The Emerging Field of Signal Processing on Graphs: Extending High-Dimensional Data Analysis to Networks and Other Irregular Domains

Abstract

In applications such as social, energy, transportation, sensor, and neuronal networks, high-dimensional data naturally reside on the vertices of weighted graphs. The emerging field of signal processing on graphs merges algebraic and spectral graph theoretic concepts with computational harmonic analysis to process such signals on graphs. In this tutorial overview, we outline the main challenges of the area, discuss different ways to define graph spectral domains, which are the analogues to the classical frequency domain, and highlight the importance of incorporating the irregular structures of graph data domains when processing signals on graphs. We then review methods to generalize fundamental operations such as filtering, translation, modulation, dilation, and downsampling to the graph setting, and survey the localized, multiscale transforms that have been proposed to efficiently extract information from high-dimensional data on graphs. We conclude with a brief discussion of open issues and possible extensions.

To appear in the IEEE Signal Processing Magazine

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• Graph Saliency Maps through Spectral Convolutional Networks: Application to Sex Classification with Brain Connectivity
• Comparing Graph Spectra of Adjacency and Laplacian Matrices
• Pesticide concentration monitoring: investigating spatio-temporal patterns in left censored data
• Supervised Linear Regression for Graph Learning from Graph Signals
• Joint Estimation of Low-Rank Components and Connectivity Graph in High-Dimensional Graph Signals: Application to Brain Imaging
• VGAI: End-to-End Learning of Vision-Based Decentralized Controllers for Robot Swarms
• Predictive Generalized Graph Fourier Transform for Attribute Compression of Dynamic Point Clouds
• Robust Deep Graph Based Learning for Binary Classification
• Assessment of Spatio-Temporal Predictors in the Presence of Missing and Heterogeneous Data
• Multi-Graph Tensor Networks
• Continuous-Depth Neural Models for Dynamic Graph Prediction
• Large Graph Signal Denoising with Application to Differential Privacy
• The Atlas for the Aspiring Network Scientist
• Simultaneous Low-rank Component and Graph Estimation for High-dimensional Graph Signals: Application to Brain Imaging
• Adaptive Gaussian Processes on Graphs via Spectral Graph Wavelets
• Scalable Semi-Supervised Learning over Networks using Nonsmooth Convex Optimization
• Physics-Constrained Predictive Molecular Latent Space Discovery with Graph Scattering Variational Autoencoder
• Graph Convolutional Neural Networks Sensitivity under Probabilistic Error Model
• A Flexible Convolutional Solver with Application to Photorealistic Style Transfer
• Duality between Temporal Networks and Signals: Extraction of the Temporal Network Structures
• Filter Design for Autoregressive Moving Average Graph Filters
• Improved Visual Localization via Graph Smoothing
• Signal Processing on Directed Graphs
• Generalizing diffuse interface methods on graphs: non-smooth potentials and hypergraphs
• Spectral Graph Convolutions for Population-based Disease Prediction
• Compact Graph Architecture for Speech Emotion Recognition
• On the Supermodularity of Active Graph-based Semi-supervised Learning with Stieltjes Matrix Regularization
• Gradient and Passive Circuit Structure in a Class of Non-linear Dynamics on a Graph
• AMOS: An Automated Model Order Selection Algorithm for Spectral Graph Clustering
• Dynamic Polygon Clouds: Representation and Compression for VR/AR
• 3D Dynamic Point Cloud Inpainting via Temporal Consistency on Graphs
• Low-dimensional controllability of brain networks
• A Time-Vertex Signal Processing Framework
• Graph Fourier Transform based Audio Zero-watermarking
• Improving J-divergence of brain connectivity states by graph Laplacian denoising
