AI Insight
This paper introduces LieBN, a novel framework for batch normalization in deep neural networks that operate on Lie groups and manifolds. The authors propose a mathematically rigorous approach using left- and right-invariant metrics that provides theoretical guarantees for controlling mean and variance on curved geometric spaces. They demonstrate the framework's applicability across nine different geometric structures, including symmetric positive definite matrices, rotation matrices, and correlation matrices.
Why it matters
Many real-world machine learning applications involve data that naturally lives on curved manifolds rather than flat Euclidean spaces, such as covariance matrices in computer vision or rotational data in robotics. This work provides a unified and theoretically grounded method for normalizing such data in deep learning models, potentially improving performance in applications involving geometric data structures.
Understand the Science
arXiv:2607.08783v2 Announce Type: replace-cross
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.