LieBN: Batch normalization over Lie groups for 9 geometries

Manifold-valued measurements are common in modern ML tasks like computer vision and robotics, but normalizing such data has been a challenge. Existing Riemannian normalization methods are often tailored to specific manifolds or fail to properly center and scale sample distributions. To address this, researchers from multiple institutions propose LieBN, a unified batch normalization framework designed for Lie groups. The key innovation is the use of left- and right-invariant metrics — mathematically natural choices that exist in every Lie group — which provide theoretical guarantees for controlling the Riemannian mean and variance. The framework is instantiated over nine different geometries, including four formulations on the Symmetric Positive Definite (SPD) manifold, one on the group of rotation matrices (SO(3)), and four on the manifold of full-rank correlation matrices. Notably, the authors introduce a novel right-invariant metric for the SPD manifold and extend three existing Lie group structures using matrix power deformation, an algebraic trick that preserves group properties.

Extensive experiments across these geometries demonstrate the effectiveness of LieBN over previous ad-hoc normalization techniques. The open-source implementation gives practitioners a drop-in replacement for standard batch normalization when working with manifold-valued features. This could significantly improve training stability and convergence in geometric deep learning models, such as those used for 3D pose estimation, covariance descriptors, or diffusion MRI analysis. By providing a generic, theoretically grounded normalization method for Lie groups, LieBN removes a major barrier in applying deep neural networks to non-Euclidean data. Code is available on GitHub.