Graph Filter Research Articles

Graphs, Convolutions, and Neural Networks: From Graph Filters to Graph Neural Networks

Network data can be conveniently modeled as a graph signal, where data values are assigned to nodes of a graph that describes the underlying network topology. Successful learning from network data is built upon methods that effectively exploit this graph structure. In this work, we leverage graph signal processing to characterize the representation space of graph neural networks (GNNs). We discuss the role of graph convolutional filters in GNNs and show that any architecture built with such filters has the fundamental properties of permutation equivariance and stability to changes in the topology. These two properties offer insight about the workings of GNNs and help explain their scalability and transferability properties which, coupled with their local and distributed nature, make GNNs powerful tools for learning in physical networks. We also introduce GNN extensions using edge-varying and autoregressive moving average graph filters and discuss their properties. Finally, we study the use of GNNs in recommender systems and learning decentralized controllers for robot swarms.

Localized Spectral Graph Filter Frames: A Unifying Framework, Survey of Design Considerations, and Numerical Comparison

Representing data residing on a graph as a linear combination of building block signals can enable efficient and insightful visual or statistical analysis of the data, and such representations prove useful as regularizers in signal processing and machine learning tasks. Designing collections of building block signals -- or more formally, dictionaries of atoms -- that specifically account for the underlying graph structure as well as any available representative training signals has been an active area of research over the last decade. In this article, we survey a particular class of dictionaries called localized spectral graph filter frames, whose atoms are created by localizing spectral patterns to different regions of the graph. After showing how this class encompasses a variety of approaches from spectral graph wavelets to graph filter banks, we focus on the two main questions of how to design the spectral filters and how to select the center vertices to which the patterns are localized. Throughout, we emphasize computationally efficient methods that ensure the resulting transforms and their inverses can be applied to data residing on large, sparse graphs. We demonstrate how this class of transform methods can be used in signal processing tasks such as denoising and non-linear approximation, and provide code for readers to experiment with these methods in new application domains.

Design of Time-Vertex Node-Variant Graph Filters

This paper addresses the node-variant graph filter on the joint time-vertex domain. An efficient algorithm is proposed to design the filter that can provide a good approximation to desirable linear operators for the time-varying signals on graphs. As a case study, the application of the designed filter on the signal denoise is addressed, where the filter is designed to approximate an inverse filtering operator. Numerical experiments conducted on the synthetic and real-world datasets demonstrate that the proposed filter is superior to the polynomial counterpart.

A Modified Spline Graph Filter Bank

Two-channel critically sampled spline graph filter bank (SGFB) is one of the existing solutions for signal processing on arbitrary graphs. This paper addresses an approach to modify the filter design for SGFB. Our method improves the shape of analysis filters in the spectral domain and also reduces the computational complexity for the synthesis section. Additionally, the proposed method exploits polynomial filters and does not need to compute eigendecomposition for the Laplacian matrix of the underlying graph. Numerical results show the efficacy of our approach by comparing its performance in graph signal decomposition and denoising with the existing solutions.

Fast Spectral Approximation of Structured Graphs with Applications to Graph Filtering

To analyze and synthesize signals on networks or graphs, Fourier theory has been extended to irregular domains, leading to a so-called graph Fourier transform. Unfortunately, different from the traditional Fourier transform, each graph exhibits a different graph Fourier transform. Therefore to analyze the graph-frequency domain properties of a graph signal, the graph Fourier modes and graph frequencies must be computed for the graph under study. Although to find these graph frequencies and modes, a computationally expensive, or even prohibitive, eigendecomposition of the graph is required, there exist families of graphs that have properties that could be exploited for an approximate fast graph spectrum computation. In this work, we aim to identify these families and to provide a divide-and-conquer approach for computing an approximate spectral decomposition of the graph. Using the same decomposition, results on reducing the complexity of graph filtering are derived. These results provide an attempt to leverage the underlying topological properties of graphs in order to devise general computational models for graph signal processing.

Malfunction Detection and Localization Algorithm for Wireless Sensor Network

Recently, the graph signal processing paves a new way to analyze the data residing in the networks. In this letter, the graph signal processing theory is applied to detect and localize the malfunctioning sensors caused by the sheer quality and complicated deploying environment of wireless sensor networks. Based on the high-pass graph filter and the history data information, a two-channel graph filtering structure is constructed to not only detect the malfunctioning phenomenon but also determine the positions of the malfunctioning sensors. The proposed scheme is operated on vertex domain, in which the graph Fourier transform is not required. Numerical examples conducted on the real-world data on sensor networks and artificial attack on IEEE bus grids demonstrate the effectiveness of the proposed algorithm.

