMMSE) estimator, which minimizes the mean-squared-error (MSE) among estimators that are represented as an output of a graph filter. The GSP-LMMSE estimator is based on <i>diagonal</i> covariance matrices in the graph frequency domain, and thus, has reduced complexity compared with the LMMSE estimator. This property is especially important when using the sample-mean estimators that are based on a training dataset. We then state conditions under which the low-complexity GSP-LMMSE estimator coincides with the optimal LMMSE estimator. Next, we develop an approximate parametrization of the GSP-LMMSE estimator by graph filters. We present three implementations of the parametric GSP-LMMSE estimator for typical graph filters. These parametric graph filters are more robust to outliers and to network topology changes. In our simulations, we evaluate the performance of the proposed GSP-LMMSE estimators for the problem of state estimation in power systems, which can be interpreted as a graph signal recovery task. We show that the proposed sample-GSP estimators outperform the sample-LMMSE estimator for a limited training dataset and that the parametric GSP-LMMSE estimators are more robust to topology changes in the form of adding/removing vertices/edges.

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.

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.

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.

An intuitive and accessible text explaining the fundamentals and applications of graph signal processing. Requiring only an elementary understanding of linear algebra, it covers both basic and advanced topics, including node domain processing, graph signal frequency, sampling, and graph signal representations, as well as how to choose a graph. Understand the basic insights behind key concepts and learn how graphs can be associated to a range of specific applications across physical, biological and social networks, distributed sensor networks, image and video processing, and machine learning. With numerous exercises and Matlab examples to help put knowledge into practice, and a solutions manual available online for instructors, this unique text is essential reading for graduate and senior undergraduate students taking courses on graph signal processing, signal processing, information processing, and data analysis, as well as researchers and industry professionals.

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

Chapter 8 - Graph Signal Processing

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.

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.

We study the optimal design of graph filters (GFs) to implement arbitrary linear transformations between graph signals. GFs can be represented by matrix polynomials of the graph-shift operator (GSO). Since this operator captures the local structure of the graph, GFs naturally give rise to distributed linear network operators. In most setups, the GSO is given so that GF design consists fundamentally in choosing the (filter) coefficients of the matrix polynomial to resemble desired linear transformations. We determine spectral conditions under which a specific linear transformation can be implemented perfectly using GFs. For the cases where perfect implementation is infeasible, we address the optimization of the filter coefficients to approximate the desired transformation. Additionally, for settings where the GSO itself can be modified, we study its optimal design as well. After this, we introduce the notion of a node-variant GF, which allows the simultaneous implementation of multiple (regular) GFs in different nodes of the graph. This additional flexibility enables the design of more general operators without undermining the locality in implementation. Perfect and approximate designs are also studied for this new type of GFs. To showcase the relevance of the results in the context of distributed linear network operators, this paper closes with the application of our framework to two particular distributed problems: finite-time consensus and analog network coding.

In this paper, an adaptive Kalman filter algorithm is proposed for simultaneous graph topology learning and graph signal recovery from noisy time series. Each time series corresponds to one node of the graph and underlying graph edges express the causality among nodes. We assume that graph signals are generated via a multivariate auto-regressive processes (MAR), generated by an innovation noise and graph weight matrices. Then we relate the state transition matrix of Kalman filter to the graph weight matrices since both of them can play the role of signal propagation and transition. Our proposed Kalman filter for MAR processes, called KF-MAR, runs three main steps; prediction, update, and learn. In prediction and update steps, we fix the previously learned graph weight matrices and follow a regular Kalman algorithm for graph signal recovery. Then in the learning step, we use the last update of graph signal estimates and keep track of topology changes. Simulation results show that our proposed graph Kalman filter outperforms the available online algorithms for graph topology inference and also it can achieve the same performance of the batch method, when the number of observations increase.

Signal processing on graph is attracting more and more attentions. For a graph signal in the low-frequency subspace, the missing data associated with unsampled vertices can be reconstructed through the sampled data by exploiting the smoothness of the graph signal. In this paper, the concept of local set is introduced and two local-set-based iterative methods are proposed to reconstruct bandlimited graph signal from sampled data. In each iteration, one of the proposed methods reweights the sampled residuals for different vertices, while the other propagates the sampled residuals in their respective local sets. These algorithms are built on frame theory and the concept of local sets, based on which several frames and contraction operators are proposed. We then prove that the reconstruction methods converge to the original signal under certain conditions and demonstrate the new methods lead to a significantly faster convergence compared with the baseline method. Furthermore, the correspondence between graph signal sampling and time-domain irregular sampling is analyzed comprehensively, which may be helpful to future works on graph signals. Computer simulations are conducted. The experimental results demonstrate the effectiveness of the reconstruction methods in various sampling geometries, imprecise priori knowledge of cutoff frequency, and noisy scenarios.