2026-07-13 04:00 UTCOriginal source2 min readUpdated: 2026-07-13 08:15 UTC

LieBN: Batch Normalization over Lie Groups

This paper proposes LieBN, a framework for Riemannian Batch Normalization over Lie groups, leveraging left- and right-invariant metrics for theoretical guarantees. It instantiates across nine geometries, including SPD manifold, rotation matrices, and full-rank correlation matrices, with extensive experimental validation.

SourcearXiv Machine LearningAuthor: Ziheng Chen, Yue Song, Rui Wang, Xiao-Jun Wu, Nicu Sebe

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[Submitted on 13 Jun 2026]

Title:LieBN: Batch Normalization over Lie Groups

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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 this https URL.

Comments: arXiv admin note: text overlap with arXiv:2403.11261

Subjects:

Machine Learning (cs.LG); Artificial Intelligence (cs.AI)

Cite as: arXiv:2607.08783 [cs.LG]

(or arXiv:2607.08783v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2607.08783

arXiv-issued DOI via DataCite

Submission history

From: Ziheng Chen [view email] [v1] Sat, 13 Jun 2026 07:06:42 UTC (6,087 KB)

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