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Siegel Neural Networks
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Builds discriminative Siegel neural networks with new MLR layers (via quotient-structure point-to-hyperplane distances and vector-valued distances) and affine/dimensionality-reduction FC layers, evaluated on radar clutter and UCI node classification.
Problem
Discriminative neural networks with manifold-valued outputs on Siegel spaces, a family of Riemannian symmetric spaces that has been little explored for classification; existing Siegel-based work only learns embeddings.
Approach
Derives closed-form point-to-hyperplane distances on Siegel spaces using the quotient structure (Sp_2m/SpO_2m) and proposes a second MLR formulation based on vector-valued distances (with an upper bound), plus two FC layers (affine group-action AFC and Stiefel-based DFC) that always output Siegel points.
Key results
- On radar dataset 1, SiegelNet-AFC-QMLR reaches 80.94% vs 76.22% for kNN (second best), a 4.71-point margin.
- On node classification Glass, SiegelNet-AFC-QMLR_SH_m reaches 45.79% vs 41.54% for the LogEig classifier.
- SiegelNet-QMLR_SH_m improves over SiegelNet-BMLR by 2.06%, 0.58%, and 5.57% on Glass, Iris, Zoo.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SiegelNet-AFC-QMLR instead of kNN | Radar clutter classification · Dataset 1 | accuracy: 76.22$\pm$0.0 -> 80.94$\pm$0.14 | Helped |
| Use SiegelNet-AFC-QMLR instead of kNN | Radar clutter classification · Dataset 2 | accuracy: 93.00$\pm$0.0 -> 96.50$\pm$0.12 | Helped |
| Use SiegelNet-AFC-QMLR instead of kNN | Radar clutter classification · Dataset 3 | accuracy: 76.75$\pm$0.0 -> 91.00$\pm$0.18 | Helped |
| Use SiegelNet-AFC-QMLR instead of kNN | Radar clutter classification · Dataset 4 | accuracy: 73.20$\pm$0.0 -> 85.60$\pm$0.06 | Helped |
| Replace AFC with DFC (ablation) | SiegelNet-QMLR on radar clutter classification · Datasets 1–4 | accuracy: 80.94$\pm$0.14 -> 40.78$\pm$0.23; 96.50$\pm$0.12 -> 82.70$\pm$0.18; 91.00$\pm$0.18 -> 74.88$\pm$0.21; 85.60$\pm$0.06 -> | Hurt |
| Add AFC layer to SiegelNet-QMLR (ablation) | SiegelNet-QMLR · Glass | accuracy: 42.61$\pm$3.26 -> 45.79$\pm$4.66 | Helped |
| Add AFC layer to SiegelNet-QMLR (ablation) | SiegelNet-QMLR · Iris | accuracy: 37.52$\pm$2.54 -> 38.20$\pm$3.03 | Helped |
| Add AFC layer to SiegelNet-QMLR (ablation) | SiegelNet-QMLR · Zoo | accuracy: 52.00$\pm$4.78 -> 53.37$\pm$4.23 | Helped |
| Use QMLR instead of BMLR (ablation) | SiegelNet without FC layers · Glass | accuracy: 40.55$\pm$3.50 -> 42.61$\pm$3.26 | Helped |
| Use QMLR instead of BMLR (ablation) | SiegelNet without FC layers · Iris | accuracy: 36.94$\pm$2.09 -> 37.52$\pm$2.54 | Helped |
| Use QMLR instead of BMLR (ablation) | SiegelNet without FC layers · Zoo | accuracy: 46.43$\pm$3.64 -> 52.00$\pm$4.78 | Helped |
| Use vector-valued-distance MLR instead of quotient-distance MLR (ablation) | SiegelNet node classifier · Glass, Iris, and Zoo | accuracy | Mixed |
Surprising findings
- The DFC layer performs notably worse than the AFC layer on radar data (e.g., 40.78% vs 80.94% on dataset 1).
- The reparameterized vector-valued-distance MLR (VMLR) with upper-bound distance is outperformed by the QMLR formulation despite its smaller model size.
Limitations
- Siegel hyperplane formulation suffers from over-parameterization; the reparameterized VMLR variant is less accurate.
- Operations on Siegel spaces (eigendecomposition-based distances) are computationally expensive, especially for high-dimensional signals and large graphs.
- Only shallow networks evaluated; convolution, batch normalization, pooling and attention layers are not studied.
Experimental setup
Datasets: simulated radar clutter (AR Gaussian time series) UCI Glass UCI Iris UCI Zoo
Benchmarks: radar clutter classification (4 simulated datasets) UCI Glass UCI Iris UCI Zoo node classification
Metrics: mean accuracy standard deviation over 10 runs
Baselines: kNN SPDNet SPDNetBN MLR-AI GyroSpd++ LogEig classifier SiegelNet-BFC-BMLR SiegelNet-BMLR
Builds on: gyrovector hyperplanes (Ganea et al.) SPDNet Bimap layer Busemann-function MLR Kähler distance kNN
Closest papers by meaning
- Riemannian Multinomial Logistics Regression for SPD Neural Networks (CVPR 2024 · similarity 0.90)
- Neural networks on Symmetric Spaces of Noncompact Type (ICLR 2025 · similarity 0.88)
- RMLR: Extending Multinomial Logistic Regression into General Geometries (NeurIPS 2024 · similarity 0.86)
- Matrix Manifold Neural Networks++ (ICLR 2024 · similarity 0.86)
- Building Transformation Layers for Riemannian Neural Networks (NeurIPS 2026 · similarity 0.86)
- Hyperbolic-PDE GNN: Spectral Graph Neural Networks in the Perspective of A System of Hyperbolic Partial Differential Equations (ICML 2025 · similarity 0.85)
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