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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• Convolutional Geometric Matrix Completion
• Embed Me If You Can: A Geometric Perceptron
• SITReg: Multi-resolution architecture for symmetric, inverse consistent, and topology preserving image registration
• Harnessing Scale and Physics: A Multi-Graph Neural Operator Framework for PDEs on Arbitrary Geometries
• ANAE: Learning Node Context Representation for Attributed Network Embedding
• Synthesizing facial photometries and corresponding geometries using generative adversarial networks
• Continuous Geodesic Convolutions for Learning on 3D Shapes
• HCGR: Hyperbolic Contrastive Graph Representation Learning for Session-based Recommendation
• Geometric Scattering for Graph Data Analysis
• Adversarial Diffusion Attacks on Graph-based Traffic Prediction Models
• Dense 3D Face Decoding over 2500FPS: Joint Texture & Shape Convolutional Mesh Decoders
• Fast Quantum Property Prediction via Deeper 2D and 3D Graph Networks
• Symmetric Spaces for Graph Embeddings: A Finsler-Riemannian Approach
• Graph Convolution with Low-rank Learnable Local Filters
• Augmenting photometric redshift estimates using spectroscopic nearest neighbours
• Recognizing Unseen Objects via Multimodal Intensive Knowledge Graph Propagation
• R-VGAE: Relational-variational Graph Autoencoder for Unsupervised Prerequisite Chain Learning
• Markov-Lipschitz Deep Learning
• Graph Homomorphism Convolution
• Enhancing ensemble learning and transfer learning in multimodal data analysis by adaptive dimensionality reduction
• Weisfeiler and Lehman Go Cellular: CW Networks
• GMLP: Building Scalable and Flexible Graph Neural Networks with Feature-Message Passing
• An Overview on the Application of Graph Neural Networks in Wireless Networks
• Deep Hypergraph U-Net for Brain Graph Embedding and Classification
• Geometric Scattering on Manifolds
• Towards Multimodal Graph Large Language Model
• Normal Estimation for 3D Point Clouds via Local Plane Constraint and Multi-scale Selection
• LightSAL: Lightweight Sign Agnostic Learning for Implicit Surface Representation
• Fast Spectral Ranking for Similarity Search
• Dr-COVID: Graph Neural Networks for SARS-CoV-2 Drug Repurposing
• NOMU: Neural Optimization-based Model Uncertainty
• Traffic4cast 2020 -- Graph Ensemble Net and the Importance of Feature And Loss Function Design for Traffic Prediction
• RealPoint3D: Point Cloud Generation from a Single Image with Complex Background
• Direct Feedback Alignment Scales to Modern Deep Learning Tasks and Architectures
• Learning Product Graphs Underlying Smooth Graph Signals
• Domain-informed neural networks for interaction localization within astroparticle experiments
• NAS-Count: Counting-by-Density with Neural Architecture Search
• Gait-based Age Group Classification with Adaptive Graph Neural Network
• GCN-SL: Graph Convolutional Networks with Structure Learning for Graphs under Heterophily
• Generalization bounds for graph convolutional neural networks via Rademacher complexity
• Mesh Variational Autoencoders with Edge Contraction Pooling
• Geometric Wavelet Scattering Networks on Compact Riemannian Manifolds
• Equivariant Entity-Relationship Networks
• 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
• Interpretable Stability Bounds for Spectral Graph Filters
• Graph Convolutions on Spectral Embeddings: Learning of Cortical Surface Data
• 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
• Graph Saliency Maps through Spectral Convolutional Networks: Application to Sex Classification with Brain Connectivity
• Rational Neural Networks for Approximating Jump Discontinuities of Graph Convolution Operator
• 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
• Geometric Convolutional Neural Network for Analyzing Surface-Based Neuroimaging Data
• HodgeNet: Learning Spectral Geometry on Triangle Meshes
• Learning Compressed Transforms with Low Displacement Rank
