Manifold-valued measurements exist in numerous applications within computer vision and machine learning. Recent studies have extended Deep Neural Networks (DNNs) to manifolds, and concomitantly, normalization techniques have also been adapted to several manifolds, referred to as Riemannian normalization. Nonetheless, most of the existing Riemannian normalization methods have been derived in an ad hoc manner and only apply to specific manifolds. This paper establishes a unified framework for Riemannian Batch Normalization (RBN) techniques on Lie groups. Our framework offers the theoretical guarantee of controlling both the Riemannian mean and variance. Empirically, we focus on Symmetric Positive Definite (SPD) manifolds, which possess three distinct types of Lie group structures. Using the deformation concept, we generalize the existing Lie groups on SPD manifolds into three families of parameterized Lie groups. Specific normalization layers induced by these Lie groups are then proposed for SPD neural networks. We demonstrate the effectiveness of our approach through three sets of experiments: radar recognition, human action recognition, and electroencephalography (EEG) classification. The code is available at https://github.com/GitZH-Chen/LieBN.git.

翻译：在计算机视觉和机器学习领域的众多应用中，存在流形值测量。近期研究已将深度神经网络扩展至流形，同时归一化技术也被适配到多种流形上，称为黎曼归一化。然而，现有的大多数黎曼归一化方法都是针对特定流形临时推导的。本文建立了一个基于李群的黎曼批量归一化统一框架。该框架提供了同时控制黎曼均值和方差的理论保障。在实验层面，我们聚焦于具有三种不同李群结构的对称正定流形。利用形变概念，我们将对称正定流形上的现有李群推广为三类参数化李群，并提出由这些李群导出的特定归一化层用于对称正定神经网络。通过雷达识别、人体动作识别和脑电图分类三组实验验证了方法的有效性。代码见https://github.com/GitZH-Chen/LieBN.git。