Graph Signal Processing Techniques Research Articles

Rating prediction based on the graph Fourier basis and PSD estimation from the perspective of graph signal reconstruction

Accurately and quickly predicting unknown ratings from fewer known ratings has always been a challenging problem for recommender systems. Inspired by emerging graph signal processing techniques, this paper proposes a rating prediction method via a graph signal reconstruction technique. The most novel point is that the proposed method does not need to construct (or learn) the graph structures in advance and then estimate the power spectral density (PSD), but directly estimates the graph Fourier basis and the PSD from data. Furthermore, we present an approximate solution strategy to significantly reduce the computational complexity of the reconstructed model. Finally, we conduct experiments with two public databases to test the proposed method and compare it with existing methods. The experiments give an encouraging result in both prediction accuracy and efficiency.

T-HGSP: Hypergraph Signal Processing Using t-Product Tensor Decompositions

Graph signal processing (GSP) techniques are powerful tools that model complex relationships within large datasets, being now used in a myriad of applications in different areas including data science, communication networks, epidemiology, and sociology. Simple graphs can only model pairwise relationships among data which prevents their application in modeling networks with higher-order relationships. For this reason, some efforts have been made to generalize well-known graph signal processing techniques to more complex graphs such as hypergraphs, which allow capturing higher-order relationships among data. In this paper, we provide a new hypergraph signal processing framework (t-HGSP) based on a novel tensor-tensor product algebra that has emerged as a powerful tool for preserving the intrinsic structures of tensors. The proposed framework allows the generalization of traditional GSP techniques while keeping the dimensionality characteristic of the complex systems represented by hypergraphs. To this end, the core elements of the t-HGSP framework are introduced, including the shifting operators and the hypergraph signal. The hypergraph Fourier space is also defined, followed by the concept of bandlimited signals and sampling. In our experiments, we demonstrate the benefits of our approach in applications such as clustering and denoising.

A Multilevel Spectral Framework for Scalable Vectorless Power/Thermal Integrity Verification

Vectorless integrity verification is becoming increasingly critical to the robust design of nanoscale integrated circuits. This article introduces a general vectorless integrity verification framework that allows computing the worst-case voltage drops or temperature (gradient) distributions across the entire chip under a set of local and global workload (power density) constraints. To address the computational challenges introduced by the large power grids and three-dimensional mesh-structured thermal grids, we propose a novel spectral approach for highly scalable vectorless verification of large chip designs by leveraging a hierarchy of almost linear-sized spectral sparsifiers of input grids that can well retain effective resistances between nodes. As a result, the vectorless integrity verification solution obtained on coarse-level problems can effectively help compute the solution of the original problem. Our approach is based on emerging spectral graph theory and graph signal processing techniques, which consists of a graph topology sparsification and graph coarsening phase, an edge weight scaling phase, as well as a solution refinement procedure. Extensive experimental results show that the proposed vectorless verification framework can efficiently and accurately obtain worst-case scenarios in even very large designs.

Graph Normalized-LMP Algorithm for Signal Estimation Under Impulsive Noise

In this paper, we introduce an adaptive graph normalized least mean pth power (GNLMP) algorithm for graph signal processing (GSP) that utilizes GSP techniques, including bandlimited filtering and node sampling, to estimate sampled graph signals under impulsive noise. Different from least-squares-based algorithms, such as the adaptive GSP Least Mean Squares (GLMS) algorithm and the normalized GLMS (GNLMS) algorithm, the GNLMP algorithm has the ability to reconstruct a graph signal that is corrupted by non-Gaussian noise with heavy-tailed characteristics. Compared to the recently introduced adaptive GSP least mean pth power (GLMP) algorithm, the GNLMP algorithm reduces the number of iterations to converge to a steady graph signal. The convergence condition of the GNLMP algorithm is derived, and the ability of the GNLMP algorithm to process multidimensional time-varying graph signals with multiple features is demonstrated as well. Simulations show the performance of the GNLMP algorithm in estimating steady-state and time-varying graph signals is faster than GLMP and more robust in comparison to GLMS and GNLMS.

