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Fully Hyperbolic Convolutional Neural Networks for Computer Vision
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Proposes HCNN, a fully hyperbolic CNN in the Lorentz model with new formulations of convolution, batch normalization, and MLR, evaluated on ResNet-18 image classification and VAE generation.
Problem
Existing hyperbolic neural networks in computer vision use Euclidean backbones and only map features to hyperbolic space in task heads, and missing Lorentz-model components (2D convolution, batch norm, MLR) prevent fully hyperbolic vision encoders.
Approach
Generalize CNN components to the Lorentz model: a Lorentz convolutional layer (linearized kernel over HCat of receptive field), batch normalization using the Lorentzian centroid plus parallel transport and tangent-space rescaling, and a Lorentz MLR based on distances to hyperplanes, enabling per-layer hyperbolic features in a ResNet and VAE.
Key results
- HCNN Lorentz ResNet-18 reaches 78.07% on CIFAR-100 and 65.71% on Tiny-ImageNet, versus 77.72% and 65.19% for the Euclidean baseline; the hybrid HECNN reaches 78.76% and 65.96%.
- Under PGD attack at epsilon 3.2/255 on CIFAR-100, HCNN achieves 31.77% accuracy versus 26.30% for the Euclidean model.
- HCNN-VAE achieves the best generation FID on CIFAR-10 (89.20 vs 92.40 Euclidean) and CelebA (78.11 vs 79.25), but not on CIFAR-100 where the hybrid Poincaré is better.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace the Euclidean ResNet encoder blocks with Lorentz blocks 1 and 3 while using a Lorentz decoder (HECNN) | ResNet-18 · CIFAR-100 | accuracy (%): 77.72_{\pm0.15} -> 78.76_{\pm0.24} | Helped |
| Use the hybrid Lorentz encoder and decoder instead of the Euclidean model | ResNet-18 · Tiny-ImageNet | accuracy (%): 65.19_{ \pm0.12} -> 65.63_{\pm0.10} | Helped |
| Use a fully hyperbolic Lorentz ResNet instead of the Euclidean ResNet | ResNet-18 · CIFAR-100 | accuracy (%): 77.72_{\pm0.15} -> 78.07_{\pm0.17} | Helped |
| Replace the hybrid Lorentz decoder-only model with the fully hyperbolic Lorentz model | ResNet-18 · CIFAR-100 | accuracy (%): 78.03_{\pm0.21} -> 78.07_{\pm0.17} | No clear effect |
| Use a Lorentz hyperbolic VAE instead of a Euclidean VAE | VAE · CIFAR-10 generation | FID: 92.40_{ \pm 0.80} -> 89.20_{\pm0.85} | Helped |
| Use a Lorentz hyperbolic VAE instead of the strongest baseline | VAE · CIFAR-100 generation | FID: 98.19_{\pm0.57} -> 100.27_{\pm0.84} | Hurt |
| Use hyperbolic latent distributions in a hybrid VAE instead of a Euclidean latent space | VAE · CIFAR-10 generation | FID: 92.40_{ \pm 0.80} -> 90.13_{\pm0.77} | Helped |
| Use a Lorentz hyperbolic ResNet instead of a Euclidean ResNet under FGSM ε=3.2/255 | ResNet-18 · CIFAR-100 FGSM ε=3.2/255 | accuracy (%): 39.97_{\pm 0.43} -> 43.51_{\pm 0.35} | Helped |
| Use a Lorentz hyperbolic ResNet instead of a Euclidean ResNet under PGD ε=3.2/255 | ResNet-18 · CIFAR-100 PGD ε=3.2/255 | accuracy (%): 26.30_{\pm 0.40} -> 31.77_{\pm 0.55} | Helped |
| Use a hybrid Lorentz decoder-only model instead of a Euclidean model under CIFAR-100 attacks | ResNet-18 · CIFAR-100 adversarial attacks | accuracy (%) | Mixed |
| Reduce the final ResNet block and embedding dimensionality | ResNet-18 · CIFAR-100 low-dimensional classification | accuracy | Helped |
Surprising findings
- Poincaré-based hyperbolic ResNets are consistently worse than the Euclidean baseline, and hybrid Poincaré decoders are more susceptible to adversarial attacks than Euclidean models.
- The hybrid encoder HECNN outperforms the fully hyperbolic HCNN on CIFAR-100 with low embedding dimensions, contrary to the hypothesis that the fully hyperbolic model would perform best.
Limitations
- Hyperbolic CNNs introduce mathematical complexity and computational overhead.
- Generalized layers were designed for Euclidean geometry and may not fully capture hyperbolic properties.
- Evaluation is limited to small-scale image classification and VAE generation; optimization and scalability are open questions.
Experimental setup
Datasets: CIFAR-10 CIFAR-100 Tiny-ImageNet CelebA MNIST
Benchmarks: CIFAR-10 CIFAR-100 Tiny-ImageNet CelebA
Metrics: classification accuracy FGSM accuracy PGD accuracy reconstruction FID generation FID
Baselines: Euclidean ResNet-18 Hybrid Poincaré Hybrid Lorentz Poincaré ResNet Euclidean VAE Hybrid Poincaré VAE Hybrid Lorentz VAE
Builds on: Hyperbolic Neural Networks Poincaré MLR Lorentz fully-connected layer Riemannian batch normalization wrapped normal distribution
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