Riemannian Batch Normalization: A Gyro Approach

• URL: http://arxiv.org/abs/2509.07115v2
• Date: Fri, 19 Sep 2025 17:45:21 GMT
• Title: Riemannian Batch Normalization: A Gyro Approach
• Authors: Ziheng Chen, Xiao-Jun Wu, Bernhard Schölkopf, Nicu Sebe,
• Abstract summary: We introduce GyroBN, a principled Riemannian batch normalization framework for gyrogroups.<n>We show that GyroBN can be used to extend Euclidean neural networks to non-Euclidean domains.<n>We further instantiate GyroBN on seven representative geometries, including the Grassmannian, five constant curvature spaces, and the correlation manifold.
• Score: 102.84784139051526
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: Normalization layers are crucial for deep learning, but their Euclidean formulations are inadequate for data on manifolds. On the other hand, many Riemannian manifolds in machine learning admit gyro-structures, enabling principled extensions of Euclidean neural networks to non-Euclidean domains. Inspired by this, we introduce GyroBN, a principled Riemannian batch normalization framework for gyrogroups. We establish two necessary conditions, namely \emph{pseudo-reduction} and \emph{gyroisometric gyrations}, that guarantee GyroBN with theoretical control over sample statistics, and show that these conditions hold for all known gyrogroups in machine learning. Our framework also incorporates several existing Riemannian normalization methods as special cases. We further instantiate GyroBN on seven representative geometries, including the Grassmannian, five constant curvature spaces, and the correlation manifold, and derive novel gyro and Riemannian structures to enable these instantiations. Experiments across these geometries demonstrate the effectiveness of GyroBN. The code is available at https://github.com/GitZH-Chen/GyroBN.git.

Related papers

• RMLR: Extending Multinomial Logistic Regression into General Geometries [64.16104856124029]
  Our framework only requires minimal geometric properties, thus exhibiting broad applicability. We develop five families of SPD MLRs under five types of power-deformed metrics. On rotation matrices we propose Lie MLR based on the popular bi-invariant metric.
  arXiv  Detail & Related papers  (2024-09-28T18:38:21Z)
• A Lie Group Approach to Riemannian Batch Normalization [59.48083303101632]
  This paper establishes a unified framework for normalization techniques on Lie groups. We focus on Symmetric Positive Definite (SPD), which possess three distinct types of Lie group structures. Specific normalization layers induced by these Lie groups are then proposed for SPD neural networks.
  arXiv  Detail & Related papers  (2024-03-17T16:24:07Z)
• DeepRicci: Self-supervised Graph Structure-Feature Co-Refinement for Alleviating Over-squashing [72.70197960100677]
  Graph Structure Learning (GSL) plays an important role in boosting Graph Neural Networks (GNNs) with a refined graph. GSL solutions usually focus on structure refinement with task-specific supervision (i.e., node classification) or overlook the inherent weakness of GNNs themselves. We propose to study self-supervised graph structure-feature co-refinement for effectively alleviating the issue of over-squashing in typical GNNs.
  arXiv  Detail & Related papers  (2024-01-23T14:06:08Z)
• Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approach [8.003578990152945]
  We propose new models and layers for building neural networks on SPD and Grassmann manifold. We show the effectiveness of our approach in two applications, i.e., human action recognition and knowledge graph completion.
  arXiv  Detail & Related papers  (2023-05-08T09:10:11Z)
• A singular Riemannian geometry approach to Deep Neural Networks I. Theoretical foundations [77.86290991564829]
  Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. We study a particular sequence of maps between manifold, with the last manifold of the sequence equipped with a Riemannian metric. We investigate the theoretical properties of the maps of such sequence, eventually we focus on the case of maps between implementing neural networks of practical interest.
  arXiv  Detail & Related papers  (2021-12-17T11:43:30Z)
• Vector-valued Gaussian Processes on Riemannian Manifolds via Gauge Equivariant Projected Kernels [108.60991563944351]
  We present a recipe for constructing gauge equivariant kernels, which induce vector-valued Gaussian processes coherent with geometry. We extend standard Gaussian process training methods, such as variational inference, to this setting.
  arXiv  Detail & Related papers  (2021-10-27T13:31:10Z)
• Semi-Riemannian Graph Convolutional Networks [36.09315878397234]
  We develop a principled Semi-Riemannian GCN that first models data in semi-Riemannian manifold of constant nonzero curvature. Our method provides a geometric inductive bias that is sufficiently flexible to model mixed heterogeneous topologies like hierarchical graphs with cycles.
  arXiv  Detail & Related papers  (2021-06-06T14:23:34Z)
• Nested Grassmannians for Dimensionality Reduction with Applications [7.106986689736826]
  We propose a novel framework for constructing a nested sequence of homogeneous Riemannian manifold. We focus on applying the proposed framework to the Grassmann manifold, giving rise to the nested Grassmannians (NG) Specifically, each planar (2D) shape can be represented as a point in the complex projective space which is a complex Grass-mann manifold. With the proposed NG structure, we develop algorithms for the supervised and unsupervised dimensionality reduction problems respectively.
  arXiv  Detail & Related papers  (2020-10-27T20:09:12Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.