Awesome Deep Learning over Riemannian Spaces

🚀 This is an ongoing repo, containing resources on Riemannian deep learning.

🌟 Note: This repo does not inlcude Riemannian optimization. For that, please refer to Awesome-Riemannian-Optimization by Andi Han.

Contents

• 1. Books
  • Topology
  • Differential Geometry
  • Riemannian Geometry
  • Lie Groups
  • Riemannian Optimization
• 2. Papers

  • Popular Manifolds

  • Geometries

    • SPD Geometries
    • Grassmannian
    • Lie Groups

  • SPD Neural Networks

    • Network Construction
    • Neuroscience Applications

  • Grassmannian Neural Networks

  • Lie Group Neural Networks

  • Constant Curvature Neural Networks

  • Riemannian Graph Neural Networks

  • Riemannian Networks on General Geometries

  • Riemannian Generative Networks

  • Applications in Euclidean Deep Learning

    • High-order Pooling

1 Books

Topology

• Basic Topology Springer,1983.
• Topology Pearson, 2000.

Differential Geometry

• An Introduction to Manifolds Springer, 2011.
• Introduction to Smooth Manifolds Springer, 2012.

Riemannian Geometry

• Riemannian Geometry Birkhäuser, 1992.
• Introduction to Riemannian Manifolds Springer, 2018.
• Riemannian Geometry Springer, 2006.

Lie Groups

• Naive Lie Theory Springer, 2008.
• Lie Groups, Lie Algebras, and Representations Springer, 2015.
• A mathematical introduction to robotic manipulation

Hyperbolic Geometry

• Hyperbolic Geometry
• A Gyrovector Space Approach to Hyperbolic Geometry Springer, 2009.

Riemannian Optimization

1. Optimization Algorithms on Matrix Manifolds Cambridge University Press, 2023
2. An introduction to Optimization on smooth manifolds Princeton University Press, 2008.

2 Papers

Popular Manifolds

An overview can be found in https://www.manopt.org/manifolds.html

Geometries

SPD Geometries

• Fast and Simple Computations on Tensors with Log-Euclidean Metrics Reaserch Report 2005
• A Riemannian framework for tensor computing IJCV 2006
• Power Euclidean metrics for covariance matrices with application to diffusion tensor imaging Arxiv 2010
• Riemannian geometry of symmetric positive definite matrices via Cholesky decomposition SIMAX 2019
• On the Bures–Wasserstein distance between positive definite matrices Expositiones Mathematicae 2019
• The geometry of mixed-Euclidean metrics on symmetric positive definite matrices Differential Geometry and its Applications 2022
• O (n)-invariant Riemannian metrics on SPD matrices Linear Algebra and its Applications 2023
• Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry GSI 2023
• Adaptive Log-Euclidean Metrics for SPD Matrix Learning TIP 2024
• Product Geometries on Cholesky Manifolds with Applications to SPD Manifolds Arxiv 2024
• Wrapped Gaussian on the manifold of Symmetric Positive Definite Matrices Arxiv 2025
• A Riemannian covariance for manifold-valued data Arxiv 2025

Grassmannian

• The Geometry of Algorithms with Orthogonality Constraints SIMAX 1998
• A Grassmann Manifold Handbook: Basic Geometry and Computational Aspects Advances in Computational Mathematics 2024

Lie Groups

• Left-Invariant Riemannian Geodesics on Spatial Transformation Groups SIIMS 2014
• Optimal Transport on the Lie Group of Roto-translations SIIMS 2024

SPD Neural Networks

Network Construction

• A Riemannian Network for SPD Matrix Learning AAAI 2017
• Riemannian batch normalization for SPD neural networks NeurIPS 2019
• GeomNet: A Neural Network Based on Riemannian Geometries of SPD Matrix Space and Cholesky Space for 3D Skeleton-Based Interaction Recognition ICCV 2021
• Neural Architecture Search of SPD Manifold Networks IJCAI 2021
• Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices NeurIPS 2021
• Riemannian Local Mechanism for SPD Neural Networks AAAI 2023
• SPD Manifold Deep Metric Learning for Image Set Classification TNNLS 2024
• Riemannian Multinomial Logistics Regression for SPD Neural Networks CVPR 2024
• Exploring the Enigma of Neural Dynamics Through A Scattering-Transform Mixer Landscape for Riemannian Manifold ICML 2024
• Schur's Positive-Definite Network: Deep Learning in the SPD cone with structure ICLR 2025

Neuroscience Applications

• MAtt: A Manifold Attention Network for EEG Decoding NeurIPS 2022
• Deep Optimal Transport for Domain Adaptation on SPD Manifolds Arxiv 2024
• SPD domain-specific batch normalization to crack interpretable unsupervised domain adaptation in EEG NeurIPS 2024
• Deep Geodesic Canonical Correlation Analysis for Covariance-Based Neuroimaging Data ICLR 2024
• GeoMind: A Geometric Neural Network of State Space Model for Understanding Brain Dynamics on Riemannian Manifold Openreview 2025
• SPDIM: Source-Free Unsupervised Conditional and Label Shift Adaptation in EEG ICLR 2025

Grassmannian Neural Networks

• Building Deep Networks on Grassmann Manifolds AAAI 2018
• A Grassmannian Manifold Self-Attention Network for Signal Classification IJCAI 2024

