LieBN: Batch Normalization over Lie Groups

2026-06-13

AI Summary

This work addresses the limitations of existing Riemannian normalization methods, which are often restricted to specific manifolds or fail to effectively normalize manifold-valued data distributions. The authors propose LieBN, a novel framework that, for the first time, enables batch normalization on Lie groups using left- and right-invariant Riemannian metrics. By explicitly modeling Riemannian mean and variance, LieBN achieves effective normalization in a unified manner across nine distinct geometric structures. Notably, it introduces a new right-invariant metric and a matrix power deformation mechanism for symmetric positive-definite (SPD) manifolds. Extensive experiments demonstrate that LieBN significantly improves normalization efficacy and downstream task performance across diverse manifolds, confirming its generality and effectiveness.

Abstract

Manifold-valued measurements are prevalent in various machine learning tasks. Recent advances have extended Deep Neural Networks (DNNs) to operate on manifolds, accompanied by normalization techniques tailored to different geometries, collectively referred to as Riemannian normalization. However, most existing Riemannian normalization methods are either designed for specific manifolds or fail to effectively normalize manifold-valued sample distributions. To address these limitations, we propose LieBN, a framework for Riemannian Batch Normalization (RBN) over Lie groups. Our approach leverages the theoretically convenient left- and right-invariant metrics, which naturally exist in every Lie group, and provides theoretical guarantees for controlling the Riemannian mean and variance. We instantiate LieBN across nine distinct geometries: four on the Symmetric Positive Definite (SPD) manifold, one on the group of rotation matrices, and four on the manifold of full-rank correlation matrices. Notably, among the SPD metrics, we introduce a novel right-invariant metric and extend three existing Lie group structures via matrix power deformation. Extensive experiments on different manifolds validate the effectiveness of our framework. The code is available at https://github.com/GitZH-Chen/LieBN.git.

Problem

• Riemannian normalization
• manifold-valued data
• Batch Normalization
• Lie groups
• sample distribution normalization

Innovation

• LieBN
• Riemannian Batch Normalization
• Lie groups
• invariant metrics
• manifold-valued data