  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.
