Graph Signal Processing Meets Blind Source Separation

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

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.

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.

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.

In the field of camera-based photoplethysmography the application of blind source separation (BSS) techniques has extensively stressed to cope with frequently occurring artifacts and noise. Although said techniques can help to extract the cardiac component from a mixture of input sources, permutation indeterminacy inherit to BSS techniques often introduces inaccuracies or requires manual intervention. The current contribution focuses on methods to automatically select the cardiac component from the output of BSS techniques applied to camera-based photoplethysmograms. To that end, we propose simple Markov models to describe and subsequently identify cardiac components. It is shown that good results can be obtained by combining different simple Markov models.

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.

Signal estimation from functional magnetic resonance imaging data (fMRI) is a difficult and challenging task that involves carefully chosen models that can be validated by domain experts. Blind source separation (BSS) techniques such as principal component analysis (PCA), independent component analysis (ICA), and tensor decomposition have been proposed for estimating the task-fMRI signals. However, few studies have investigated their efficacy for classifying different tasks. This paper proposes yet another novel method to validate the signal estimation using predictive performance of the methods. Three commonly used BSS techniques are used for estimating signals (spatio-temporal maps) from task-fMRI. The dataset (consisting of 4 tasks) is taken from the Human Connectome Project (HCP). The extracted temporal signals are used to train a classifier that classifies two different tasks. For a binary classification, a Constrained-PARAFAC tensor decomposition achieves 92% accuracy for differentiating gambling vs. relational tasks. In addition, results for other binary task classifications are also presented. Overall, the proposed Constrained-PARAFAC performs better in prediction performance compared to PCA and ICA.

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.

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.

With a change of signal notion to graph signal, new means of performing blind source separation (BSS) appear. Particularly, existing independent component analysis (ICA) methods exploit the non-Gaussianity of the signals or other types of prior information. For graph signals, such prior information is present in a graph of dependencies in the signals. We propose BSS of graph signals which uses the prior information presented by the signal graph together with nonGaussianity. We derive the identifiability conditions for the proposed method and compare them to the conditions when only graph or non-Gaussianity approach is used. In simulation studies, we verify that the new method can separate a broader range of graph signals and show that it is also more efficient when both approaches are useful.

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 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.

In this paper, we propose a new algorithm that efficiently separates a directional source and diffuse background noise based on independent low-rank matrix analysis (ILRMA). ILRMA is one of the state-of-the-art techniques of blind source separation (BSS) and is based on a rank-1 spatial model. Although such a model does not hold for diffuse noise, ILRMA can accurately estimate the spatial parameters of the directional source. Motivated by this fact, we utilize these estimates to restore the lost spatial basis of diffuse noise, which can be considered as an efficient full-rank spatial covariance estimation. BSS experiments show the efficacy of the proposed method in terms of the computational cost and separation performance.

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.