Geometric deep learning: going beyond Euclidean data

Abstract

Many scientific fields study data with an underlying structure that is a non-Euclidean space. Some examples include social networks in computational social sciences, sensor networks in communications, functional networks in brain imaging, regulatory networks in genetics, and meshed surfaces in computer graphics. In many applications, such geometric data are large and complex (in the case of social networks, on the scale of billions), and are natural targets for machine learning techniques. In particular, we would like to use deep neural networks, which have recently proven to be powerful tools for a broad range of problems from computer vision, natural language processing, and audio analysis. However, these tools have been most successful on data with an underlying Euclidean or grid-like structure, and in cases where the invariances of these structures are built into networks used to model them. Geometric deep learning is an umbrella term for emerging techniques attempting to generalize (structured) deep neural models to non-Euclidean domains such as graphs and manifolds. The purpose of this paper is to overview different examples of geometric deep learning problems and present available solutions, key difficulties, applications, and future research directions in this nascent field.

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• Graph Homomorphism Convolution
• R-VGAE: Relational-variational Graph Autoencoder for Unsupervised Prerequisite Chain Learning
• GCN-SL: Graph Convolutional Networks with Structure Learning for Graphs under Heterophily
• RealPoint3D: Point Cloud Generation from a Single Image with Complex Background
• Enhancing ensemble learning and transfer learning in multimodal data analysis by adaptive dimensionality reduction
• Weisfeiler and Lehman Go Cellular: CW Networks
• An Overview on the Application of Graph Neural Networks in Wireless Networks
• Gait-based Age Group Classification with Adaptive Graph Neural Network
• Geometric Scattering on Manifolds
• Augmenting photometric redshift estimates using spectroscopic nearest neighbours
• Direct Feedback Alignment Scales to Modern Deep Learning Tasks and Architectures
• Normal Estimation for 3D Point Clouds via Local Plane Constraint and Multi-scale Selection
• Fast Spectral Ranking for Similarity Search
• Learning Product Graphs Underlying Smooth Graph Signals
• Domain-informed neural networks for interaction localization within astroparticle experiments
• Dr-COVID: Graph Neural Networks for SARS-CoV-2 Drug Repurposing
• NAS-Count: Counting-by-Density with Neural Architecture Search
• LightSAL: Lightweight Sign Agnostic Learning for Implicit Surface Representation
• Traffic4cast 2020 -- Graph Ensemble Net and the Importance of Feature And Loss Function Design for Traffic Prediction
• Deep Hypergraph U-Net for Brain Graph Embedding and Classification
• NOMU: Neural Optimization-based Model Uncertainty
• Graph Convolution with Low-rank Learnable Local Filters
• Markov-Lipschitz Deep Learning
• Symmetric Spaces for Graph Embeddings: A Finsler-Riemannian Approach
• GMLP: Building Scalable and Flexible Graph Neural Networks with Feature-Message Passing
• Recognizing Unseen Objects via Multimodal Intensive Knowledge Graph Propagation
• Graph Neural Processes: Towards Bayesian Graph Neural Networks
• Approximation of Riemannian measures by Stein's method
• Sampling theory for spatial field sensing: Application to electro- and magnetoencephalography
• Mining urban lifestyles: urban computing, human behavior and recommender systems
• Human Action Recognition with Multi-Laplacian Graph Convolutional Networks
• A fast method for particle tracking and triggering using small-radius silicon detectors
• Fully Convolutional Graph Neural Networks for Parametric Virtual Try-On
• Mix Dimension in Poincaré Geometry for 3D Skeleton-based Action Recognition
• Self-Constructing Graph Convolutional Networks for Semantic Labeling
• Generalization bounds for graph convolutional neural networks via Rademacher complexity
• Interpretable Stability Bounds for Spectral Graph Filters
• Hop-Hop Relation-aware Graph Neural Networks
• SB-MTL: Score-based Meta Transfer-Learning for Cross-Domain Few-Shot Learning
• LayoutGMN: Neural Graph Matching for Structural Layout Similarity
• Sampling and Recovery of Graph Signals based on Graph Neural Networks
• End-to-end Stroke imaging analysis, using reservoir computing-based effective connectivity, and interpretable Artificial intelligence