• Graph Convolutional Neural Networks Sensitivity under Probabilistic Error Model
• Large Graph Signal Denoising with Application to Differential Privacy
• Ordinal Motifs in Lattices
• KMF: Knowledge-Aware Multi-Faceted Representation Learning for Zero-Shot Node Classification
• Decision Support for Intoxication Prediction Using Graph Convolutional Networks
• Multi-granular Software Annotation using File-level Weak Labelling
• Foundations of automatic feature extraction at LHC--point clouds and graphs
• Dirac-Equation Signal Processing: Physics Boosts Topological Machine Learning
• Performance of a Geometric Deep Learning Pipeline for HL-LHC Particle Tracking
• Topological based classification of paper domains using graph convolutional networks
• Mapped Convolutions
• Top-philic Machine Learning
• Deep Spectral Meshes: Multi-Frequency Facial Mesh Processing with Graph Neural Networks
• AMA-GCN: Adaptive Multi-layer Aggregation Graph Convolutional Network for Disease Prediction
• Multivariate Boosted Trees and Applications to Forecasting and Control
• Graph Attentional Autoencoder for Anticancer Hyperfood Prediction
• Simulating Execution Time of Tensor Programs using Graph Neural Networks
• An In-depth Summary of Recent Artificial Intelligence Applications in Drug Design
• Machine Learning Analysis of Complex Networks in Hyperspherical Space
• A Survey on Deep Geometry Learning: From a Representation Perspective
• Reward Propagation Using Graph Convolutional Networks
• Breaking the Expressive Bottlenecks of Graph Neural Networks
• Hyperbolic Generative Adversarial Network
• Deep learning reveals hidden interactions in complex systems
• Stability of Neural Networks on Riemannian Manifolds
• Graph Classification by Mixture of Diverse Experts
• Multi-view Self-Constructing Graph Convolutional Networks with Adaptive Class Weighting Loss for Semantic Segmentation
• DeepRetinotopy: Predicting the Functional Organization of Human Visual Cortex from Structural MRI Data using Geometric Deep Learning
• Learning Combined Set Covering and Traveling Salesman Problem
• Hierarchical Protein Function Prediction with Tail-GNNs
• Signal Processing on Directed Graphs
• Predicting Animation Skeletons for 3D Articulated Models via Volumetric Nets
• Neural Execution of Graph Algorithms
• Dilated Convolutional Neural Networks for Sequential Manifold-valued Data
• MANet: Multimodal Attention Network based Point- View fusion for 3D Shape Recognition
• Physics-Constrained Predictive Molecular Latent Space Discovery with Graph Scattering Variational Autoencoder
• A Biased Graph Neural Network Sampler with Near-Optimal Regret
• Learning Interpretable Disease Self-Representations for Drug Repositioning
• Actional-Structural Graph Convolutional Networks for Skeleton-based Action Recognition
• 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
• Anomaly and Change Detection in Graph Streams through Constant-Curvature Manifold Embeddings
• Permutation-Invariant Variational Autoencoder for Graph-Level Representation Learning
• Dynamic Graph Modules for Modeling Object-Object Interactions in Activity Recognition
• Disentangled Representation Learning for 3D Face Shape
• Fast and Accurate: Structure Coherence Component for Face Alignment
• A Graph Deep Learning Framework for High-Level Synthesis Design Space Exploration
• Deep Generative Modeling in Network Science with Applications to Public Policy Research
• Building Function Approximators on top of Haar Scattering Networks
• KGTN-ens: Few-Shot Image Classification with Knowledge Graph Ensembles
• Structured agents for physical construction
• Biologically Inspired Hexagonal Deep Learning for Hexagonal Image Generation
• On the Stability of Graph Convolutional Neural Networks under Edge Rewiring
• Progressive Relation Learning for Group Activity Recognition
• CHILI: Chemically-Informed Large-scale Inorganic Nanomaterials Dataset for Advancing Graph Machine Learning
• AdvectiveNet: An Eulerian-Lagrangian Fluidic reservoir for Point Cloud Processing
