Research map / Learning theory and algorithms / Matrix and tensor methods
Learning to Normalize on the SPD Manifold under Bures-Wasserstein Geometry
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Proposes GBWBN, a Riemannian batch normalization on the SPD manifold using a learnable generalized Bures-Wasserstein metric with matrix-power deformation, evaluated as a plug-in for SPDNet and RResNet on EEG and skeleton action recognition.
Problem
Existing Riemannian batch normalization on SPD manifolds (AIM-based) struggles with ill-conditioned SPD matrices, which are prevalent in covariance-based representations.
Approach
Derives batch normalization under the Bures-Wasserstein metric and generalizes it to a learnable GBWM parameterized by an SPD matrix M, further deformed by matrix power, so the normalization geometry adapts to the data and handles ill-conditioning better than AIM.
Key results
- SPDNet-GBWBN reaches 65.47% accuracy on MAMEM-SSVEP-II vs 62.76% for SPDNet-BN(m) and 62.30% for vanilla SPDNet.
- RResNet-GBWBN with theta=0.5 achieves 69.10% on HDM05 vs 61.09% for vanilla RResNet.
- RResNet-GBWBN with theta=1 reaches 59.45% on NTU RGB+D vs 52.54% for vanilla RResNet and 53.86% for RResNet-BN(m).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add GBWBN to SPDNet | SPDNet · MAMEM-SSVEP-II | accuracy: 62.30 -> 65.47 | Helped |
| Add GBWBN with θ=0.5 to RResNet | RResNet · HDM05 | accuracy: 61.09 -> 69.10 | Helped |
| Add GBWBN with θ=0.5 to RResNet | RResNet · NTU RGB+D | accuracy: 52.54 -> 59.72 | Helped |
| Use θ=0.5 rather than θ=1 (ablation) | RResNet-GBWBN · HDM05 | accuracy: 62.01 -> 69.10 | Helped |
| Use θ=0.5 rather than θ=1 (ablation) | RResNet-GBWBN · NTU RGB+D | accuracy: 59.45 -> 59.72 | Helped |
| Remove the BiMap layer (ablation) | RResNet-GBWBN · NTU RGB+D | accuracy: 59.45 ± 0.38 -> 57.32 ± 0.43 | Hurt |
| Replace GBWBN with fixed BWM-based BWBN (ablation) | RResNet · HDM05 | accuracy: 62.01 ± 1.23 -> 61.94 ± 0.62 | Hurt |
| Replace GBWBN with fixed BWM-based BWBN (ablation) | RResNet · NTU RGB+D | accuracy: 59.45 ± 0.38 -> 57.40 ± 0.52 | Hurt |
| Set θ=0.25 rather than θ=1 (ablation) | RResNet-GBWBN · HDM05 | accuracy: 62.01 ± 1.23 -> 65.28 ± 1.03 | Helped |
| Set θ=0.75 rather than θ=1 (ablation) | RResNet-GBWBN · HDM05 | accuracy: 62.01 ± 1.23 -> 62.89 ± 0.74 | Helped |
| Set θ=0.25 rather than θ=1 (ablation) | RResNet-GBWBN · NTU RGB+D | accuracy: 59.45 ± 0.38 -> 58.23 ± 0.28 | Hurt |
| Set θ=0.75 rather than θ=1 (ablation) | RResNet-GBWBN · NTU RGB+D | accuracy: 59.45 ± 0.38 -> 59.82 ± 0.27 | Helped |
Surprising findings
- Ill-conditioned SPD features (condition number above 1e3) remain prevalent after standard regularization in many settings, with HDM05 and NTU RGB+D showing near 100% ill-conditioned features before and after the RBN layer under the tested lambda.
Limitations
- Slightly higher training cost from SVD operations than other RBN methods
- Best power parameter theta varies by dataset and must be tuned (e.g., 0.5 for HDM05 vs 1 for SPDNet on EEG)
- Evaluated only on two signal classification tasks with two SPD backbones
Experimental setup
Datasets: MAMEM-SSVEP-II HDM05 NTU RGB+D
Benchmarks: MAMEM-SSVEP-II HDM05 NTU RGB+D
Metrics: accuracy training time (s/epoch) condition number statistics
Baselines: SPDNet-BN SPDNet-BN(m+v) SPDNet-Manifoldnorm SPDNet-LieBN RResNet-LieBN EEGNet ShallowConvNet SCCNet EEG-TCNet FBCNet TCNet-Fusion MBEEGSE
Builds on: SPDNet RResNet LieBN Riemannian batch normalization (SPDBN) generalized Bures-Wasserstein metric
Closest papers by meaning
- Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry (ICLR 2025 · similarity 0.88)
- A Lie Group Approach to Riemannian Batch Normalization (ICLR 2024 · similarity 0.88)
- RMLR: Extending Multinomial Logistic Regression into General Geometries (NeurIPS 2024 · similarity 0.87)
- Building Transformation Layers for Riemannian Neural Networks (NeurIPS 2026 · similarity 0.86)
- Matrix Manifold Neural Networks++ (ICLR 2024 · similarity 0.86)
- Riemannian Multinomial Logistics Regression for SPD Neural Networks (CVPR 2024 · similarity 0.85)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.