• Analysis vs Synthesis with Structure - An Investigation of Union of Subspace Models on Graphs
• Spatially Focused Attack against Spatiotemporal Graph Neural Networks
• Nonsubsampled Graph Filter Banks and Distributed Implementation
• Stacked Graph Filter
• Graph Equivalence Classes for Spectral Projector-Based Graph Fourier Transforms
• Online Inference for Mixture Model of Streaming Graph Signals with Non-White Excitation
• A Hierarchical Graph Signal Processing Approach to Inference from Spatiotemporal Signals
• Learning hidden influences in large-scale dynamical social networks: A data-driven sparsity-based approach
• Vertex-Frequency Graph Signal Processing: A review
• Steering Macro-Scale Network Community Structure by Micro-Scale Features
• From Spectrum Wavelet to Vertex Propagation: Graph Convolutional Networks Based on Taylor Approximation
• Scalable $M$-Channel Critically Sampled Filter Banks for Graph Signals
• Predicting the Accuracy of a Few-Shot Classifier
• Accelerated graph-based spectral polynomial filters
• Message Passing in Graph Convolution Networks via Adaptive Filter Banks
• Classifying the Cosmic-Ray Proton and Light Groups on the LHAASO-KM2A Experiment with the Graph Neural Network
• An Example-Driven Introduction to Data Analytics on Graphs
• Fast Incremental von Neumann Graph Entropy Computation: Theory, Algorithm, and Applications
• Sound Event Detection Using Graph Laplacian Regularization Based on Event Co-occurrence
• Hybrid graph convolutional neural networks for landmark-based anatomical segmentation
• Scalable Perception-Action-Communication Loops with Convolutional and Graph Neural Networks
• Blind Community Detection from Low-rank Excitations of a Graph Filter
• Extending the Davis-Kahan theorem for comparing eigenvectors of two symmetric matrices I: Theory
• Capturing and Explaining Trajectory Singularities using Composite Signal Neural Networks
• Robust Graph Learning Under Wasserstein Uncertainty
• Isometric Transformation Invariant Graph-based Deep Neural Network
• State Estimation in Unobservable Power Systems via Graph Signal Processing Tools
• Tensor Networks for Multi-Modal Non-Euclidean Data
• Graph signal denoising using $t$-shrinkage priors
• Polynomial Selection in Spectral Graph Neural Networks: An Error-Sum of Function Slices Approach
• A Bayesian Perspective on Uncertainty Quantification for Estimated Graph Signals
• On the Stability of Graph Convolutional Neural Networks under Edge Rewiring
• Principal Patterns on Graphs: Discovering Coherent Structures in Datasets
• GRASS: Graph Spectral Sparsification Leveraging Scalable Spectral Perturbation Analysis
• Extreme Values of the Fiedler Vector on Trees
• Smoothness Sensor: Adaptive Smoothness-Transition Graph Convolutions for Attributed Graph Clustering
• Folded Graph Signals: Sensing with Unlimited Dynamic Range
• Topological Neural Networks over the Air
• Guided Signal Reconstruction Theory
• Graph learning under spectral sparsity constraints
• A unified framework for manifold landmarking
• A new decomposition of the graph Laplacian and the binomial structure of mass-action systems
• Optimized Quantization in Distributed Graph Signal Filtering
• Graph Blind Deconvolution with Sparseness Constraint
• Learning Graph Laplacian with MCP
• Robust recovery of bandlimited graph signals via randomized dynamical sampling
• Graph Fourier Transform with Negative Edges for Depth Image Coding
• Graph Signal Processing over a Probability Space of Shift Operators
• Inferring Graph Signal Translations as Invariant Transformations for Classification Tasks
• Learning Parametrised Graph Shift Operators
• Blind Demixing of Diffused Graph Signals
• Space-Time Graph Neural Networks
• Dual Geometric Graph Network (DG2N) -- Iterative network for deformable shape alignment
• DPC: Unsupervised Deep Point Correspondence via Cross and Self Construction
• Laplacian Constrained Precision Matrix Estimation: Existence and High Dimensional Consistency