• Leveraging Interpolation Models and Error Bounds for Verifiable Scientific Machine Learning
• nSimplex Zen: A Novel Dimensionality Reduction for Euclidean and Hilbert Spaces
• Reconciling collider signals, dark matter, and the muon anomalous magnetic moment in the supersymmetric $U(1{)}_R\times U(1{)}_{B-L}$ model
• Projection-based Classification of Surfaces for 3D Human Mesh Sequence Retrieval
• A graph complexity measure based on the spectral analysis of the Laplace operator
• Manifold Topology Divergence: a Framework for Comparing Data Manifolds
• Graph Mixture Density Networks
• Randomised Wasserstein Barycenter Computation: Resampling with Statistical Guarantees
• A Dynamic Reduction Network for Point Clouds
• Fusion-Aware Point Convolution for Online Semantic 3D Scene Segmentation
• PushNet: Efficient and Adaptive Neural Message Passing
• COPT: Coordinated Optimal Transport for Graph Sketching
• A Computationally Efficient Neural Network Invariant to the Action of Symmetry Subgroups
• Learning Flat Latent Manifolds with VAEs
• GraphTER: Unsupervised Learning of Graph Transformation Equivariant Representations via Auto-Encoding Node-wise Transformations
• SurReal: Complex-Valued Learning as Principled Transformations on a Scaling and Rotation Manifold
• Mesh Variational Autoencoders with Edge Contraction Pooling
• Geometric Wavelet Scattering Networks on Compact Riemannian Manifolds
• Equivariant Entity-Relationship Networks
• Rational Neural Networks for Approximating Jump Discontinuities of Graph Convolution Operator
• Graph Saliency Maps through Spectral Convolutional Networks: Application to Sex Classification with Brain Connectivity
• Graph Convolutions on Spectral Embeddings: Learning of Cortical Surface Data
• Dilated Convolutional Neural Networks for Sequential Manifold-valued Data
• Decision Support for Intoxication Prediction Using Graph Convolutional Networks
• Stability of Neural Networks on Riemannian Manifolds
• Breaking the Expressive Bottlenecks of Graph Neural Networks
• A Survey on Deep Geometry Learning: From a Representation Perspective
• AMA-GCN: Adaptive Multi-layer Aggregation Graph Convolutional Network for Disease Prediction
• Dirac-Equation Signal Processing: Physics Boosts Topological Machine Learning
• Reward Propagation Using Graph Convolutional Networks
• Machine Learning Analysis of Complex Networks in Hyperspherical Space
• Simulating Execution Time of Tensor Programs using Graph Neural Networks
• An In-depth Summary of Recent Artificial Intelligence Applications in Drug Design
• Predicting Animation Skeletons for 3D Articulated Models via Volumetric Nets
• Graph Convolutional Neural Networks Sensitivity under Probabilistic Error Model
• Mapped Convolutions
• Deep learning reveals hidden interactions in complex systems
• Large Graph Signal Denoising with Application to Differential Privacy
• Signal Processing on Directed Graphs
• Performance of a Geometric Deep Learning Pipeline for HL-LHC Particle Tracking
• Hierarchical Protein Function Prediction with Tail-GNNs
• Hyperbolic Generative Adversarial Network
• HodgeNet: Learning Spectral Geometry on Triangle Meshes
• Physics-Constrained Predictive Molecular Latent Space Discovery with Graph Scattering Variational Autoencoder
• Ordinal Motifs in Lattices
• Top-philic Machine Learning
• Deep Spectral Meshes: Multi-Frequency Facial Mesh Processing with Graph Neural Networks
• Multivariate Boosted Trees and Applications to Forecasting and Control
• Neural Execution of Graph Algorithms
• Geometric Convolutional Neural Network for Analyzing Surface-Based Neuroimaging Data
• Learning Combined Set Covering and Traveling Salesman Problem
• Multi-view Self-Constructing Graph Convolutional Networks with Adaptive Class Weighting Loss for Semantic Segmentation
• KMF: Knowledge-Aware Multi-Faceted Representation Learning for Zero-Shot Node Classification
• A Biased Graph Neural Network Sampler with Near-Optimal Regret
• Topological based classification of paper domains using graph convolutional networks
• Learning Compressed Transforms with Low Displacement Rank
• DeepRetinotopy: Predicting the Functional Organization of Human Visual Cortex from Structural MRI Data using Geometric Deep Learning
• Multi-granular Software Annotation using File-level Weak Labelling
• Graph Classification by Mixture of Diverse Experts
• Foundations of automatic feature extraction at LHC--point clouds and graphs
• MANet: Multimodal Attention Network based Point- View fusion for 3D Shape Recognition
• Graph Attentional Autoencoder for Anticancer Hyperfood Prediction
• Structured agents for physical construction
