# Neural Networks on Symmetric Spaces of Noncompact Type

arXiv: https://arxiv.org/abs/2601.01097 · HTML: https://mlsanity.com/paper/2601.01097/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 43 of its 78 references that are in the map.
- First posted on arXiv 3 Jan 2026.

## Builds on (most cited first, 30 of 43)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Improved Regularization of Convolutional Neural Networks with Cutout](https://mlsanity.com/paper/1708.04552/) (2017)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [EEGNet: A Compact Convolutional Network for EEG-based Brain-Computer Interfaces](https://mlsanity.com/paper/1611.08024/) (2016)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [Hyperbolic Neural Networks](https://mlsanity.com/paper/1805.09112/) (2018)
- [Hyperbolic Graph Convolutional Neural Networks](https://mlsanity.com/paper/1910.12933/) (2019)
- [Hyperbolic Graph Neural Networks](https://mlsanity.com/paper/1910.12892/) (2019)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Hyperbolic Entailment Cones for Learning Hierarchical Embeddings](https://mlsanity.com/paper/1804.01882/) (2018)
- [Hyperbolic Attention Networks](https://mlsanity.com/paper/1805.09786/) (2018)
- [Geoopt: Riemannian Optimization in PyTorch](https://mlsanity.com/paper/2005.02819/) (2020)
- [Trivializations for Gradient-Based Optimization on Manifolds](https://mlsanity.com/paper/1909.09501/) (2019)
- [Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds](https://mlsanity.com/paper/2106.06020/) (2021)
- [Riemannian batch normalization for SPD neural networks](https://mlsanity.com/paper/1909.02414/) (2019)
- [EEG-TCNet: An Accurate Temporal Convolutional Network for Embedded Motor-Imagery Brain-Machine Interfaces](https://mlsanity.com/paper/2006.00622/) (2020)
- [Building Deep Networks on Grassmann Manifolds](https://mlsanity.com/paper/1611.05742/) (2016)
- [FBCNet: A Multi-view Convolutional Neural Network for Brain-Computer Interface](https://mlsanity.com/paper/2104.01233/) (2021)
- [Clipped Hyperbolic Classifiers Are Super-Hyperbolic Classifiers](https://mlsanity.com/paper/2107.11472/) (2021)
- [SPD domain-specific batch normalization to crack interpretable unsupervised domain adaptation in EEG](https://mlsanity.com/paper/2206.01323/) (2022)
- [Fully Hyperbolic Convolutional Neural Networks for Computer Vision](https://mlsanity.com/paper/2303.15919/) (2023)
- [HoroPCA: Hyperbolic Dimensionality Reduction via Horospherical Projections](https://mlsanity.com/paper/2106.03306/) (2021)
- [A neural network based on SPD manifold learning for skeleton-based hand gesture recognition](https://mlsanity.com/paper/1904.12970/) (2019)
- [MAtt: A Manifold Attention Network for EEG Decoding](https://mlsanity.com/paper/2210.01986/) (2022)
- [Deep Learning on Lie Groups for Skeleton-based Action Recognition](https://mlsanity.com/paper/1612.05877/) (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)

## Built on by (most cited first, 6 of 6)

- [Hyperbolic Busemann Neural Networks](https://mlsanity.com/paper/2602.18858/) (2026)
- [Intrinsic Lorentz Neural Network](https://mlsanity.com/paper/2602.23981/) (2026)
- [LAtte: Hyperbolic Lorentz Attention for Cross-Subject EEG Classification](https://mlsanity.com/paper/2603.10881/) (2026)
- [ReManNet: A Riemannian Manifold Network for Monocular 3D Lane Detection](https://mlsanity.com/paper/2603.19776/) (2026)
- [Batch Normalization for Neural Networks on Complex Domains](https://mlsanity.com/paper/2605.00467/) (2026)
- [The Implicit Bias of Hyperbolic Representation Learning for Multiclass Data: A Busemann Risk Perspective](https://mlsanity.com/paper/2610.07131/) (2026)

## Abstract (arXiv)

Recent works have demonstrated promising performances of neural networks on hyperbolic spaces and symmetric positive definite (SPD) manifolds. These spaces belong to a family of Riemannian manifolds referred to as symmetric spaces of noncompact type. In this paper, we propose a novel approach for developing neural networks on such spaces. Our approach relies on a unified formulation of the distance from a point to a hyperplane on the considered spaces. We show that some existing formulations of the point-to-hyperplane distance can be recovered by our approach under specific settings. Furthermore, we derive a closed-form expression for the point-to-hyperplane distance in higher-rank symmetric spaces of noncompact type equipped with G-invariant Riemannian metrics. The derived distance then serves as a tool to design fully-connected (FC) layers and an attention mechanism for neural networks on the considered spaces. Our approach is validated on challenging benchmarks for image classification, electroencephalogram (EEG) signal classification, image generation, and natural language inference.