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 this https URL

本文在李群上建立了一个统一的黎曼均值归一化 (RBN) 技术框架，理论上保证了控制黎曼均值和方差；在实验中，我们针对对称正定流形展示了该方法的有效性，应用于雷达识别、人类动作识别和脑电图分类。

一个针对 Riemannian Batch Normalization 的 Lie 群方法

A Lie Group Approach to Riemannian Batch Normalization

The domain adaptation (DA) problem on symmetric positive definite (SPD)
manifolds has raised interest in the machine learning community because of the
growing potential for the SPD-matrix representations across many cross-domain
applicable scenarios. However, due to the different underlying space, the
previous experience and solution to the DA problem cannot benefit this new
scenario directly. This study addresses a specific DA problem: the marginal and
conditional distributions differ in the source and target domains on SPD
manifolds. We then formalize this problem from an optimal transport perspective
and derive an optimal transport framework on SPD manifolds for supervised
learning. In addition, we propose a computational scheme under the optimal
transport framework, Deep Optimal Transport (DOT), for general computation,
using the generalized joint distribution adaptation approach and the existing
Riemannian-based network architectures on SPD manifolds. DOT is applied to the
real-world scenario and becomes a specific EEG-BCI classifier against the
cross-session motor-imagery classification from the calibration phase to the
feedback phase. In the experiments, DOT exhibits a marked improvement in the
average accuracy in two highly non-stationary cross-session scenarios in the
EEG-BCI classification, respectively, indicating the proposed methodology's
validity.

该研究提出了一种基于对称正定流形上的最优传输框架的迁移学习方法，并应用于 EEG-BCI 领域的跨会话运动想象分类问题，有效提升了分类的准确率。