• Disentangled Representation Learning for 3D Face Shape
• Embedding Graphs on Grassmann Manifold
• Hypergraph Pre-training with Graph Neural Networks
• Unsupervised Geometric Disentanglement for Surfaces via CFAN-VAE
• Real-Time Oil Leakage Detection on Aftermarket Motorcycle Damping System with Convolutional Neural Networks
• Geometrically Principled Connections in Graph Neural Networks
• ManifoldNorm: Extending normalizations on Riemannian Manifolds
• Learning Interpretable Disease Self-Representations for Drug Repositioning
• Progressive Relation Learning for Group Activity Recognition
• Learning to Optimize Non-Rigid Tracking
• Unsupervised Resource Allocation with Graph Neural Networks
• Hybrid graph convolutional neural networks for landmark-based anatomical segmentation
• Learning Hyperbolic Representations of Topological Features
• A Graph Deep Learning Framework for High-Level Synthesis Design Space Exploration
• Dynamic Graph Modules for Modeling Object-Object Interactions in Activity Recognition
• Building Function Approximators on top of Haar Scattering Networks
• Geometric feature performance under downsampling for EEG classification tasks
• Learning the Implicit Semantic Representation on Graph-Structured Data
• Quantifying the Reproducibility of Graph Neural Networks using Multigraph Brain Data
• Adaptive Neural Message Passing for Inductive Learning on Hypergraphs
• NPTC-net: Narrow-Band Parallel Transport Convolutional Neural Network on Point Clouds
• Generative-Discriminative Complementary Learning
• Permutation-Invariant Variational Autoencoder for Graph-Level Representation Learning
• KGTN-ens: Few-Shot Image Classification with Knowledge Graph Ensembles
• From Spectrum Wavelet to Vertex Propagation: Graph Convolutional Networks Based on Taylor Approximation
• CHILI: Chemically-Informed Large-scale Inorganic Nanomaterials Dataset for Advancing Graph Machine Learning
• Amortized Probabilistic Detection of Communities in Graphs
• Anomaly and Change Detection in Graph Streams through Constant-Curvature Manifold Embeddings
• Application of Graph Networks to a wide-field Water-Cherenkov-based Gamma-Ray Observatory
• Group-Convolutional Extended Dynamic Mode Decomposition
• Scalable and Interpretable Verification of Image-based Neural Network Controllers for Autonomous Vehicles
• Data-Driven Learning of Geometric Scattering Networks
• VC-Net: Deep Volume-Composition Networks for Segmentation and Visualization of Highly Sparse and Noisy Image Data
• AdvectiveNet: An Eulerian-Lagrangian Fluidic reservoir for Point Cloud Processing
• Practical Implementation of an End-to-End Methodology for SPC of 3-D Part Geometry: A Case Study
• On the Stability of Graph Convolutional Neural Networks under Edge Rewiring
• Latent Space Representation for Shape Analysis and Learning
• Deep Generative Modeling in Network Science with Applications to Public Policy Research
• Biologically Inspired Hexagonal Deep Learning for Hexagonal Image Generation
• Graph-Based Physics-Guided Urban PM2.5 Air Quality Imputation with Constrained Monitoring Data
• Modeling Edge Features with Deep Bayesian Graph Networks
• Actional-Structural Graph Convolutional Networks for Skeleton-based Action Recognition
• Isometric Transformation Invariant Graph-based Deep Neural Network
• Fast and Accurate: Structure Coherence Component for Face Alignment
• Multi-intersection Traffic Optimisation: A Benchmark Dataset and a Strong Baseline
• Graph Partitioning and Sparse Matrix Ordering using Reinforcement Learning and Graph Neural Networks
• Wide and Deep Graph Neural Networks with Distributed Online Learning
• Neural Trees for Learning on Graphs
• Deep Constraint-based Propagation in Graph Neural Networks
• Space-Time Graph Neural Networks
• Laplacian Constrained Precision Matrix Estimation: Existence and High Dimensional Consistency
• Deep Dynamic Effective Connectivity Estimation from Multivariate Time Series
• ${\text{H}}^2\text{TNE}$: Temporal Heterogeneous Information Network Embedding in Hyperbolic Spaces
• HypLL: The Hyperbolic Learning Library
• Graph-Based Method for Anomaly Prediction in Brain Network
• LBS Autoencoder: Self-supervised Fitting of Articulated Meshes to Point Clouds
• Learning nuclear cross sections across the chart of nuclides with graph neural networks
• Beyond holography: the entropic quantum gravity foundations of image processing
• Conditional Graph Neural Network for Predicting Soft Tissue Deformation and Forces
• SPARK: Adaptive Low-Rank Knowledge Graph Modeling in Hybrid Geometric Spaces for Recommendation
