发表机构
Department of Information Engineering and Computer Science, University of Trento; Computing and Mathematical Sciences, Caltech; School of Artificial Intelligence and Computer Science, Jiangnan University（信息工程与计算机科学系，特伦托大学; 计算与数学科学系，加州理工学院; 人工智能与计算机科学学院，江南大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对流形值测量在机器学习任务中的问题，提出LieBN框架用于李群上的黎曼批量归一化，利用左右不变度量，在多种几何结构上实例化并经实验验证有效。
AI 中文摘要
流形值测量在各种机器学习任务中很普遍。近期进展将深度神经网络扩展到在流形上运行，同时有针对不同几何结构的归一化技术，即黎曼归一化。但现有多数方法专为特定流形设计或无法有效归一化流形值样本分布。为此提出LieBN，一种李群上的黎曼批量归一化框架。利用李群中自然存在的左右不变度量，为控制黎曼均值和方差提供理论保证。在九种不同几何结构上实例化LieBN并通过实验验证其有效性。
英文摘要
Manifold-valued measurements are prevalent in various machine learning tasks. Recent advances have extended Deep Neural Networks (DNNs) to operate on manifolds. These extensions have been 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. Experiments on different manifolds validate the effectiveness of our framework. The code is available at https://github.com/GitZH-Chen/LieBN.git.
CommentsExtended version of the ICLR 2024 paper: A Lie Group Approach to Riemannian Batch Normalization. arXiv admin note: substantial text overlap with arXiv:2403.11261
