# Learning to Normalize on the SPD Manifold under Bures-Wasserstein Geometry

arXiv: https://arxiv.org/abs/2504.00660 · HTML: https://mlsanity.com/paper/2504.00660/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 57 references that are in the map.
- First posted on arXiv 1 Apr 2025.

## Builds on (most cited first, 24 of 24)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [NTU RGB+D: A Large Scale Dataset for 3D Human Activity Analysis](https://mlsanity.com/paper/1604.02808/) (2016)
- [A Riemannian Network for SPD Matrix Learning](https://mlsanity.com/paper/1608.04233/) (2016)
- [Riemannian batch normalization for SPD neural networks](https://mlsanity.com/paper/1909.02414/) (2019)
- [EEG-TCNet: An Accurate Temporal Convolutional Network for Embedded Motor-Imagery Brain-Machine Interfaces](https://mlsanity.com/paper/2006.00622/) (2020)
- [FBCNet: A Multi-view Convolutional Neural Network for Brain-Computer Interface](https://mlsanity.com/paper/2104.01233/) (2021)
- [SPD domain-specific batch normalization to crack interpretable unsupervised domain adaptation in EEG](https://mlsanity.com/paper/2206.01323/) (2022)
- [MAtt: A Manifold Attention Network for EEG Decoding](https://mlsanity.com/paper/2210.01986/) (2022)
- [Hypformer: Exploring Efficient Transformer Fully in Hyperbolic Space](https://mlsanity.com/paper/2407.01290/) (2024)
- [On Riemannian Optimization over Positive Definite Matrices with the Bures-Wasserstein Geometry](https://mlsanity.com/paper/2106.00286/) (2021)
- [Training Deep Networks with Structured Layers by Matrix Backpropagation](https://mlsanity.com/paper/1509.07838/) (2015)
- [GeomNet: A Neural Network Based on Riemannian Geometries of SPD Matrix Space and Cholesky Space for 3D Skeleton-Based Interaction Recognition](https://mlsanity.com/paper/2111.13089/) (2021)
- [Riemannian Local Mechanism for SPD Neural Networks](https://mlsanity.com/paper/2201.10145/) (2022)
- [Riemannian Residual Neural Networks](https://mlsanity.com/paper/2310.10013/) (2023)
- [A Statistical Recurrent Model on the Manifold of Symmetric Positive Definite Matrices](https://mlsanity.com/paper/1805.11204/) (2018)
- [Building Neural Networks on Matrix Manifolds: A Gyrovector Space Approach](https://mlsanity.com/paper/2305.04560/) (2023)
- [Adaptive Log-Euclidean Metrics for SPD Matrix Learning](https://mlsanity.com/paper/2303.15477/) (2023)
- [Matrix Manifold Neural Networks++](https://mlsanity.com/paper/2405.19206/) (2024)
- [Learning with symmetric positive definite matrices via generalized Bures-Wasserstein geometry](https://mlsanity.com/paper/2110.10464/) (2021)
- [Analyzing Dynamical Brain Functional Connectivity As Trajectories on Space of Covariance Matrices](https://mlsanity.com/paper/1904.05449/) (2019)
- [Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry](https://mlsanity.com/paper/2407.10484/) (2024)
- [RMLR: Extending Multinomial Logistic Regression into General Geometries](https://mlsanity.com/paper/2409.19433/) (2024)

## Built on by (most cited first, 7 of 7)

- [ReManNet: A Riemannian Manifold Network for Monocular 3D Lane Detection](https://mlsanity.com/paper/2603.19776/) (2026)
- [Batch Normalization for Neural Networks on Complex Domains](https://mlsanity.com/paper/2605.00467/) (2026)
- [Riemannian Networks over Full-Rank Correlation Matrices](https://mlsanity.com/paper/2605.19073/) (2026)
- [A Sliced-Wasserstein Framework on Correlation Matrices for EEG Decoding](https://mlsanity.com/paper/2606.06104/) (2026)
- [Towards Robust EEG Decoding Based on Riemannian Self-Attention](https://mlsanity.com/paper/2606.25456/) (2026)
- [ITSPACE: Monotone Gaussian Optimal Transport Updates](https://mlsanity.com/paper/2606.30523/) (2026)
- [Geometric Feature Learning for Functional Data Valued on the Symmetric Positive Definite Manifold](https://mlsanity.com/paper/2609.30487/) (2026)

## Abstract (arXiv)

Covariance matrices have proven highly effective across many scientific fields. Since these matrices lie within the Symmetric Positive Definite (SPD) manifold - a Riemannian space with intrinsic non-Euclidean geometry, the primary challenge in representation learning is to respect this underlying geometric structure. Drawing inspiration from the success of Euclidean deep learning, researchers have developed neural networks on the SPD manifolds for more faithful covariance embedding learning. A notable advancement in this area is the implementation of Riemannian batch normalization (RBN), which has been shown to improve the performance of SPD network models. Nonetheless, the Riemannian metric beneath the existing RBN might fail to effectively deal with the ill-conditioned SPD matrices (ICSM), undermining the effectiveness of RBN. In contrast, the Bures-Wasserstein metric (BWM) demonstrates superior performance for ill-conditioning. In addition, the recently introduced Generalized BWM (GBWM) parameterizes the vanilla BWM via an SPD matrix, allowing for a more nuanced representation of vibrant geometries of the SPD manifold. Therefore, we propose a novel RBN algorithm based on the GBW geometry, incorporating a learnable metric parameter. Moreover, the deformation of GBWM by matrix power is also introduced to further enhance the representational capacity of GBWM-based RBN. Experimental results on different datasets validate the effectiveness of our proposed method.