• Descriptive power and predictive limits of a discrete Hasimoto--DNLS model of protein backbone structure
• PoissonNet: A Local-Global Approach for Learning on Surfaces
• CageNet: A Meta-Framework for Learning on Wild Meshes
• Graph convolutional networks for learning with few clean and many noisy labels
• MeshCNN Fundamentals: Geometric Learning through a Reconstructable Representation
• Estimating Fund-Raising Performance for Start-up Projects from a Market Graph Perspective
• On the Universality of Graph Neural Networks on Large Random Graphs
• Experimental performance of graph neural networks on random instances of max-cut
• Neural Consciousness Flow
• OD-GCN: Object Detection Boosted by Knowledge GCN
• Geometric Machine Learning for Channel Covariance Estimation in Vehicular Networks
• Graph Attention Network For Microwave Imaging of Brain Anomaly
• Graph-Convolutional Deep Learning to Identify Optimized Molecular Configurations
• Augmenting the User-Item Graph with Textual Similarity Models
• Graph Denoising with Framelet Regularizer
• Revisiting convolutional neural network on graphs with polynomial approximations of Laplace-Beltrami spectral filtering
• OperatorNet: Recovering 3D Shapes From Difference Operators
• Graph Neural Networks for Distributed Linear-Quadratic Control
• Fast Hierarchical Neural Network for Feature Learning on Point Cloud
• Learning geometry-image representation for 3D point cloud generation
• Spectral Algorithms for Temporal Graph Cuts
• Flow-based Generative Models for Learning Manifold to Manifold Mappings
• Particle Track Reconstruction using Geometric Deep Learning
• Catastrophic Forgetting in Deep Graph Networks: an Introductory Benchmark for Graph Classification
• Controllability, Multiplexing, and Transfer Learning in Networks using Evolutionary Learning
• Weighted Spectral Embedding of Graphs
• Novel Perception Algorithmic Framework For Object Identification and Tracking In Autonomous Navigation
• Multi-Kernel Diffusion CNNs for Graph-Based Learning on Point Clouds
• Graph Neural Networks for Node-Level Predictions
• A Tutorial on Graph Theory for Brain Signal Analysis
• Relational dynamic memory networks
• A curvature and density-based generative representation of shapes
• Go Wider: An Efficient Neural Network for Point Cloud Analysis via Group Convolutions
• Graph Analysis and Graph Pooling in the Spatial Domain
• Learnable Pooling in Graph Convolution Networks for Brain Surface Analysis
• Spectral Graph Transformer Networks for Brain Surface Parcellation
• An Ontology-Aware Framework for Audio Event Classification
• Critical Percolation as a Framework to Analyze the Training of Deep Networks
• Multifold Acceleration of Diffusion MRI via Slice-Interleaved Diffusion Encoding (SIDE)
• Sim-to-Real Optimization of Complex Real World Mobile Network with Imperfect Information via Deep Reinforcement Learning from Self-play
• Deep Learning on Attributed Graphs: A Journey from Graphs to Their Embeddings and Back
• PanRep: Graph neural networks for extracting universal node embeddings in heterogeneous graphs
• CorrNet3D: Unsupervised End-to-end Learning of Dense Correspondence for 3D Point Clouds
• Matching Distributions via Optimal Transport for Semi-Supervised Learning
• Blind Demixing of Diffused Graph Signals
• Picasso: A CUDA-based Library for Deep Learning over 3D Meshes
• Universal-RCNN: Universal Object Detector via Transferable Graph R-CNN
• Multi-modal Entity Alignment in Hyperbolic Space
• Scaling up graph homomorphism for classification via sampling
• Interferometric Graph Transform for Community Labeling
• Gaussian Processes on Hypergraphs
• Pooling in Graph Convolutional Neural Networks
• Translation Operator in Graph Signal Processing: A Generalized Approach
• Geometric Brain Surface Network For Brain Cortical Parcellation
• Interpolation of surfaces with asymptotic curves in Euclidean 3-space
• Graph Neural Network for Hamiltonian-Based Material Property Prediction
• SimPool: Towards Topology Based Graph Pooling with Structural Similarity Features
• Learning pose variations within shape population by constrained mixtures of factor analyzers
• HyperLearn: A Distributed Approach for Representation Learning in Datasets With Many Modalities
• Enhancing Geometric Deep Learning via Graph Filter Deconvolution
• Linear Classifiers in Product Space Forms
• Diffusion Mean Estimation on the Diagonal of Product Manifolds
• Stationarity of Time-Series on Graph via Bivariate Translation Invariance
• Co-embedding of Nodes and Edges with Graph Neural Networks