Three-Dimensional Spatial Analyses of Cholinergic Neuronal Distributions Across The Mouse Septum, Nucleus Basalis, Globus Pallidus, Nucleus Accumbens, and Caudate-Putamen

Neuronal networks are regulated by three-dimensional spatial and structural properties. Despite robust evidence of functional implications in the modulation of cognition, little is known about the three-dimensional internal organization of cholinergic networks in the forebrain. Cholinergic networks in the forebrain primarily occur in subcortical nuclei, specifically the septum, nucleus basalis, globus pallidus, nucleus accumbens, and the caudate-putamen. Therefore, the present investigation analyzed the three-dimensional spatial organization of 14,000 cholinergic neurons that expressed choline acetyltransferase (ChAT) in these subcortical nuclei of the mouse forebrain. Point process theory and graph signal processing techniques identified three topological principles of organization. First, cholinergic interneuronal distance is not uniform across brain regions. Specifically, in the septum, globus pallidus, nucleus accumbens, and the caudate-putamen, the cholinergic neurons were clustered compared with a uniform random distribution. In contrast, in the nucleus basalis, the cholinergic neurons had a spatial distribution of greater regularity than a uniform random distribution. Second, a quarter of the caudate-putamen is composed of axonal bundles, yet the spatial distribution of cholinergic neurons remained clustered when axonal bundles were accounted for. However, comparison with an inhomogeneous Poisson distribution showed that the nucleus basalis and caudate-putamen findings could be explained by density gradients in those structures. Third, the number of cholinergic neurons varies as a function of the volume of a specific brain region but cell body volume is constant across regions. The results of the present investigation provide topographic descriptions of cholinergic somata distribution and axonal conduits, and demonstrate spatial differences in cognitive control networks. The study provides a comprehensive digital database of the total population of ChAT-positive neurons in the reported structures, with the x,y,z coordinates of each neuron at micrometer resolution. This information is important for future digital cellular atlases and computational models of the forebrain cholinergic system enabling models based on actual spatial geometry.

A Unified Approach for Simultaneous Graph Learning and Blind Separation of Graph Signal Sources

Blind separation of the sources supported by graphs, i.e., graph signals, is a nascent and challenging Blind Source Separation (BSS) problem. In these cases, along with the statistical independence of the sources, additional dependency information can be interpreted from their graph structure, which provides exploiting different Graph Signal Processing (GSP) tools to improve the separation performance. To the best of our knowledge, in these cases, only GraDe and GraphJADE methods have been proposed to exploit the graph dependencies and/or GSP techniques to improve the separation quality rather than the conventional BSS methods. Despite the significant advantages of these graph-based methods, assuming that the underlying graphs are known is a serious drawback, especially in many real-world applications. To address this issue, in this paper, we propose a Unified objective function for GraphJADE with Graph Learning, namely U-GraphJADE-GL, and use the Block Coordinate Descent (BCD) to optimize it. In our proposed method, along with the separation task, the underlying graphs are learned simultaneously in an efficient manner. We compare the performance of the U-GraphJADE-GL with that of the GraDe with Graph Learning (U-GraDe-GL) and the conventional BSS methods in the BSS task and also analyze the GL performance. Besides, as well as the theoretical and experimental convergence analysis, we derive/state the Cramér-Rao bound (CRB) on the estimation of the mixing and unmixing matrices and also on the attainable Interference-to-Source Ratio (ISR) and compare the asymptotic performance of the proposed method with that of the optimal CRB estimators. To investigate the applicability in real applications, the performance of the proposed method is also compared with that of the abovementioned algorithms for denoising the epileptic Electroencephalogram (EEG) signals and also for the audio speech source separation task. The results show the superiority of the proposed algorithm compared to the other methods in the case of BSS of graph signals with unknown graphs.

