# Riemannian Batch Normalization: A Gyro Approach

arXiv: https://arxiv.org/abs/2509.07115 · HTML: https://mlsanity.com/paper/2509.07115/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Sep 2025.

## 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)
- [HEEGNet: Hyperbolic Embeddings for EEG](https://mlsanity.com/paper/2601.03322/) (2026)
- [Geodesic Gradient Descent: A Generic and Learning-rate-free Optimizer on Objective Function-induced Manifolds](https://mlsanity.com/paper/2603.06651/) (2026)
- [Riemannian Deep Learning: Modules, Networks, and Geometries](https://mlsanity.com/paper/2607.19305/) (2026)
- [Building Transformation Layers for Riemannian Neural Networks](https://mlsanity.com/paper/2609.35436/) (2026)

## Abstract (arXiv)

Normalization layers are crucial for deep learning, but their Euclidean formulations are inadequate for data on manifolds. On the other hand, many Riemannian manifolds in machine learning admit gyro-structures, enabling principled extensions of Euclidean neural networks to non-Euclidean domains. Inspired by this, we introduce GyroBN, a principled Riemannian batch normalization framework for gyrogroups. We establish two necessary conditions, namely pseudo-reduction and gyroisometric gyrations, that guarantee GyroBN with theoretical control over sample statistics, and show that these conditions hold for all known gyrogroups in machine learning. Our framework also incorporates several existing Riemannian normalization methods as special cases. We further instantiate GyroBN on seven representative geometries, including the Grassmannian, five constant curvature spaces, and the correlation manifold, and derive novel gyro and Riemannian structures to enable these instantiations. Experiments across these geometries demonstrate the effectiveness of GyroBN. The code is available at https://github.com/GitZH-Chen/GyroBN.git.