LieBN Introduces Batch Normalization for Lie Group Manifolds.

Researchers have introduced LieBN, a novel framework for Riemannian Batch Normalization (RBN) specifically designed for operations over Lie groups. This innovation addresses a gap in machine learning where manifold-valued measurements are common, but existing normalization techniques are often limited to specific geometries or fail to effectively normalize sample distributions on these complex manifolds. LieBN capitalizes on the inherent left- and right-invariant metrics present in every Lie group, providing theoretical guarantees for controlling Riemannian mean and variance. This principled approach allows for robust normalization across diverse geometric structures. The framework has been instantiated and validated across nine distinct geometries, including four on the Symmetric Positive Definite (SPD) manifold, one on the group of rotation matrices, and four on full-rank correlation matrices. This includes the introduction of a new right-invariant metric for SPD manifolds and extensions of existing Lie group structures. Extensive experiments confirm its effectiveness.