Graph Signal Processing Techniques for Analyzing Aviation Disruptions

Understanding the characteristics of air-traffic delays and disruptions is critical for developing ways to mitigate their significant economic and environmental impacts. Conventional delay-performance metrics reflect only the magnitude of incurred flight delays at airports; in this work, we show that it is also important to characterize the spatial distribution of delays across a network of airports. We analyze graph-supported signals, leveraging techniques from spectral theory and graph-signal processing to compute analytical and simulation-driven bounds for identifying outliers in spatial distribution. We then apply these methods to the case of airport-delay networks and demonstrate the applicability of our methods by analyzing U.S. airport delays from 2008 through 2017. We also perform an airline-specific analysis, deriving insights into the delay dynamics of individual airline subnetworks. Through our analysis, we highlight key differences in delay dynamics between different types of disruptions, ranging from nor’easters and hurricanes to airport outages. We also examine delay interactions between airline subnetworks and the system-wide network and compile an inventory of outlier days that could guide future aviation operations and research. In doing so, we demonstrate how our approach can provide operational insights in an air-transportation setting. Our analysis provides a complementary metric to conventional aviation-delay benchmarks and aids airlines, traffic-flow managers, and transportation-system planners in quantifying off-nominal system performance.

Graph Signal Processing for Machine Learning: A Review and New Perspectives

The effective representation, processing, analysis, and visualization of large-scale structured data, especially those related to complex domains such as networks and graphs, are one of the key questions in modern machine learning. Graph signal processing (GSP), a vibrant branch of signal processing models and algorithms that aims at handling data supported on graphs, opens new paths of research to address this challenge. In this article, we review a few important contributions made by GSP concepts and tools, such as graph filters and transforms, to the development of novel machine learning algorithms. In particular, our discussion focuses on the following three aspects: exploiting data structure and relational priors, improving data and computational efficiency, and enhancing model interpretability. Furthermore, we provide new perspectives on future development of GSP techniques that may serve as a bridge between applied mathematics and signal processing on one side, and machine learning and network science on the other. Cross-fertilization across these different disciplines may help unlock the numerous challenges of complex data analysis in the modern age.

Generalized Euclidean Measure to Estimate Network Distances

Estimating the distance covered by a propagation phenomenon on a network is an important task: it can help us estimating the infectiousness of a disease or the effectiveness of an online viral marketing campaign. However, so far the only way to make such an estimate relies on solving the optimal transportation problem, or by adapting graph signal processing techniques. Such solutions are either inefficient, because they require solving a complex optimization problem; or fragile, because they were not designed with this problem in mind. In this paper, we propose a new generalized Euclidean approach to estimate distances between weighted groups of nodes in a network. We do so by adapting the Mahalanobis distance, incorporating the graph's topology via the pseudoinverse of its Laplacian. In experiments we see that this measure returns intuitive distances which agree with the ones a human would estimate. We also show that the measure is able to recover the infection parameter in an epidemic model, or the activation threshold in a cascade model. We conclude by showing that the measure can be used in online social media settings to identify fast-spreading behaviors. Our measure is also less computationally expensive.

GRASS: Graph Spectral Sparsification Leveraging Scalable Spectral Perturbation Analysis

Spectral graph sparsification aims to find ultra-sparse subgraphs whose Laplacian matrix can well approximate the original Laplacian eigenvalues and eigenvectors. In recent years, spectral sparsification techniques have been extensively studied for accelerating various numerical and graph-related applications. Prior nearly-linear-time spectral sparsification methods first extract low-stretch spanning tree from the original graph to form the backbone of the sparsifier, and then recover small portions of spectrally-critical off-tree edges to the spanning tree to significantly improve the approximation quality. However, it is not clear how many off-tree edges should be recovered for achieving a desired spectral similarity level within the sparsifier. Motivated by recent graph signal processing techniques, this work proposes a similarity-aware spectral graph sparsification framework that leverages efficient spectral off-tree edge embedding and filtering schemes to construct spectral sparsifiers with guaranteed spectral similarity (relative condition number) level. An iterative graph densification scheme is also introduced to facilitate efficient and effective filtering of off-tree edges for highly ill-conditioned problems. The proposed method has been validated using various kinds of graphs obtained from public domain sparse matrix collections relevant to very large-scale integration computer-aided design, finite element analysis, as well as social and data networks frequently studied in many machine learning and data mining applications. For instance, a sparse SDD matrix with 40 million unknowns and 180 million nonzeros can be solved (1E-3 accuracy level) within 2 min using a single CPU core and about 6-GB memory.

