# Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approach

arXiv: https://arxiv.org/abs/2305.04560 · HTML: https://mlsanity.com/paper/2305.04560/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 55 references that are in the map.
- First posted on arXiv 8 May 2023.

## Builds on (most cited first, 19 of 19)

- [Spatial Temporal Graph Convolutional Networks for Skeleton-Based Action Recognition](https://mlsanity.com/paper/1801.07455/) (2018)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [NTU RGB+D: A Large Scale Dataset for 3D Human Activity Analysis](https://mlsanity.com/paper/1604.02808/) (2016)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [First-Person Hand Action Benchmark with RGB-D Videos and 3D Hand Pose Annotations](https://mlsanity.com/paper/1704.02463/) (2017)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Multi-relational Poincar'e Graph Embeddings](https://mlsanity.com/paper/1905.09791/) (2019)
- [Skeleton-based Action Recognition via Spatial and Temporal Transformer Networks](https://mlsanity.com/paper/2008.07404/) (2020)
- [Mixed-curvature Variational Autoencoders](https://mlsanity.com/paper/1911.08411/) (2019)
- [Riemannian batch normalization for SPD neural networks](https://mlsanity.com/paper/1909.02414/) (2019)
- [Building Deep Networks on Grassmann Manifolds](https://mlsanity.com/paper/1611.05742/) (2016)
- [A neural network based on SPD manifold learning for skeleton-based hand gesture recognition](https://mlsanity.com/paper/1904.12970/) (2019)
- [Dimensionality Reduction on SPD Manifolds: The Emergence of Geometry-Aware Methods](https://mlsanity.com/paper/1605.06182/) (2016)
- [GeomNet: A Neural Network Based on Riemannian Geometries of SPD Matrix Space and Cholesky Space for 3D Skeleton-Based Interaction Recognition](https://mlsanity.com/paper/2111.13089/) (2021)
- [Vector-valued Distance and Gyrocalculus on the Space of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/2110.13475/) (2021)
- [Unified Fourier-based Kernel and Nonlinearity Design for Equivariant Networks on Homogeneous Spaces](https://mlsanity.com/paper/2206.08362/) (2022)
- [A Statistical Recurrent Model on the Manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/1805.11204/) (2018)
- [VolterraNet: A higher order convolutional network with group equivariance for homogeneous manifolds](https://mlsanity.com/paper/2106.15301/) (2021)
- [Skeleton-Based Hand Gesture Recognition by Learning SPD Matrices with Neural Networks](https://mlsanity.com/paper/1905.07917/) (2019)

## Built on by (most cited first, 18 of 18)

- [Matrix Manifold Neural Networks++](https://mlsanity.com/paper/2405.19206/) (2024)
- [Riemannian Multinomial Logistics Regression for SPD Neural Networks](https://mlsanity.com/paper/2305.11288/) (2023)
- [Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry](https://mlsanity.com/paper/2407.10484/) (2024)
- [A Lie Group Approach to Riemannian Batch Normalization](https://mlsanity.com/paper/2403.11261/) (2024)
- [RMLR: Extending Multinomial Logistic Regression into General Geometries](https://mlsanity.com/paper/2409.19433/) (2024)
- [Learning to Normalize on the SPD Manifold under Bures-Wasserstein Geometry](https://mlsanity.com/paper/2504.00660/) (2025)
- [SPD Matrix Learning for Neuroimaging Analysis: Perspectives, Methods, and Challenges](https://mlsanity.com/paper/2504.18882/) (2025)
- [Neural Networks on Symmetric Spaces of Noncompact Type](https://mlsanity.com/paper/2601.01097/) (2026)
- [Fast and Stable Riemannian Metrics on SPD Manifolds via Cholesky Product Geometry](https://mlsanity.com/paper/2407.02607/) (2024)
- [ASIL: Augmented Structural Information Learning for Deep Graph Clustering in Hyperbolic Space](https://mlsanity.com/paper/2504.09970/) (2025)
- [Intrinsic Lorentz Neural Network](https://mlsanity.com/paper/2602.23981/) (2026)
- [Sheaf Neural Networks on SPD Manifolds: Second-Order Geometric Representation Learning](https://mlsanity.com/paper/2604.20308/) (2026)
- [Siegel Neural Networks](https://mlsanity.com/paper/2511.09577/) (2025)
- [Batch Normalization for Neural Networks on Complex Domains](https://mlsanity.com/paper/2605.00467/) (2026)
- [Riemannian Networks over Full-Rank Correlation Matrices](https://mlsanity.com/paper/2605.19073/) (2026)
- [Expressivity of congruence-based architectures for DNNs on positive-definite matrices](https://mlsanity.com/paper/2606.02490/) (2026)
- [Towards Robust EEG Decoding Based on Riemannian Self-Attention](https://mlsanity.com/paper/2606.25456/) (2026)
- [1-Lipschitz Neural Networks on Hadamard Manifolds](https://mlsanity.com/paper/2607.19335/) (2026)

## 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.