• GraSSNet: Graph Soft Sensing Neural Networks
• Gasper: GrAph Signal ProcEssing in R
• PanRep: Graph neural networks for extracting universal node embeddings in heterogeneous graphs
• Diffusion and Superposition Distances for Signals Supported on Networks
• Wide and Deep Graph Neural Networks with Distributed Online Learning
• Spline-Like Wavelet Filterbanks with Perfect Reconstruction on Arbitrary Graphs
• Perfect Reconstruction Two-Channel Filter Banks on Arbitrary Graphs
• Learning Graph Filters for Structure-Function Coupling based Hub Node Identification
• Entrywise convergence of iterative methods for eigenproblems
• Graph Metric Learning via Gershgorin Disc Alignment
• Error Adjustment Based on Spatiotemporal Correlation Fusion for Traffic Forecasting
• A Gated Graph Neural Network Approach to Fast-Convergent Dynamic Average Estimation
• From Nodes to Edges: Edge-Based Laplacians for Brain Signal Processing
• A Probabilistic Interpretation of Sampling Theory of Graph Signals
• Bayesian Design of Sampling Set for Bandlimited Graph Signals
• Pure Spectral Graph Embeddings: Reinterpreting Graph Convolution for Top-N Recommendation
• Smoothed Multi-View Subspace Clustering
• Semi-Supervised Classification on Non-Sparse Graphs Using Low-Rank Graph Convolutional Networks
• 3D Dynamic Point Cloud Denoising via Spatial-Temporal Graph Learning
• Orthogonal Transforms for Signals on Directed Graphs
• Detecting Anomalous Activity on Networks with the Graph Fourier Scan Statistic
• GFCN: A New Graph Convolutional Network Based on Parallel Flows
• Graph Denoising with Framelet Regularizer
• Graph Neural Networks for Distributed Linear-Quadratic Control
• GAIN: Graph Attention & Interaction Network for Inductive Semi-Supervised Learning over Large-scale Graphs
• On Sparse Graph Fourier Transform
• Local Frequency Interpretation and Non-Local Self-Similarity on Graph for Point Cloud Inpainting
• Performance Analysis of Plug-and-Play ADMM: A Graph Signal Processing Perspective
• Spectrum-Adapted Polynomial Approximation for Matrix Functions
• Identifying First-order Lowpass Graph Signals using Perron Frobenius Theorem
• An information-geometric approach to feature extraction and moment reconstruction in dynamical systems
• Extreme Learning Machine for Graph Signal Processing
• Domain Adaptation on Graphs by Learning Aligned Graph Bases
• Eigen-GNN: A Graph Structure Preserving Plug-in for GNNs
• A Discrete Probabilistic Approach to Dense Flow Visualization
• A Tutorial on Graph Theory for Brain Signal Analysis
• Framework for Designing Filters of Spectral Graph Convolutional Neural Networks in the Context of Regularization Theory
• Network Topology Inference Using Information Cascades with Limited Statistical Knowledge
• Neural Embedding Propagation on Heterogeneous Networks
• Spectral Graph Wavelet Transform as Feature Extractor for Machine Learning in Neuroimaging
• Spectrum degeneracy for functions on branching lines and impact on extrapolation and sampling
• A Spectral Nonlocal Block for Neural Networks
• Node Embedding via Word Embedding for Network Community Discovery
• Adaptive Graph-based Total Variation for Tomographic Reconstructions
• Unified Functorial Signal Representation I: From Grothendieck fibration to Base structured categories
• Online Distributed Learning over Graphs with Multitask Graph-Filter Models
• Distributed Network Privacy using Error Correcting Codes
• Tracking Time-Vertex Propagation using Dynamic Graph Wavelets
• Graph Signal Processing of Indefinite and Complex Graphs using Directed Variation
• Graph Signal Representation with Wasserstein Barycenters
• Feature Preserving and Uniformity-controllable Point Cloud Simplification on Graph
• Deep Learning on Attributed Graphs: A Journey from Graphs to Their Embeddings and Back
• Active Sampling for Approximately Bandlimited Graph Signals
• Mesh Learning Using Persistent Homology on the Laplacian Eigenfunctions
• Compressive PCA for Low-Rank Matrices on Graphs