• Application of Graph Networks to a wide-field Water-Cherenkov-based Gamma-Ray Observatory
• Group-Convolutional Extended Dynamic Mode Decomposition
• Learning the Implicit Semantic Representation on Graph-Structured Data
• Scalable and Interpretable Verification of Image-based Neural Network Controllers for Autonomous Vehicles
• Unsupervised Geometric Disentanglement for Surfaces via CFAN-VAE
• Data-Driven Learning of Geometric Scattering Networks
• Practical Implementation of an End-to-End Methodology for SPC of 3-D Part Geometry: A Case Study
• Isometric Transformation Invariant Graph-based Deep Neural Network
• Geometric feature performance under downsampling for EEG classification tasks
• Learning to Optimize Non-Rigid Tracking
• Graph-Based Physics-Guided Urban PM2.5 Air Quality Imputation with Constrained Monitoring Data
• VC-Net: Deep Volume-Composition Networks for Segmentation and Visualization of Highly Sparse and Noisy Image Data
• From Spectrum Wavelet to Vertex Propagation: Graph Convolutional Networks Based on Taylor Approximation
• Learning Hyperbolic Representations of Topological Features
• Modeling Edge Features with Deep Bayesian Graph Networks
• Latent Space Representation for Shape Analysis and Learning
• Real-Time Oil Leakage Detection on Aftermarket Motorcycle Damping System with Convolutional Neural Networks
• Embedding Graphs on Grassmann Manifold
• Hypergraph Pre-training with Graph Neural Networks
• Amortized Probabilistic Detection of Communities in Graphs
• ManifoldNorm: Extending normalizations on Riemannian Manifolds
• Geometrically Principled Connections in Graph Neural Networks
• Unsupervised Resource Allocation with Graph Neural Networks
• Hybrid graph convolutional neural networks for landmark-based anatomical segmentation
• Generative-Discriminative Complementary Learning
• Deep Dynamic Effective Connectivity Estimation from Multivariate Time Series
• Fast Hierarchical Neural Network for Feature Learning on Point Cloud
• Neural Consciousness Flow
• LBS Autoencoder: Self-supervised Fitting of Articulated Meshes to Point Clouds
• Graph-Based Method for Anomaly Prediction in Brain Network
• Deep Learning on Attributed Graphs: A Journey from Graphs to Their Embeddings and Back
• Multifold Acceleration of Diffusion MRI via Slice-Interleaved Diffusion Encoding (SIDE)
• An Ontology-Aware Framework for Audio Event Classification
• Spectral Graph Transformer Networks for Brain Surface Parcellation
• Learnable Pooling in Graph Convolution Networks for Brain Surface Analysis
• Graph Analysis and Graph Pooling in the Spatial Domain
• Go Wider: An Efficient Neural Network for Point Cloud Analysis via Group Convolutions
• A curvature and density-based generative representation of shapes
• A Tutorial on Graph Theory for Brain Signal Analysis
• Graph Neural Networks for Node-Level Predictions
• Novel Perception Algorithmic Framework For Object Identification and Tracking In Autonomous Navigation
• Catastrophic Forgetting in Deep Graph Networks: an Introductory Benchmark for Graph Classification
• Particle Track Reconstruction using Geometric Deep Learning
• Flow-based Generative Models for Learning Manifold to Manifold Mappings
• Learning geometry-image representation for 3D point cloud generation
• Graph Neural Networks for Distributed Linear-Quadratic Control
• Revisiting convolutional neural network on graphs with polynomial approximations of Laplace-Beltrami spectral filtering
• Graph Denoising with Framelet Regularizer
• Augmenting the User-Item Graph with Textual Similarity Models
• Graph-Convolutional Deep Learning to Identify Optimized Molecular Configurations
• Graph Attention Network For Microwave Imaging of Brain Anomaly
• Geometric Machine Learning for Channel Covariance Estimation in Vehicular Networks
• On the Universality of Graph Neural Networks on Large Random Graphs
• Estimating Fund-Raising Performance for Start-up Projects from a Market Graph Perspective
