mlsanityData to 7 Oct 2026

Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approach

arXiv 2305.04560 ↗First posted 8 May 2023stat.ML cs.LGcited by 18

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Abstract (arXiv)

Matrix manifolds, such as manifolds of Symmetric Positive Definite (SPD) matrices and Grassmann manifolds, appear in many applications. Recently, by applying the theory of gyrogroups and gyrovector spaces that is a powerful framework for studying hyperbolic geometry, some works have attempted to build principled generalizations of Euclidean neural networks on matrix manifolds. However, due to the lack of many concepts in gyrovector spaces for the considered manifolds, e.g., the inner product and gyroangles, techniques and mathematical tools provided by these works are still limited compared to those developed for studying hyperbolic geometry. In this paper, we generalize some notions in gyrovector spaces for SPD and Grassmann manifolds, and propose new models and layers for building neural networks on these manifolds. We show the effectiveness of our approach in two applications, i.e., human action recognition and knowledge graph completion.

Key facts. Cited by 18 papers in the map (data through 7 Oct 2026). Builds on 19 of its 55 references that are in the map. First posted on arXiv 8 May 2023.

Builds on

19 of 55 references.

References · 19

  1. 2018Spatial Temporal Graph Convolutional Networks for Skeleton-Based Action Recognitioncited by 1,199
  2. 2016Geometric deep learning: going beyond Euclidean datacited by 1,042
  3. 2016NTU RGB+D: A Large Scale Dataset for 3D Human Activity Analysiscited by 683
  4. 2018Hyperbolic Neural Networkscited by 393
  5. 2017First-Person Hand Action Benchmark with RGB-D Videos and 3D Hand Pose Annotationscited by 229
  6. 2016A Riemannian Network for SPD Matrix Learningcited by 162
  7. 2019Multi-relational Poincar'e Graph Embeddingscited by 126
  8. 2020Skeleton-based Action Recognition via Spatial and Temporal Transformer Networkscited by 98
  9. 2019Mixed-curvature Variational Autoencoderscited by 85
  10. 2019Riemannian batch normalization for SPD neural networkscited by 72
  11. 2016Building Deep Networks on Grassmann Manifoldscited by 58
  12. 2019A neural network based on SPD manifold learning for skeleton-based hand gesture recognitioncited by 46
7 more
  1. 2016Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methodscited by 29
  2. 2021GeomNet: A Neural Network Based on Riemannian Geometries of SPD Matrix Space and Cholesky Space for 3D Skeleton-Based Interaction Recognitioncited by 27
  3. 2021Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matricescited by 22
  4. 2022Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spacescited by 20
  5. 2018A Statistical Recurrent Model on the Manifold of Symmetric Positive Definite Matricescited by 18
  6. 2021VolterraNet: A higher order convolutional network with group equivariance for homogeneous manifoldscited by 11
  7. 2019Skeleton-Based Hand Gesture Recognition by Learning SPD Matrices with Neural Networkscited by 5

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 1; 2024: 5; 2025: 4; 2026: 8.

18 papers.

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  1. 2024Matrix Manifold Neural Networks++cited by 13
  2. 2023Riemannian Multinomial Logistics Regression for SPD Neural Networkscited by 11
  3. 2024Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometrycited by 9
  4. 2024A Lie Group Approach to Riemannian Batch Normalizationcited by 8
  5. 2024RMLR: Extending Multinomial Logistic Regression into General Geometriescited by 8
  6. 2025Learning to Normalize on the SPD Manifold under Bures-Wasserstein Geometrycited by 7
  7. 2025SPD Matrix Learning for Neuroimaging Analysis: Perspectives, Methods, and Challengescited by 5
  8. 2026Neural Networks on Symmetric Spaces of Noncompact Typecited by 5
  9. 2024Fast and Stable Riemannian Metrics on SPD Manifolds via Cholesky Product Geometrycited by 4
  10. 2025ASIL: Augmented Structural Information Learning for Deep Graph Clustering in Hyperbolic Spacecited by 4
  11. 2026Intrinsic Lorentz Neural Networkcited by 3
  12. 2026Sheaf Neural Networks on SPD Manifolds: Second-Order Geometric Representation Learningcited by 2
6 more
  1. 2025Siegel Neural Networkscited by 1
  2. 2026Batch Normalization for Neural Networks on Complex Domainscited by 0
  3. 2026Riemannian Networks over Full-Rank Correlation Matricescited by 0
  4. 2026Expressivity of congruence-based architectures for DNNs on positive-definite matricescited by 0
  5. 2026Towards Robust EEG Decoding Based on Riemannian Self-Attentioncited by 0
  6. 20261-Lipschitz Neural Networks on Hadamard Manifoldscited by 0