黎曼批量归一化：一种陀螺方法
Riemannian Batch Normalization: A Gyro Approach
AI总结：
本文提出 GyroBN，一种基于陀螺群的黎曼批量归一化框架，通过伪约化和陀螺等距陀螺变换保证统计量理论控制，并在七种几何上验证其有效性。
摘要：
归一化层对深度学习至关重要，但其欧氏形式并不适用于流形上的数据。另一方面，机器学习中的许多黎曼流形具有陀螺结构，从而能够将欧氏神经网络有原则地扩展到非欧域。受此启发，我们提出了 GyroBN，一个面向陀螺群的有原则的黎曼批量归一化框架。我们建立了两个必要条件，即伪约化和陀螺等距陀螺变换，以保证 GyroBN 对样本统计量具有理论控制，并证明这些条件对机器学习中所有已知的陀螺群均成立。我们的框架还将若干现有的黎曼归一化方法作为特例纳入其中。我们进一步在七种代表性几何上实例化 GyroBN，包括格拉斯曼流形、五种常曲率空间和相关流形，并推导出新的陀螺结构和黎曼结构以实现这些实例化。在这些几何上的实验证明了 GyroBN 的有效性。代码可在 https://github.com/GitZH-Chen/GyroBN.git 获取。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
