提出基于几何群的流形批归一化方法，解决非欧数据归一化难题。
Riemannian Batch Normalization: A Gyro Approach
- 用几何群结构设计流形批归一化，理论可控制统计量
- 在7种几何空间验证有效，包括格拉斯曼流形和相关性流形
- 兼容已有流形归一化方法，适合处理非欧神经网络
归一化层对深度学习至关重要，但其欧式形式不适用于流形上的数据。许多机器学习中的黎曼流形具有几何群结构，可将欧式神经网络合理扩展至非欧域。受此启发，我们提出一种基于几何群的流形批归一化框架——GyroBN。我们建立两个必要条件：伪简化与几何等距旋转，确保GyroBN对样本统计量有理论控制，并证明这些条件在所有已知机器学习几何群中均成立。该框架还包含多种现有流形归一化方法作为特例。我们在七种代表性几何空间（包括格拉斯曼流形、五类常曲率空间及相关性流形）上实例化GyroBN，推导出新的几何与黎曼结构以支持实现。跨几何实验验证了其有效性。代码已公开于 https://github.com/GitZH-Chen/GyroBN.git。
原文摘要 · Abstract (English)
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 \emph{pseudo-reduction} and \emph{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.
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