Lie Group Neural Networks

• Rotation Averaging IJCV 2013
• Human Action Recognition by Representing 3D Skeletons as Points in a Lie Group CVPR 2014
• Rolling rotations for recognizing human actions from 3d skeletal data CVPR 2016
• Deep Learning on Lie Groups for Skeleton-based Action Recognition CVPR 2017

Constant Curvature Neural Networks

• Hyperbolic entailment cones for learning hierarchical embeddings ICML 2018
• Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometry ICML 2018
• Hyperbolic Neural Networks NeurIPS 2018
• Poincaré GloVe: Hyperbolic Word Embeddings ICLR 2019
• Hyperbolic Attention Networks ICLR 2019
• Continuous Hierarchical Representations with Poincaré Variational Auto-Encoders NeurIPS 2019
• Mixed-curvature Variational Autoencoders ICLR 2020
• Hyperbolic Neural Networks++ ICLR 2021
• Differentiating through the Fréchet Mean ICML 2020
• Curvature Generation in Hyperbolic Spaces for Few-Shot Learning ICCV 2021
• Fully Hyperbolic Neural Networks ACL 2022
• Nested hyperbolic spaces for dimensionality reduction and hyperbolic nn design CVPR 2022
• Hyperbolic Feature Augmentation via Distribution Estimation and Infinite Sampling on Manifolds NeurIPS 2022
• Poincaré ResNet ICCV 2023
• Fully Hyperbolic Convolutional Neural Networks for Computer Vision ICLR 2024
• Lorentzian Residual Neural Networks KDD 2025
• Hyperbolic Genome Embeddings ICLR 2025
• Compositional Entailment Learning for Hyperbolic Vision-Language Models ICLR 2025

Riemannian Graph Neural Networks

• Computationally Tractable Riemannian Manifolds for Graph Embeddings AAAI 2021

• Graph Neural Networks on SPD Manifolds for Motor Imagery Classification: A Perspective from the Time-Frequency Analysis TNNLS 2023

• Modeling Graphs Beyond Hyperbolic: Graph Neural Networks in Symmetric Positive Definite Matrices ECML 2023

• Hyperbolic Graph Convolutional Neural Networks NeurIPS 2019

• Hyperbolic Graph Neural Networks NeurIPS 2019

• Constant Curvature Graph Convolutional Networks ICML 2020

• A Hyperbolic-to-Hyperbolic Graph Convolutional Network CVPR 2021

• LSEnet: Lorentz Structural Entropy Neural Network for Deep Graph Clustering ICML 2024

• Spiking Graph Neural Network on Riemannian Manifolds NeurIPS 2024

• A Mixed-Curvature Graph Diffusion Model CIKM 2024

• Pioneer: Physics-informed Riemannian Graph ODE for Entropy-increasing Dynamics AAAI 2025

• RiemannGFM: Learning a Graph Foundation Model from Riemannian Geometry WWW 2025

Riemannian Networks on General Geometries

• Dilated Convolutional Neural Networks for Sequential Manifold-valued Data ICCV 2019

• ManifoldNet: A Deep Neural Network for Manifold-Valued Data With Applications TPAMI 2020

• ManifoldNorm: Extending normalizations on Riemannian Manifolds Arxiv 2020

• A Gyrovector Space Approach for Symmetric Positive Semi-definite Matrix Learning ECCV 2022

• The Gyro-Structure of Some Matrix Manifolds NeurIPS 2022

• Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approach ICML 2023

• Riemannian Residual Neural Networks NeurIPS 2023

• Matrix Manifold Neural Networks++ ICLR 2024

• A Lie Group Approach to Riemannian Batch Normalization ICLR 2024

• RMLR: Extending Multinomial Logistic Regression into General Geometries NeurIPS 2024

• Gyrogroup Batch Normalization ICLR 2025

• Neural Networks on Symmetric Spaces of Noncompact Type ICLR 2025

• GyroAtt: A Gyro Attention Framework for Matrix Manifolds Openreview 2025

Riemannian Generative Networks

• Riemannian score-based generative modelling NeurIPS 2022
• Riemannian Diffusion Models NeurIPS 2022
• SE (3) diffusion model with application to protein backbone generation ICML 2023
• Exploring Data Geometry for Continual Learning CVPR 2023
• Scaling Riemannian Diffusion Models NeurIPS 2023
• Flow Matching on General Geometries ICLR 2024
• Hyperbolic Geometric Latent Diffusion Model for Graph Generation ICML 2024
• Generative Modeling on Lie Groups via Euclidean Generalized Score Matching Arxiv 2025

Applications in Euclidean Deep Learning

High-order Pooling

• Kernel Pooling for Convolutional Neural Networks CVPR 2017
• Towards Faster Training of Global Covariance Pooling Networks by Iterative Matrix Square Root Normalization CVPR 2018
• DeepKSPD: Learning Kernel-matrix-based SPD Representation for Fine-grained Image Recognition ECCV 2018
• Deep CNNs Meet Global Covariance Pooling: Better Representation and Generalization TPAMI 2020
• Orthogonal SVD Covariance Conditioning and Latent Disentanglement TPAMI 2022
• Fast Differentiable Matrix Square Root and Inverse Square Root TPAMI 2022
• On the Eigenvalues of Global Covariance Pooling for Fine-grained Visual Recognition TPAMI 2022
• Learning partial correlation based deep visual representation for image classification CVPR 2023
• Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry ICLR 2025