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The Lie Derivative for Measuring Learned Equivariance
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add spatial aliasing-aware design? No: finetune a pretrained MAE ViT on rotated MNIST instead of using rotation-equivariant architecture | ViT (MAE pretrained) · RotMNIST | test error (%): 1.14 vs 0.68 for best equivariant CNN | No clear effect |
| Apply pretrained large-scale ViT finetuning instead of architectural rotation equivariance | ViT · RotMNIST | test error (%): beats G-CNN, H-NET, ORN, TI-Pooling, RotEqNet but not E(2)-CNN | Mixed |
| Measure translation equivariance with Lie derivative (LEE): larger, more accurate models are more equivariant regardless of architecture | 410 pretrained classification models · ImageNet test set | translation LEE vs test accuracy | Helped |
| Add BlurPool low-pass filtering to ResNet-50 (ablation) | ResNet-50 · ImageNet test set | translation equivariance error | Helped |
| Improve training recipe (Mixup/CutMix-style augmentation) for ResNeXt-50 instead of architecture change (ablation) | ResNeXt-50 · ImageNet test set | translation equivariance error | Helped |
| Pretrain ResNeXt-50 with semi-supervised learning on more data (ablation) | ResNeXt-50 · ImageNet test set | translation equivariance error | Helped |
| Increase training dataset size for ResMLP Big (ablation) | ResMLP Big · ImageNet test set | translation equivariance error | Helped |
| Increase hyperbolic-rotation/brightening accuracy-linked equivariance: more accurate models more equivariant to hyperbolic rotation | pretrained classification models · ImageNet test set | hyperbolic rotation Lie derivative norm | Helped |
| Stretch and brightening transformations: LEE shows no significant correlation with generalization | pretrained classification models · ImageNet test set | Lie derivative norm vs accuracy | No clear effect |
| Use integer translation metric instead of Lie derivative for layerwise analysis (ablation) | layerwise equivariance analysis · ResNet-50 ImageNet | layer contribution to equivariance error | Hurt |
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