• Structural Landmarking and Interaction Modelling: on Resolution Dilemmas in Graph Classification
• On Inductive Biases for Machine Learning in Data Constrained Settings
• Online Graph Dictionary Learning
• Functional Maps Representation on Product Manifolds
• Spaceland Embedding of Sparse Stochastic Graphs
• Classification of vertices on social networks by multiple approaches
• Learning Nonlinear Responses in PET Bottle Buckling with a Hybrid DeepONet-Transolver Framework
• FC2T2: The Fast Continuous Convolutional Taylor Transform with Applications in Vision and Graphics
• Attributed-graphs kernel implementation using local detuning of neutral-atoms Rydberg Hamiltonian
• Adaptive directional Haar tight framelets on bounded domains for digraph signal representations
• NuGraph2 with Context-Aware Inputs: Physics-Inspired Improvements in Semantic Segmentation
• Computational Analysis of Deformable Manifolds: from Geometric Modelling to Deep Learning
• LFGCN: Levitating over Graphs with Levy Flights
• Pseudoinverse Graph Convolutional Networks: Fast Filters Tailored for Large Eigengaps of Dense Graphs and Hypergraphs
• Heterogeneous Graph Neural Networks for Short-term State Forecasting in Power Systems across Domains and Time Scales: A Hydroelectric Power Plant Case Study
• Deep Modeling of Growth Trajectories for Longitudinal Prediction of Missing Infant Cortical Surfaces
• Adapted and Oversegmenting Graphs: Application to Geometric Deep Learning
• Learning flexible representations of stochastic processes on graphs
• Discriminative and Generative Models for Anatomical Shape Analysison Point Clouds with Deep Neural Networks
• SStaGCN: Simplified stacking based graph convolutional networks
• Graph-Preserving Grid Layout: A Simple Graph Drawing Method for Graph Classification using CNNs
• Parameter Estimation on Homogeneous Spaces
• Spherical U-Net on Cortical Surfaces: Methods and Applications
• Recurrent Graph Tensor Networks: A Low-Complexity Framework for Modelling High-Dimensional Multi-Way Sequence
• Learning Mechanically Driven Emergent Behavior with Message Passing Neural Networks
• Decoupling feature propagation from the design of graph auto-encoders
• On the Generalization of Agricultural Drought Classification from Climate Data
• Controlling Meshes via Curvature: Spin Transformations for Pose-Invariant Shape Processing
• Space-Time-Separable Graph Convolutional Network for Pose Forecasting
• Using NASA Satellite Data Sources and Geometric Deep Learning to Uncover Hidden Patterns in COVID-19 Clinical Severity
• FDGATII : Fast Dynamic Graph Attention with Initial Residual and Identity Mapping
• Towards a Taxonomy of Graph Learning Datasets
• Adaptive Hierarchical Similarity Metric Learning with Noisy Labels
• Rotation Equivariant 3D Hand Mesh Generation from a Single RGB Image
• Learning-based Real-time Detection of Intrinsic Reflectional Symmetry
• Deep Iterative Surface Normal Estimation
• StairwayGraphNet for Inter- and Intra-modality Multi-resolution Brain Graph Alignment and Synthesis
• All-Weather Object Recognition Using Radar and Infrared Sensing
• One Representative-Shot Learning Using a Population-Driven Template with Application to Brain Connectivity Classification and Evolution Prediction
• Untrained Graph Neural Networks for Denoising
• ADAVI: Automatic Dual Amortized Variational Inference Applied To Pyramidal Bayesian Models
• Point Cloud Processing via Recurrent Set Encoding
• RaWaNet: Enriching Graph Neural Network Input via Random Walks on Graphs
• Exploring Deep 3D Spatial Encodings for Large-Scale 3D Scene Understanding
• ChebLieNet: Invariant Spectral Graph NNs Turned Equivariant by Riemannian Geometry on Lie Groups
• Estimating Early Fundraising Performance of Innovations via Graph-based Market Environment Model
• Spectral Transform Forms Scalable Transformer
• Learning Equivariant Representations
• Deep Bayesian Optimization on Attributed Graphs
• Comparison of Syntactic and Semantic Representations of Programs in Neural Embeddings
• Training Graph Neural Networks by Graphon Estimation
• Automatic design of novel potential 3CL${}^{\text{pro}}$ and PL${}^{\text{pro}}$ inhibitors
• Length Learning for Planar Euclidean Curves
• Edge-featured Graph Neural Architecture Search
• Using Topological Framework for the Design of Activation Function and Model Pruning in Deep Neural Networks
• Surrogate Modelling for Injection Molding Processes using Machine Learning
• Fast Haar Transforms for Graph Neural Networks
• Non-isomorphic Inter-modality Graph Alignment and Synthesis for Holistic Brain Mapping
• Hermitian Symmetric Spaces for Graph Embeddings