Graph Signal Processing-Based Network Health Estimation for Next Generation Wireless Systems

In this letter, we propose a novel network health estimation technique for wireless cellular networks. The proposed scheme makes use of graph signal processing techniques to estimate network health over the entire coverage area with sparse availability of measured data. To achieve this objective, we solve an optimization problem on graph using proximal splitting method. The results show that the proposed technique outperforms existing methods such as kriging for network health estimation with an improved accuracy of more than 60% along with a reduced time complexity of $\mathcal {O}(n)$ compared with $\mathcal {O}(n^{4})$ for kriging. Thereafter, coverage holes present in the network are found with an extremely high detection rate and extremely low false positive rate. The results, unlike kriging, are devoid of any spatial bias present in the training data.

Nonlinear Polynomial Graph Filter for Signal Processing With Irregular Structures

Processing of signals with irregular structures is a fundamental challenge to the conventional signal processing due to the complex structures. Since most of the relationships among such data from engineering applications are inherently nonlinear, those tools that do not take the nonlinearity into account may have limitations in dealing with such relationships. In this paper, a novel nonlinear polynomial graph filter (NPGF) is proposed to facilitate the nonlinear processing of signals with irregular structures. First of all, the nonlinear relationship between the input and output graph signals is constructed, which is further developed to utilize the structure information of the irregular structures (i.e., the graph shift ${\boldsymbol S}$ ). Specifically, the nonlinear kernels and the nonlinear input terms are both designed to be polynomials of the graph shift ${\boldsymbol S}$ . Such nonlinear polynomial relationship is then defined as the NPGF. Note that shift invariance is a fundamental property in signal processing but is not straightforward in the existing nonlinear graph signal processing techniques. Due to the special expression of the NPGF, the shift-invariance is extended to our NPGF, which can conveniently facilitate the analysis and design of graph signals using NPGF and is one of the most important contributions in this paper. Numerical studies that characterize the nonlinear temperature distribution and GDP prediction based on real datasets are presented. Compared with other existing techniques, our NPGF achieves superior capability in characterizing the nonlinear relationship between the input and output graph signals, especially when only very few graph signals can be observed.

Cross-view human action recognition from depth maps using spectral graph sequences

We present a method for view-invariant action recognition from depth cameras based on graph signal processing techniques. Our framework leverages a novel graph representation of an action as a temporal sequence of graphs, onto which we apply a spectral graph wavelet transform for creating our feature descriptor. We evaluate two view-invariant graph types: skeleton-based and keypoint-based. The skeleton-based descriptor captures the spatial pose of the subject, whereas the keypoint-based is able to capture complementary information about human-object interaction and the shape of the point cloud. We investigate the effectiveness of our method by experiments on five publicly available datasets. By the graph structure, our method captures the temporal interaction between depth map interest points and achieves a 19.8% increase in performance compared to state-of-the-art results for cross-view action recognition, and competing results for frontal-view action recognition and human-object interaction. Namely, our method results in 90.8% accuracy on the cross-view N-UCLA Multiview Action3D dataset and 91.4% accuracy on the challenging MSRAction3D dataset in the cross-subject setting. For human-object interaction, our method achieves 72.3% accuracy on the Online RGBD Action dataset.We also achieve 96.0% and 98.8% accuracy on the MSRActionPairs3D and UCF-Kinect datasets, respectively.