• MeshCNN Fundamentals: Geometric Learning through a Reconstructable Representation
• CageNet: A Meta-Framework for Learning on Wild Meshes
• PoissonNet: A Local-Global Approach for Learning on Surfaces
• Descriptive power and predictive limits of a discrete Hasimoto--DNLS model of protein backbone structure
• SPARK: Adaptive Low-Rank Knowledge Graph Modeling in Hybrid Geometric Spaces for Recommendation
• Conditional Graph Neural Network for Predicting Soft Tissue Deformation and Forces
• Beyond holography: the entropic quantum gravity foundations of image processing
• Learning nuclear cross sections across the chart of nuclides with graph neural networks
• HypLL: The Hyperbolic Learning Library
• ${\text{H}}^2\text{TNE}$: Temporal Heterogeneous Information Network Embedding in Hyperbolic Spaces
• Spectral Algorithms for Temporal Graph Cuts
• Laplacian Constrained Precision Matrix Estimation: Existence and High Dimensional Consistency
• Space-Time Graph Neural Networks
• Neural Trees for Learning on Graphs
• Graph Partitioning and Sparse Matrix Ordering using Reinforcement Learning and Graph Neural Networks
• Multi-intersection Traffic Optimisation: A Benchmark Dataset and a Strong Baseline
• Blind Demixing of Diffused Graph Signals
• Matching Distributions via Optimal Transport for Semi-Supervised Learning
• CorrNet3D: Unsupervised End-to-end Learning of Dense Correspondence for 3D Point Clouds
• PanRep: Graph neural networks for extracting universal node embeddings in heterogeneous graphs
• Wide and Deep Graph Neural Networks with Distributed Online Learning
• Deep Constraint-based Propagation in Graph Neural Networks
• Graph convolutional networks for learning with few clean and many noisy labels
• Experimental performance of graph neural networks on random instances of max-cut
• OD-GCN: Object Detection Boosted by Knowledge GCN
• OperatorNet: Recovering 3D Shapes From Difference Operators
• Controllability, Multiplexing, and Transfer Learning in Networks using Evolutionary Learning
• Weighted Spectral Embedding of Graphs
• Multi-Kernel Diffusion CNNs for Graph-Based Learning on Point Clouds
• Relational dynamic memory networks
• Critical Percolation as a Framework to Analyze the Training of Deep Networks
• Sim-to-Real Optimization of Complex Real World Mobile Network with Imperfect Information via Deep Reinforcement Learning from Self-play
• Comparison of Syntactic and Semantic Representations of Programs in Neural Embeddings
• Pseudoinverse Graph Convolutional Networks: Fast Filters Tailored for Large Eigengaps of Dense Graphs and Hypergraphs
• Recurrent Graph Tensor Networks: A Low-Complexity Framework for Modelling High-Dimensional Multi-Way Sequence
• Co-embedding of Nodes and Edges with Graph Neural Networks
• Online Graph Dictionary Learning
• Linear Classifiers in Product Space Forms
• Interpolation of surfaces with asymptotic curves in Euclidean 3-space
• Hermitian Symmetric Spaces for Graph Embeddings
• Universal-RCNN: Universal Object Detector via Transferable Graph R-CNN
• ADAVI: Automatic Dual Amortized Variational Inference Applied To Pyramidal Bayesian Models
• Untrained Graph Neural Networks for Denoising
• SStaGCN: Simplified stacking based graph convolutional networks
• FC2T2: The Fast Continuous Convolutional Taylor Transform with Applications in Vision and Graphics
• Enhancing Geometric Deep Learning via Graph Filter Deconvolution
• Spectral Transform Forms Scalable Transformer
• ChebLieNet: Invariant Spectral Graph NNs Turned Equivariant by Riemannian Geometry on Lie Groups
• Controlling Meshes via Curvature: Spin Transformations for Pose-Invariant Shape Processing
• Rotation Equivariant 3D Hand Mesh Generation from a Single RGB Image
• On the Generalization of Agricultural Drought Classification from Climate Data
• Learning Mechanically Driven Emergent Behavior with Message Passing Neural Networks
• Parameter Estimation on Homogeneous Spaces
• Functional Maps Representation on Product Manifolds