• Fast Hierarchical Neural Network for Feature Learning on Point Cloud
• Semi-Supervised Graph Embedding for Multi-Label Graph Node Classification
• Unsupervised Learning of Spike Patterns for Seizure Detection and Wavefront Estimation of High Resolution Micro Electrocorticographic (μECoG) Data
• Deep Haar Scattering Networks
• Conjugate Gradient Acceleration of Non-Linear Smoothing Filters
• Design of Sampling Set for Bandlimited Graph Signal Estimation
• Blind Image Deblurring via Reweighted Graph Total Variation
• Boosting of Image Denoising Algorithms
• Graph-Based Manifold Frequency Analysis for Denoising
• On the rational approximation of Markov functions,with applications to the computation of Markovfunctions of Toeplitz matrices
• Community-preserving Graph Convolutions for Structural and Functional Joint Embedding of Brain Networks
• Time-Varying Graph Learning with Constraints on Graph Temporal Variation
• An Introduction to Robust Graph Convolutional Networks
• Network Representation Learning: From Traditional Feature Learning to Deep Learning
• Robust Semi-Supervised Graph Classifier Learning with Negative Edge Weights
• Recursive Prediction of Graph Signals with Incoming Nodes
• Sampling Policy Design for Tracking Time-Varying Graph Signals with Adaptive Budget Allocation
• Untrained Graph Neural Networks for Denoising
• PET Image Reconstruction with Multiple Kernels and Multiple Kernel Space Regularizers
• Constructing Frequency Domains on Graphs in Near-Linear Time
• Regularized Recovery by Multi-order Partial Hypergraph Total Variation
• A Novel Scheme for Support Identification and Iterative Sampling of Bandlimited Graph Signals
• Stationarity of Time-Series on Graph via Bivariate Translation Invariance
• Average Consensus by Graph Filtering: New Approach, Explicit Convergence Rate and Optimal Design
• AN-GCN: An Anonymous Graph Convolutional Network Defense Against Edge-Perturbing Attack
• Joint Network Topology Inference via Structured Fusion Regularization
• Intrinsic Geometric Information Transfer Learning on Multiple Graph-Structured Datasets
• Subgraph Signal Processing
• Contributions to Representation Learning with Graph Autoencoders and Applications to Music Recommendation
• Asymptotic Justification of Bandlimited Interpolation of Graph signals for Semi-Supervised Learning
• Graphs for deep learning representations
• Offline detection of change-points in the mean for stationary graph signals
• Signal processing with a distribution of graph operators
• EEG-based video identification using graph signal modeling and graph convolutional neural network
• Enhancing Geometric Deep Learning via Graph Filter Deconvolution
• Spatio-Temporal Graph Scattering Transform
• Exploring Deep 3D Spatial Encodings for Large-Scale 3D Scene Understanding
• Graph Based Sinogram Denoising for Tomographic Reconstructions
• Fast Decentralized Linear Functions Over Edge Fluctuating Graphs
• ChebLieNet: Invariant Spectral Graph NNs Turned Equivariant by Riemannian Geometry on Lie Groups
• Reconstruction-Cognizant Graph Sampling using Gershgorin Disc Alignment
• Vertex-disjoint Cycle Cover for graph signal processing
• Estimation of Shortest Path Covariance Matrices
• Conditional Attention Networks for Distilling Knowledge Graphs in Recommendation
• Spectral Transform Forms Scalable Transformer
• Weighted sampling and weighted interpolation on combinatorial graphs
• Graphs as Tools to Improve Deep Learning Methods
• When Product Search Meets Collaborative Filtering: A Hierarchical Heterogeneous Graph Neural Network Approach
• Pseudoinverse Graph Convolutional Networks: Fast Filters Tailored for Large Eigengaps of Dense Graphs and Hypergraphs
• Brain Maturation Study during Adolescence Using Graph Laplacian Learning Based Fourier Transform
• Learning Chebyshev Basis in Graph Convolutional Networks for Skeleton-based Action Recognition
• Optimization of a partial differential equation on a complex network
• How likely is a random graph shift-enabled?
