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Neural networks on Symmetric Spaces of Noncompact Type
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Proposes a unified point-to-hyperplane distance on symmetric spaces of noncompact type via Busemann functions and builds FC layers and attention from it, validated on image classification, EEG classification, image generation, and NLI.
Problem
Developing neural network building blocks (hyperplanes, FC layers, attention) on symmetric spaces of noncompact type, which generalize hyperbolic spaces and SPD manifolds, where existing point-to-hyperplane distances are restricted to specific families.
Approach
Define hyperplanes via Busemann functions and a unified point-to-hyperplane distance on noncompact-type symmetric spaces; derive a closed form for G-invariant metrics using the Iwasawa decomposition (H map), and use it to build FC layers (orthonormal boundary points as output axes) and a Riemannian-distance-based attention mechanism.
Key results
- On CIFAR-10 the b-distance hybrid ResNet-18 reaches 95.23±0.08% accuracy, above Euclidean-Poincaré-G (95.14±0.11%) and Hybrid Poincaré (95.04±0.13%).
- AttSymSpd-LE achieves 78.24±5.4% on BCIC-IV-2a, 70.96±8.6% on MAMEM-SSVEP-II and 78.02±2.3 AUC on BCI-NER, the best among compared EEG methods.
- AttSymSpd-GI reaches 78.08% on BCIC-IV-2a with about 5x fewer parameters than AttSymSpd-LE (0.007 MB vs 0.034 MB).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use the proposed b-distance instead of g-distance in the Poincaré MLR | Euclidean-Poincaré-B vs Euclidean-Poincaré-G · CIFAR-10 | accuracy: 95.14±0.11 -> 95.23±0.08 | Helped |
| use the proposed b-distance instead of g-distance in the Poincaré MLR | Euclidean-Poincaré-B vs Euclidean-Poincaré-G · CIFAR-100 | accuracy: 77.78±0.09 -> 77.78±0.15 | No clear effect |
| use the proposed b-distance instead of h-distance | Poincaré MLR · CIFAR-10 and CIFAR-100 | accuracy: 81.72±7.84 -> 95.23±0.08; 44.35±2.93 -> 77.78±0.15 | Helped |
| use the Log-Euclidean variant instead of the G-invariant variant | AttSymSpd · MAMEM-SSVEP-II | accuracy: 67.24 ± 7.4 -> 70.96 ± 8.6 | Helped |
| use the Log-Euclidean variant instead of the G-invariant variant | AttSymSpd · BCI-NER | AUC: 75.88 ± 2.2 -> 78.02 ± 2.3 | Helped |
| use the Log-Euclidean variant instead of the G-invariant variant | AttSymSpd · BCIC-IV-2a | accuracy: 78.08 ± 4.8 -> 78.24 ± 5.4 | No clear effect |
| add the attention module (ablation) | AttSymSpd-GI vs CovNet · BCIC-IV-2a | accuracy: 73.04±6.34 -> 78.08±4.8 | Helped |
| use proposed FC layers instead of Bimap layers (ablation) | AttSymSpd-GI vs AttSymSpd-GI-Bimap · BCIC-IV-2a | accuracy: 75.82±5.1 -> 78.08±4.8 | Helped |
| use the proposed b-distance instead of g-distance | Poincaré VAE image generation · CIFAR-10 and CIFAR-100 | mean performance | Helped |
| use the proposed b-distance instead of g-distance | Poincaré MLR for natural language inference · natural language inference | mean performance | Helped |
Surprising findings
- The horocycle-based h-distance performs worse than the other formulations across tasks, attributed to its PCA-like invariance property not matching the g- and b-distances.
- AttSymSpd-GI is outperformed by AttSymSpd-LE on MAMEM-SSVEP-II and BCI-NER despite being much smaller.
Limitations
- Euclidean-Poincaré-B has higher computation time than its competitors
- AttSymSpd-GI underperforms AttSymSpd-LE on two of three EEG datasets
- Image generation and NLI results are only in appendices and summarized briefly
Experimental setup
Datasets: CIFAR-10 CIFAR-100 BCIC-IV-2a MAMEM-SSVEP-II BCI-NER
Benchmarks: CIFAR-10 CIFAR-100 BCIC-IV-2a MAMEM-SSVEP-II BCI-NER
Metrics: accuracy AUC mean accuracy over runs
Baselines: Hybrid Poincaré Poincaré ResNet Euclidean-Poincaré-H Euclidean-Poincaré-G EEG-TCNet MBEEGSE MAtt Graph-CSPNet Bimap layers (AttSymSpd-GI-Bimap) CovNet
Builds on: Poincaré MLR Busemann functions Bimap layers scaled dot product attention geoopt
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