Accurate Graph Filtering in Wireless Sensor Networks

Wireless sensor networks (WSNs) are considered as a major technology enabling the Internet-of-Things (IoT) paradigm. The recent emerging graph signal processing field can also contribute to enabling the IoT by providing key tools, such as graph filters (GFs), for processing the data associated with the sensor devices. GFs can be performed over WSNs in a distributed manner by means of a certain number of communication exchanges among the nodes. But, WSNs are often affected by interferences and noise, which leads to view these networks as directed, random and time-varying graph topologies. Most of the existing works neglect this problem by considering an unrealistic assumption that claims the same probability of link activation in both directions when sending a packet between two neighboring nodes. This work focuses on the problem of operating graph filtering in random asymmetric WSNs. We show first that graph filtering with finite impulse response GFs (node-invariant and node-variant) requires having equal connectivity probabilities for all the links in order to have an unbiased filtering, which cannot be achieved in practice in random WSNs. After this, we characterize the graph filtering error and present an efficient strategy to conduct graph filtering tasks over random WSNs with node-variant GFs by maximizing accuracy, that is, ensuring a small bias-variance tradeoff. In order to enforce the desired accuracy, we optimize the filter coefficients and design a cross-layer distributed scheduling algorithm (CDSA) at the MAC layer. Extensive numerical experiments are presented to show the efficiency of the proposed solution as well as the CDSA for the denoising application.

A Sampling-based 3D Point Cloud Compression Algorithm for Immersive Communication

3D point cloud is one of the most common and basic 3D object representation model that is widely used in virtual/augmented reality applications, e.g., immersive communication. Compression of 3D point cloud is a big challenge because of its huge data volume and irregular data structure. In this paper, we propose a sampling-based compression algorithm for 3D point clouds. First, a 3D point cloud was resampled by a graph filter to obtain a subset of representative 3D points. Then, the representative points were compressed by the G-PCC (geometry-based point cloud compression) encoder software that was released by MPEG. Finally, the decoded representative points were used to reconstruct the original 3D point clouds by a CNN-based up-sampling approach. Experimental results demonstrate that a significant (73.15%) bit rate reduction can be achieved by the proposed 3D point cloud compression algorithm with minimal quality degradation of the reconstructed 3D point clouds.

Nonlinear Polynomial Graph Filter for Anomalous IoT Sensor Detection and Localization

Detecting the existence of anomaly and localizing the faulty sensors in the Internet-of-Things (IoT) systems are extremely critical, since the incorrect data could lead to catastrophic consequences in many vertical industry applications. The difficulties of such problems come from deriving an explicit error function for each sensor in IoT, and the data continuity in the temporal domain would also be seriously challenged. To overcome these difficulties, the irregular spatial information of the IoT sensors is utilized by constructing an adjacency matrix using the distances among different sensors, and the nonlinear polynomial graph filter (NPGF) is employed to characterize the relationships among the collected sensor data. The NPGF provides a more accurate model for reconstructing the sensor data by taking the data nonlinear relationships into account. The error functions at each sensor for newly detection data are theoretically derived, and it is demonstrated that the error at the anomalous sensor performs differently from that of the normal sensors if the adjacency matrix is designed appropriately. The proposed NPGF-based algorithm is illustrated and validated with a real-world data set for temperature monitoring. The simulation results demonstrate the superior performance of our scheme in both anomaly detection and faulty sensor localization when compared with existing algorithms, such as the graph frequency algorithm and oversampling PCA (OS-PCA) method, especially for the case of small sensor data deviations.

A Restricted Black-Box Adversarial Framework Towards Attacking Graph Embedding Models

With the great success of graph embedding model on both academic and industry area, the robustness of graph embedding against adversarial attack inevitably becomes a central problem in graph learning domain. Regardless of the fruitful progress, most of the current works perform the attack in a white-box fashion: they need to access the model predictions and labels to construct their adversarial loss. However, the inaccessibility of model predictions in real systems makes the white-box attack impractical to real graph learning system. This paper promotes current frameworks in a more general and flexible sense – we demand to attack various kinds of graph embedding model with black-box driven. To this end, we begin by investigating the theoretical connections between graph signal processing and graph embedding models in a principled way and formulate the graph embedding model as a general graph signal process with corresponding graph filter. As such, a generalized adversarial attacker: GF-Attack is constructed by the graph filter and feature matrix. Instead of accessing any knowledge of the target classifiers used in graph embedding, GF-Attack performs the attack only on the graph filter in a black-box attack fashion. To validate the generalization of GF-Attack, we construct the attacker on four popular graph embedding models. Extensive experimental results validate the effectiveness of our attacker on several benchmark datasets. Particularly by using our attack, even small graph perturbations like one-edge flip is able to consistently make a strong attack in performance to different graph embedding models.