• Heterogeneous Graph Neural Networks for Short-term State Forecasting in Power Systems across Domains and Time Scales: A Hydroelectric Power Plant Case Study
• NuGraph2 with Context-Aware Inputs: Physics-Inspired Improvements in Semantic Segmentation
• Attributed-graphs kernel implementation using local detuning of neutral-atoms Rydberg Hamiltonian
• Spherical U-Net on Cortical Surfaces: Methods and Applications
• Learning Nonlinear Responses in PET Bottle Buckling with a Hybrid DeepONet-Transolver Framework
• Classification of vertices on social networks by multiple approaches
• On Inductive Biases for Machine Learning in Data Constrained Settings
• Diffusion Mean Estimation on the Diagonal of Product Manifolds
• Gaussian Processes on Hypergraphs
• Interferometric Graph Transform for Community Labeling
• Multi-modal Entity Alignment in Hyperbolic Space
• Non-isomorphic Inter-modality Graph Alignment and Synthesis for Holistic Brain Mapping
• Adapted and Oversegmenting Graphs: Application to Geometric Deep Learning
• Surrogate Modelling for Injection Molding Processes using Machine Learning
• Using Topological Framework for the Design of Activation Function and Model Pruning in Deep Neural Networks
• Edge-featured Graph Neural Architecture Search
• Training Graph Neural Networks by Graphon Estimation
• RaWaNet: Enriching Graph Neural Network Input via Random Walks on Graphs
• One Representative-Shot Learning Using a Population-Driven Template with Application to Brain Connectivity Classification and Evolution Prediction
• StairwayGraphNet for Inter- and Intra-modality Multi-resolution Brain Graph Alignment and Synthesis
• 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
• All-Weather Object Recognition Using Radar and Infrared Sensing
• Exploring Deep 3D Spatial Encodings for Large-Scale 3D Scene Understanding
• Learning Equivariant Representations
• Automatic design of novel potential 3CL${}^{\text{pro}}$ and PL${}^{\text{pro}}$ inhibitors
• Length Learning for Planar Euclidean Curves
• Picasso: A CUDA-based Library for Deep Learning over 3D Meshes
• Scaling up graph homomorphism for classification via sampling
• Pooling in Graph Convolutional Neural Networks
• Graph Neural Network for Hamiltonian-Based Material Property Prediction
• SimPool: Towards Topology Based Graph Pooling with Structural Similarity Features
• Deep Bayesian Optimization on Attributed Graphs
• Learning pose variations within shape population by constrained mixtures of factor analyzers
• Structural Landmarking and Interaction Modelling: on Resolution Dilemmas in Graph Classification
• Spaceland Embedding of Sparse Stochastic Graphs
• Adaptive directional Haar tight framelets on bounded domains for digraph signal representations
• Computational Analysis of Deformable Manifolds: from Geometric Modelling to Deep Learning
• LFGCN: Levitating over Graphs with Levy Flights
• Deep Modeling of Growth Trajectories for Longitudinal Prediction of Missing Infant Cortical Surfaces
• Discriminative and Generative Models for Anatomical Shape Analysison Point Clouds with Deep Neural Networks
• Graph-Preserving Grid Layout: A Simple Graph Drawing Method for Graph Classification using CNNs
• Decoupling feature propagation from the design of graph auto-encoders
• Learning flexible representations of stochastic processes on graphs
• Learning-based Real-time Detection of Intrinsic Reflectional Symmetry
• Point Cloud Processing via Recurrent Set Encoding
• Estimating Early Fundraising Performance of Innovations via Graph-based Market Environment Model
• Deep Iterative Surface Normal Estimation
• Fast Haar Transforms for Graph Neural Networks
• Translation Operator in Graph Signal Processing: A Generalized Approach
• Geometric Brain Surface Network For Brain Cortical Parcellation
• HyperLearn: A Distributed Approach for Representation Learning in Datasets With Many Modalities
• Stationarity of Time-Series on Graph via Bivariate Translation Invariance