• Multi-dimensional graph fractional Fourier transform and its application
• Polynomial control on stability, inversion and powers of matrices on simple graphs
• SStaGCN: Simplified stacking based graph convolutional networks
• Graph-LDA: Graph Structure Priors to Improve the Accuracy in Few-Shot Classification
• Label-informed Graph Structure Learning for Node Classification
• Self-Supervised Graph Learning with Proximity-based Views and Channel Contrast
• Robust, Deep, and Reinforcement Learning for Management of Communication and Power Networks
• Comparing linear structure-based and data-driven latent spatial representations for sequence prediction
• A short-graph Fourier transform via personalized PageRank vectors
• Learning to Learn Graph Topologies
• Accelerated Spectral Clustering Using Graph Filtering Of Random Signals
• Data-Driven Tree Transforms and Metrics
• Translation Operator in Graph Signal Processing: A Generalized Approach
• A general method to compute numerical dispersion errors and its application to stretched meshes
• HyperLearn: A Distributed Approach for Representation Learning in Datasets With Many Modalities
• Graph Feature Gating Networks
• Signed Graph Learning with Hidden Nodes
• Spreading of pathological proteins through brain networks: a case study for Alzheimers disease
• Mutual Teaching for Graph Convolutional Networks
• Structured Graph Learning for Clustering and Semi-supervised Classification
• Sampling Theory of Bandlimited Continuous-Time Graph Signals
• Spectral Embedding of Graph Networks
• Estimating Network Processes via Blind Identification of Multiple Graph Filters
• Incremental Eigenpair Computation for Graph Laplacian Matrices: Theory and Applications
• Learning flexible representations of stochastic processes on graphs
• Graph Wasserstein Correlation Analysis for Movie Retrieval
• Identification of deep breath while moving forward based on multiple body regions and graph signal analysis
• Dynamic Resource Optimization for Decentralized Estimation in Energy Harvesting IoT Networks
• Time-Vertex Machine Learning for Optimal Sensor Placement in Temporal Graph Signals: Applications in Structural Health Monitoring
• Faster Inference of Cell Complexes from Flows via Matrix Factorization
• Learning Sparse Graphs for Prediction and Filtering of Multivariate Data Processes
• Graph-Time Spectral Analysis for Atrial Fibrillation
• Nonlinear Dimensionality Reduction on Graphs
• Effective Resistance Preserving Directed Graph Symmetrization
• Emerging Biometrics: Deep Inference and Other Computational Intelligence
• Joint Demosaicking / Rectification of Fisheye Camera Images using Multi-color Graph Laplacian Regularization
• Graph Pooling with Node Proximity for Hierarchical Representation Learning
• Cascading: Association Augmented Sequential Recommendation
• Joint Forecasting and Interpolation of Graph Signals Using Deep Learning
• Spectral Perturbations of the Line Graph Laplacian
• Sensor selection on graphs via data-driven node sub-sampling in network time series
• Estimating Centrality Blindly from Low-pass Filtered Graph Signals
• Network Classifiers With Output Smoothing
• Discerning media bias within a network of political allies: an analytic condition for disruption by partisans
• Predicting Station-Level Bike-Sharing Demands Using Graph Convolutional Neural Network
• Eigen component analysis: A quantum theory incorporated machine learning technique to find linearly maximum separable components
• DeepWORD: A GCN-based Approach for Owner-Member Relationship Detection in Autonomous Driving