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A Lie Group Approach to Riemannian Batch Normalization
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Proposes LieBN, a general Riemannian batch normalization framework on Lie groups with left-invariant metrics that controls both Fréchet mean and variance, instantiated on SPD manifolds via three deformed families of metrics for SPD neural networks.
Problem
Existing Riemannian batch normalization methods are ad hoc, limited to specific manifolds or metrics, and do not principled control both Riemannian mean and variance; a general normalization for manifold-valued data is needed.
Approach
Define Lie group batch normalization using left translation for centering and biasing, and scaling in the tangent space at the neutral element via the Riemannian exponential/log maps, with a Riemannian Gaussian; prove that sample mean and dispersion are controlled, and generalize SPD Lie group metrics (AIM, LEM, LCM) into deformed parameterized families computed by pullback to a Euclidean codomain
Key results
- LieBN on SPDNet improves accuracy over SPDNet by 2.22% on Radar (AIM-(1)), 11.71% on HDM05 (LCM-(0.5)), and 4.8% on FPHA (AIM-(1.5)).
- TSMNet+DSMLieBN-AIM reaches 55.10% inter-session and 53.97% inter-subject balanced accuracy, improving SPDDSMBN by 0.98% and 3.87%.
- LieBN is comparable or more efficient than SPDNetBN in fit time, with LEM- and LCM-based variants faster.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add LieBN with AIM-(1) to SPDNet | SPDNet · Radar | accuracy: +2.22% | Helped |
| Add LieBN with LCM-(0.5) to SPDNet | SPDNet · HDM05 | accuracy: +11.71% | Helped |
| Add LieBN with AIM-(1.5) to SPDNet | SPDNet · FPHA | accuracy: +4.8% | Helped |
| Use LCM-(0.5) rather than LCM-(1) (ablation) | SPDNet with LieBN on FPHA · FPHA | accuracy: 77.64±1.00 -> 86.33±0.43 | Helped |
| Use AIM-(1.5) rather than AIM-(1) (ablation) | SPDNet with LieBN on HDM05 · HDM05 | accuracy: 67.79±0.65 -> 68.16±0.68 | Helped |
| Use AIM-(1.5) rather than AIM-(1) (ablation) | SPDNet with LieBN on FPHA · FPHA | accuracy: 89.70±0.51 -> 90.39±0.66 | Helped |
| Use LCM-(0.5) rather than LCM-(1) (ablation) | SPDNet with LieBN on HDM05 · HDM05 | accuracy: 66.68±0.71 -> 70.84±0.92 | Helped |
| Use LCM-(0.5) rather than LCM-(1) (ablation) | TSMNet with DSMLieBN on Hinss2021 inter-session · Hinss2021 inter-session | balanced accuracy: 51.54±6.88 -> 53.11±5.65 | Helped |
| Use AIM-(-0.5) rather than AIM-(1) (ablation) | TSMNet with DSMLieBN on Hinss2021 inter-subject · Hinss2021 inter-subject | balanced accuracy: 50.04±8.01 -> 53.97±8.78 | Helped |
| Add DSMLieBN AIM-(1) rather than SPDDSMBN | TSMNet on Hinss2021 inter-session · Hinss2021 inter-session | balanced accuracy: +0.98% | Helped |
| Add DSMLieBN AIM-(-0.5) rather than SPDDSMBN | TSMNet on Hinss2021 inter-subject · Hinss2021 inter-subject | balanced accuracy: +3.87% | Helped |
| Vary β across the tested values (ablation) | LieBN AIM-(1.5) on SPDNet · HDM05 | accuracy: 68.18±0.86; 68.12±0.74; 68.20±0.85; 68.18±0.85; 68.16±0.80; 68.16±0.68 | No clear effect |
Surprising findings
- LCM-based LieBN at the standard setting (theta=1) impedes SPDNet learning on FPHA (77.64%) but is among the best on HDM05 with deformation.
- The best metric differs per dataset, so no single Lie group metric dominates across the three tasks.
Limitations
- Experiments restricted to SPD manifolds; the rotation-matrix (SO(n)) study is only preliminary.
- Population variance after scaling is generally agnostic; closed-form normalization holds only for some metrics (LEM, LCM).
- Best metric and deformation factor vary by dataset and must be selected by hyperparameter search.
Experimental setup
Datasets: Radar HDM05 FPHA Hinss2021
Benchmarks: Radar HDM05 FPHA Hinss2021
Metrics: accuracy balanced accuracy fit time
Baselines: SPDNet SPDNetBN SPDDSMBN TSMNet
Builds on: SPDNet TSMNet SPDNetBN (Riemannian batch normalization) Riemannian Gaussian distribution
Closest papers by meaning
- Learning to Normalize on the SPD Manifold under Bures-Wasserstein Geometry (CVPR 2025 · similarity 0.88)
- Matrix Manifold Neural Networks++ (ICLR 2024 · similarity 0.86)
- Building Transformation Layers for Riemannian Neural Networks (NeurIPS 2026 · similarity 0.86)
- RMLR: Extending Multinomial Logistic Regression into General Geometries (NeurIPS 2024 · similarity 0.86)
- Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry (ICLR 2025 · similarity 0.86)
- Riemannian Multinomial Logistics Regression for SPD Neural Networks (CVPR 2024 · similarity 0.86)
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