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Flopping for FLOPs: Leveraging Equivariance for Computational Efficiency
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Builds flopping-equivariant ResMLP, ViT and ConvNeXt by splitting features into flopping-invariant and sign-flip (-1)-equivariant irreps, block-diagonalizing linear layers to halve FLOPs and improve throughput at comparable accuracy on ImageNet-1K.
Problem
Equivariant networks improve parameter efficiency but typically cost more FLOPs per parameter than ordinary networks, so they are not compute-competitive at scale.
Approach
Parametrize every feature space by irreps of the flopping group D2 (invariant and sign-flipping features), so linear layers become block-diagonal with half the FLOPs; extend pointwise nonlinearities, layer norm, attention, positional encodings and depthwise convolutions (symmetric/antisymmetric filters) to respect this split.
Key results
- Equivariant ResMLP-L24 reaches 81.5% top-1 with 31.7 GFLOPs vs 80.4% and 63.3 GFLOPs for the non-equivariant baseline on ImageNet-1K.
- E(ConvNeXt-L iso.) matches the baseline at 82.6% top-1 while halving FLOPs (30.3 vs 60.0 GFLOPs) and increasing throughput from 1284 to 1527 im/s.
- E(ViT-H) reaches 84.4% top-1 with 87.3 GFLOPs versus 84.6% and 168.0 GFLOPs for ViT-H; E(ViT-S) failed to converge.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Enforce horizontal-flip equivariance in ResMLP-L24 | ResMLP-L24 · ImageNet-1K | Top-1 accuracy: 80.4 -> 81.5 | Helped |
| Enforce horizontal-flip equivariance in ResMLP-B24 | ResMLP-B24 · ImageNet-1K | Top-1 accuracy: 81.0 -> 80.8 | Hurt |
| Enforce horizontal-flip equivariance in smaller ResMLP models | ResMLP-S24 and ResMLP-T12 · ImageNet-1K | Top-1 accuracy: 79.4 -> 74.9; 76.6 -> 72.0 | Hurt |
| Enforce horizontal-flip equivariance throughout ViT-H | ViT-H · ImageNet-1K | Top-1 accuracy: 84.6 -> 84.4 | Hurt |
| Use equivariant layers only in the first half of ViT-H (ablation) | ViT-H · ImageNet-1K | Top-1 accuracy: 84.6 -> 85.0 | Helped |
| Keep only invariant features in the second half of ViT-H (ablation) | ViT-H · ImageNet-1K | Top-1 accuracy: 84.6 -> 84.7 | Helped |
| Use equivariant layers only in the first half of ViT-L (ablation) | ViT-L · ImageNet-1K | Top-1 accuracy: 84.2 -> 84.5 | Helped |
| Keep only invariant features in the second half of ViT-L (ablation) | ViT-L · ImageNet-1K | Top-1 accuracy: 84.2 -> 84.1 | Hurt |
| Use equivariant layers only in the first half of ViT-B (ablation) | ViT-B · ImageNet-1K | Top-1 accuracy: 83.1 -> 82.9 | Hurt |
| Keep only invariant features in the second half of ViT-B (ablation) | ViT-B · ImageNet-1K | Top-1 accuracy: 83.1 -> 82.5 | Hurt |
| Enforce horizontal-flip equivariance throughout ViT-L and ViT-B | ViT-L and ViT-B · ImageNet-1K | Top-1 accuracy: 84.2 -> 83.4; 83.1 -> 82.2 | Hurt |
| Enforce horizontal-flip equivariance in ConvNeXt-L | ConvNeXt-L (iso.) · ImageNet-1K | Top-1 accuracy: 82.6 -> 82.6 | No clear effect |
Surprising findings
- The equivariant ViT-S failed to converge despite several random seeds.
- Smaller equivariant models lose noticeable accuracy (e.g., E(ResMLP-T12) 72.0% vs 76.6%) and small equivariant models have lower throughput than their baselines.
Limitations
- Only one training run per model and no hyperparameter tuning; baselines taken from original papers with different hardware.
- Equivariant models underperform for small sizes, and some training was unstable (E(ViT-S) did not converge).
- Efficient symmetric/antisymmetric depthwise convolutions and attention kernels were not implemented, so throughput gains are limited.
Experimental setup
Datasets: ImageNet-1K
Benchmarks: ImageNet-1K
Metrics: Top-1 accuracy FLOPs per image throughput (images/s) peak memory parameter count
Baselines: ResMLP-L24 ViT-H ConvNeXt-L (iso.) ViT-L ConvNeXt-B (iso.)
Builds on: ResMLP DeiT III ConvNeXt steerable CNNs
Closest papers by meaning
- A Characterization Theorem for Equivariant Networks with Point-wise Activations (ICLR 2024 · similarity 0.87)
- A Probabilistic Approach to Learning the Degree of Equivariance in Steerable CNNs (ICML 2024 · similarity 0.86)
- On Making $SE(2)$-Invariant Networks Optimal (NeurIPS 2026 · similarity 0.86)
- A Regularization-Guided Equivariant Approach for Image Restoration (CVPR 2025 · similarity 0.85)
- Probing Equivariance and Symmetry Breaking in Convolutional Networks (NeurIPS 2025 · similarity 0.85)
- A Tale of Two Symmetries: Exploring the Loss Landscape of Equivariant Models (NeurIPS 2025 · similarity 0.85)
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