Infrared and visible image fusion based on oversampled graph filter banks

The infrared image (RI) and visible image (VI) fusion method merges complementary information from the infrared and visible imaging sensors to provide an effective way for understanding the scene. The graph filter bank-based graph wavelet transform possesses the advantages of the classic wavelet filter bank and graph representation of a signal. Therefore, we propose an RI and VI fusion method based on oversampled graph filter banks. Specifically, we consider the source images as signals on the regular graph and decompose them into the multiscale representations with M-channel oversampled graph filter banks. Then, the fusion rule for the low-frequency subband is constructed using the modified local coefficient of variation and the bilateral filter. The fusion maps of detail subbands are formed using the standard deviation-based local properties. Finally, the fusion image is obtained by applying the inverse transform on the fusion subband coefficients. The experimental results on benchmark images show the potential of the proposed method in the image fusion applications.

Research on the Knowledge Association Reasoning of Financial Reports Based on a Graph Network

With increasingly strict supervision, the complexity of enterprises’ annual reports has increased significantly, and the size of the text corpus has grown at an enormous rate. Information fusion for financial reporting has become a research hotspot. The difficulty of this problem is in filtering the massive amount of heterogeneous data and integrating related information distributed in different locations according to decision topics. This paper proposes a Graph NetWork (GNW) model that establishes the overall connection between decentralized information, as well as a graph network generation algorithm to filter large and complex data sets in financial reports and to mine key information to make it suitable for different decision situations. Finally, this paper uses the Planar Maximally Filtered Graph (PMFG) as a benchmark to show the effect of the generation algorithm.

Graph Filtering for Recommendation on Heterogeneous Information Networks

Various kinds of auxiliary data in web services have been proved to be valuable to handler data sparsity and cold-start problems of recommendation. However, it is challenging to develop effective approaches to model and utilize these various and complex information. Due to the flexibility in modelling data heterogeneity, heterogeneous information network (HIN) has been adopted to model auxiliary data for sparsity recommendation, named HIN based recommendation. But most of these HIN based methods rely on meta path-based similarity or graph embedding, which cannot fully mine global structure and semantic features of users and items. Besides, these methods, utilizing extended matrix factorization model or deep learning model, suffer expensive model-building problem and cannot treat personal latent factors carefully since their global objective functions. In this paper, we model both rate and auxiliary data through a unified graph and propose a graph filtering (GF) recommendation method on HINs. Distinct from traditional HIN based methods, GF uses a rate pair structure to represent user’s feedback information and predict the rating that says, “a predicted rating depends on its similar rating pairs.” Concretely, we design a semantic and sign value-aware similarity measure based on SimRank, named Constrained SimRank, to weight rating pair similarities on the unified graph and compute the predicting rate score for an active user via weighted average of all similar ratings. Various semantics behind edges of the unified graph have different contributions for the prediction. Thus, an adaptive framework is proposed to learn the weights of different semantic edges and products an optimized predicted rating. Finally, experimental studies on various real-world datasets demonstrate that GF is effective to handler the sparsity issue of recommendation and outperforms the state-of-the-art techniques.

Preconditioned Gradient Descent Algorithm for Inverse Filtering on Spatially Distributed Networks

Graph filters and their inverses have been widely used in denoising, smoothing, sampling, interpolating and learning. Implementation of an inverse filtering procedure on spatially distributed networks (SDNs) is a remarkable challenge, as each agent on an SDN is equipped with a data processing subsystem with limited capacity and a communication subsystem with confined range due to engineering limitations. In this letter, we introduce a preconditioned gradient descent algorithm to implement the inverse filtering procedure associated with a graph filter having small geodesic-width. The proposed algorithm converges exponentially, and it can be implemented at vertex level and applied to time-varying inverse filtering on SDNs.

Filter Design for Two-Channel Filter Banks on Directed Bipartite Graphs

Graph signal processing on directed graphs is more challenging that on undirected graph, primarily because the graph matrices, e.g., adjacency or Laplacian, associated with the latter is non-symmetric. The eigenvalues that represent the spectral frequencies are therefore complex in general, and the design of spectral filters for complex frequencies presents more challenges. In this work we consider the design of two-channel filter banks for directed bipartite graphs. By decomposing any graph into a series of bipartite graphs, the basic two-channel system can be applied in cascade for arbitrary graphs. The graph filters are constructed using ladder structures. The design of the kernels in the ladder structures is achieved via a least-squares formulation. Analytical formulas are derived for the design of the kernel coefficients.

