mlsanityData to 13 Oct 2026

Matrix Manifold Neural Networks++

arXiv 2405.19206 ↗First posted 29 May 2024stat.ML cs.LGcited by 13

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Abstract (arXiv)

Deep neural networks (DNNs) on Riemannian manifolds have garnered increasing interest in various applied areas. For instance, DNNs on spherical and hyperbolic manifolds have been designed to solve a wide range of computer vision and nature language processing tasks. One of the key factors that contribute to the success of these networks is that spherical and hyperbolic manifolds have the rich algebraic structures of gyrogroups and gyrovector spaces. This enables principled and effective generalizations of the most successful DNNs to these manifolds. Recently, some works have shown that many concepts in the theory of gyrogroups and gyrovector spaces can also be generalized to matrix manifolds such as Symmetric Positive Definite (SPD) and Grassmann manifolds. As a result, some building blocks for SPD and Grassmann neural networks, e.g., isometric models and multinomial logistic regression (MLR) can be derived in a way that is fully analogous to their spherical and hyperbolic counterparts. Building upon these works, we design fully-connected (FC) and convolutional layers for SPD neural networks. We also develop MLR on Symmetric Positive Semi-definite (SPSD) manifolds, and propose a method for performing backpropagation with the Grassmann logarithmic map in the projector perspective. We demonstrate the effectiveness of the proposed approach in the human action recognition and node classification tasks.

Key facts. Cited by 13 papers in the map (data through 13 Oct 2026). Builds on 33 of its 63 references that are in the map. First posted on arXiv 29 May 2024.

Builds on

33 of 63 references.

References · 33

  1. 2016Semi-Supervised Classification with Graph Convolutional Networkscited by 11,318
  2. 2017Graph Attention Networkscited by 8,888
  3. 2017Inductive Representation Learning on Large Graphscited by 6,303
  4. 2018Link Prediction Based on Graph Neural Networkscited by 829
  5. 2016NTU RGB+D: A Large Scale Dataset for 3D Human Activity Analysiscited by 684
  6. 2019Hyperbolic Graph Convolutional Neural Networkscited by 401
  7. 2018Hyperbolic Neural Networkscited by 396
  8. 2020Disentangling and Unifying Graph Convolutions for Skeleton-Based Action Recognitioncited by 339
  9. 2019Hyperbolic Graph Neural Networkscited by 244
  10. 2017First-Person Hand Action Benchmark with RGB-D Videos and 3D Hand Pose Annotationscited by 229
  11. 2016A Riemannian Network for SPD Matrix Learningcited by 162
  12. 2021Fully Hyperbolic Neural Networkscited by 108
21 more
  1. 2020Skeleton-based Action Recognition via Spatial and Temporal Transformer Networkscited by 98
  2. 2019Mixed-curvature Variational Autoencoderscited by 86
  3. 2019Riemannian batch normalization for SPD neural networkscited by 72
  4. 2019Hyperbolic Graph Attention Networkcited by 70
  5. 2021Lorentzian Graph Convolutional Networkscited by 67
  6. 2023Learning Discriminative Representations for Skeleton Based Action Recognitioncited by 62
  7. 2016Building Deep Networks on Grassmann Manifoldscited by 58
  8. 2021A Hyperbolic-to-Hyperbolic Graph Convolutional Networkcited by 53
  9. 2022SPD domain-specific batch normalization to crack interpretable unsupervised domain adaptation in EEGcited by 53
  10. 2019A neural network based on SPD manifold learning for skeleton-based hand gesture recognitioncited by 46
  11. 2016Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methodscited by 29
  12. 2020Computationally Tractable Riemannian Manifolds for Graph Embeddingscited by 29
  13. 2021GeomNet: A Neural Network Based on Riemannian Geometries of SPD Matrix Space and Cholesky Space for 3D Skeleton-Based Interaction Recognitioncited by 27
  14. 2021Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matricescited by 22
  15. 2023Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approachcited by 18
  16. 2022Graph Neural Networks on SPD Manifolds for Motor Imagery Classification: A Perspective from the Time-Frequency Analysiscited by 15
  17. 2023Node Embedding from Neural Hamiltonian Orbits in Graph Neural Networkscited by 12
  18. 2023Modeling Graphs Beyond Hyperbolic: Graph Neural Networks in Symmetric Positive Definite Matricescited by 10
  19. 2020Neural Architecture Search of SPD Manifold Networkscited by 6
  20. 2019Skeleton-Based Hand Gesture Recognition by Learning SPD Matrices with Neural Networkscited by 5
  21. 2022Embedding Graphs on Grassmann Manifoldcited by 2

Built on by

Cited by year (2026 to 13 Oct)

Citing papers per year: 2024: 2; 2025: 4; 2026: 7.

13 papers.

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  1. 2024RMLR: Extending Multinomial Logistic Regression into General Geometriescited by 8
  2. 2024A Riemannian Framework for Learning Reduced-order Lagrangian Dynamicscited by 7
  3. 2025Learning to Normalize on the SPD Manifold under Bures-Wasserstein Geometrycited by 7
  4. 2025SPD Matrix Learning for Neuroimaging Analysis: Perspectives, Methods, and Challengescited by 6
  5. 2026Neural Networks on Symmetric Spaces of Noncompact Typecited by 6
  6. 2026Intrinsic Lorentz Neural Networkcited by 3
  7. 2025Siegel Neural Networkscited by 1
  8. 2025Intrinsic-Metric Physics-Informed Neural Networks (IM-PINN) for Reaction-Diffusion Dynamics on Complex Riemannian Manifoldscited by 0
  9. 2026Batch Normalization for Neural Networks on Complex Domainscited by 0
  10. 2026Riemannian Networks over Full-Rank Correlation Matricescited by 0
  11. 2026Expressivity of congruence-based architectures for DNNs on positive-definite matricescited by 0
  12. 2026LieBN: Batch Normalization over Lie Groupscited by 0
1 more
  1. 2026Geometric Feature Learning for Functional Data Valued on the Symmetric Positive Definite Manifoldcited by 0