Research map / Learning theory and algorithms / Matrix and tensor methods
Understanding Matrix Function Normalizations in Covariance Pooling through the Lens of Riemannian Geometry
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Explains why matrix power and logarithm normalizations in global covariance pooling work by showing they implicitly define Riemannian (SPD MLR) classifiers rather than tangent classifiers, validated on ImageNet and three FGVC datasets.
Problem
Euclidean classifiers are applied after matrix power normalization in global covariance pooling, but the co-domain of the matrix power is the SPD manifold, so there is no intrinsic justification for this practice.
Approach
Derives Riemannian logarithmic maps under seven metric families and shows that matrix logarithm (LEM) and matrix power (PEM) in GCP implicitly respect SPD multinomial logistic regression classifiers, not tangent-space classifiers; empirically Pow-EMLR beats Pow-TMLR.
Key results
- On Cars, Pow-EMLR reaches 80.43% top-1 versus 51.14% for tangent classifier Pow-TMLR (ResNet-50 setting).
- On Birds, Pow-EMLR top-5 is 84.66% versus 77.80% for Pow-TMLR.
- Pow-EMLR (73% top-1) outperforms Pow-TMLR (71.62%) on ImageNet-1k with ResNet-18, while ScalePow-EMLR performs comparably to Pow-EMLR across datasets.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use Pow-EMLR instead of Pow-TMLR | iSQRT-COV · ImageNet-1k | Top-1 accuracy (%): 71.62 -> 73 | Helped |
| use Pow-EMLR instead of Pow-TMLR | iSQRT-COV · Birds | Top-5 accuracy (%): 77.80 -> 84.66 | Helped |
| use Pow-EMLR instead of Pow-TMLR | iSQRT-COV · Cars | Top-5 accuracy (%): 74.29 -> 94.15 | Helped |
| replace Pow-EMLR with ScalePow-EMLR (add the scalar factor before vectorization) | iSQRT-COV · Cars | Top-1 accuracy (%): 80.43 -> 80.31 | No clear effect |
| use Cho-TMLR instead of Pow-EMLR | iSQRT-COV · Aircrafts | Top-1 accuracy (%): 72.07 vs 78.97 | Mixed |
| use Pow-EMLR instead of Pow-TMLR (ablation) | iSQRT-COV with AlexNet · Cars | Top-5 accuracy (%): 59.51 -> 82.04 | Helped |
| set matrix power to 0.25 instead of 0.5 (ablation) | Pow-EMLR on ResNet-50 · Aircrafts | Top-1 accuracy (%): 74.17 -> 71.47 | Hurt |
| set matrix power to 0.25 instead of 0.5 (ablation) | Pow-EMLR on ResNet-50 · Cars | Top-1 accuracy (%): 62.83 -> 62.88 | Helped |
| set matrix power to 0.7 instead of 0.5 (ablation) | Pow-EMLR on ResNet-50 · Aircrafts | Top-1 accuracy (%): 74.17 -> 74.17 | No clear effect |
| set matrix power to 0.7 instead of 0.5 (ablation) | Pow-EMLR on ResNet-50 · Cars | Top-1 accuracy (%): 62.83 -> 61.41 | Hurt |
Surprising findings
- The tangent-classifier hypothesis, which seemed theoretically favored, is refuted: Pow-TMLR is consistently inferior to Pow-EMLR despite differing only by an affine transformation.
- Cholesky-based tangent classifier Cho-TMLR performs best on Aircrafts but worst on Cars and Birds, attributed to diagonal logarithm over-stretching.
Limitations
- Riemannian classifiers on other metrics are computationally expensive and not yet applied to large-scale GCP.
- Theoretical equivalence of ScalePow-EMLR and Pow-EMLR holds only under scaled initialization and learning rates; experiments do not state a full state-of-the-art comparison.
- Paper explicitly does not aim for state-of-the-art performance over existing GCP methods.
Experimental setup
Datasets: ImageNet-1k Caltech-UCSD Birds FGVC Aircrafts Stanford Cars
Benchmarks: ImageNet-1k FGVC Aircrafts Caltech-UCSD Birds Stanford Cars
Metrics: Top-1 accuracy Top-5 accuracy
Baselines: Pow-TMLR Cho-TMLR ScalePow-EMLR
Builds on: iSQRT-COV SPD MLR DeepO2P Log-Euclidean Metric
Closest papers by meaning
- Learning to Normalize on the SPD Manifold under Bures-Wasserstein Geometry (CVPR 2025 · similarity 0.88)
- RMLR: Extending Multinomial Logistic Regression into General Geometries (NeurIPS 2024 · similarity 0.87)
- Matrix Manifold Neural Networks++ (ICLR 2024 · similarity 0.87)
- Intrinsic Riemannian Cross-covariance for Manifold-valued Random Objects (NeurIPS 2026 · similarity 0.87)
- Building Transformation Layers for Riemannian Neural Networks (NeurIPS 2026 · similarity 0.86)
- A Lie Group Approach to Riemannian Batch Normalization (ICLR 2024 · similarity 0.86)
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