Image Denoising with Adaptive Weighted Graph Filtering

Graph filtering, which is founded on the theory of graph signal processing, is proved as a useful tool for image denoising. Most graph filtering methods focus on learning an ideal lowpass filter to remove noise, where clean images are restored from noisy ones by retaining the image components in low graph frequency bands. However, this lowpass filter has limited ability to separate the low-frequency noise from clean images such that it makes the denoising procedure less effective. To address this issue, we propose an adaptive weighted graph filtering (AWGF) method to replace the design of traditional ideal lowpass filter. In detail, we reassess the existing low-rank denoising method with adaptive regularizer learning (ARLLR) from the view of graph filtering. A shrinkage approach subsequently is presented on the graph frequency domain, where the components of noisy image are adaptively decreased in each band by calculating their component significances. As a result, it makes the proposed graph filtering more explainable and suitable for denoising. Meanwhile, we demonstrate a graph filter under the constraint of subspace representation is employed in the ARLLR method. Therefore, ARLLR can be treated as a special form of graph filtering. It not only enriches the theory of graph filtering, but also builds a bridge from the low-rank methods to the graph filtering methods. In the experiments, we perform the AWGF method with a graph filter generated by the classical graph Laplacian matrix. The results show our method can achieve a comparable denoising performance with several state-of-the-art denoising methods.

Exact Blind Community Detection From Signals on Multiple Graphs

Networks and data supported on graphs have become ubiquitous in the sciences and engineering. This paper studies the 'blind' community detection problem, where we seek to infer the community structure of a graph model given the observation of independent graph signals on a set of nodes whose connections are unknown. We model each observation as filtered white noise, where the underlying network structure varies with every observation. These varying network structures are modeled as independent realizations of a latent planted partition model (PPM), justifying our assumption of a constant underlying community structure over all observations. Under certain conditions on the graph filter and PPM parameters, we propose algorithms for determining (i) the number of latent communities and (ii) the associated partitions of the PPM. We then prove statistical guarantees in the asymptotic and non-asymptotic sampling cases. Numerical experiments on real and synthetic data demonstrate the efficacy of our algorithms.

IIR Filtering on Graphs With Random Node-Asynchronous Updates

Graph filters play an important role in graph signal processing, in which the data is analyzed with respect to the underlying network (graph) structure. As an extension to classical signal processing, graph filters are generally constructed as a polynomial (FIR), or a rational (IIR) function of the underlying graph operator, which can be implemented via successive shifts on the graph. Although the graph shift is a localized operation, it requires all nodes to communicate synchronously, which can be a limitation for large scale networks. To overcome this limitation, this study proposes a node-asynchronous implementation of rational filters on arbitrary graphs. In the proposed algorithm nodes follow a randomized collect-compute-broadcast scheme: if a node is in the passive stage it collects the data sent by its incoming neighbors and stores only the most recent data. When a node gets into the active stage at a random time instance, it does the necessary filtering computations locally, and broadcasts a state vector to its outgoing neighbors. For the analysis of the algorithm, this study first considers a general case of randomized asynchronous state recursions and presents a sufficiency condition for its convergence. Based on this result, the proposed algorithm is proven to converge to the filter output in the mean-squared sense when the filter, the graph operator and the update rate of the nodes satisfy a certain condition. The proposed algorithm is simulated using rational and polynomial filters, and its convergence is demonstrated for various different cases, which also shows the robustness of the algorithm to random communication failures.

Estimating Network Processes via Blind Identification of Multiple Graph Filters

This paper studies the problem of jointly estimating multiple network processes driven by a common unknown input, thus effectively generalizing the classical blind multi-channel identification problem to graphs. More precisely, we model network processes as graph filters and consider the observation of multiple graph signals corresponding to outputs of different filters defined on a common graph and driven by the same input. Assuming that the underlying graph is known and the input is unknown, our goal is to recover the specifications of the network processes, namely the coefficients of the graph filters, only relying on the observation of the outputs. Being generated by the same input, these outputs are intimately related and we leverage this relationship for our estimation purposes. Two settings are considered, one where the orders of the filters are known and another one where they are not known. For the former setting, we present a least-squares approach and provide conditions for recovery. For the latter scenario, we propose a sparse recovery algorithm with theoretical performance guarantees. Numerical experiments illustrate the effectiveness of the proposed algorithms, the influence of different parameter settings on the estimation performance, and the validity of our theoretical claims.

Blind Community Detection From Low-Rank Excitations of a Graph Filter

This paper considers a new framework to detect communities in a graph from the observation of signals at its nodes. We model the observed signals as noisy outputs of an unknown network process, represented as a graph filter that is excited by a set of unknown low-rank inputs/excitations. Application scenarios of this model include diffusion dynamics, pricing experiments, and opinion dynamics. Rather than learning the precise parameters of the graph itself, we aim at retrieving the community structure directly. The paper shows that communities can be detected by applying a spectral method to the covariance matrix of graph signals. Our analysis indicates that the community detection performance depends on a `low-pass' property of the graph filter. We also show that the performance can be improved via a low-rank matrix plus sparse decomposition method when the latent parameter vectors are known. Numerical experiments demonstrate that our approach is effective.