Research map / Architectures and efficiency / CNN design

On Making $SE(2)$-Invariant Networks Optimal

NeurIPS 2026 · Architectures and efficiency › CNN design · approach: Group symmetry equivariant networks

Read the paperCheck your idea against 108,967 papers

AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.

Derives provably minimal, discriminative SE(2)-invariant moment features from classical invariant theory and packages them as a plug-and-play CNN module that matches or beats E(2)-CNNs at lower cost.

Problem

Building SE(2)-invariant (rotation and translation) neural networks that are both expressive and computationally efficient, avoiding the overhead of steerable equivariant feature fields.

Approach

Project inputs onto moment bases at every pixel and combine them into invariant features using moment magnitudes and phase differences relative to a reference moment (an optimal, minimal, discriminative invariant basis), so only type-0 features are needed and standard CNN layers can be used unchanged.

Key results

• On MNIST-Rot-Test the Mom/Inv CNN reaches 99.42% accuracy vs 97.20% for E2-CNN, with RCI 0.01 vs 0.89 and 119k vs 362k parameters.
• On RESISC45 the model reaches 85.77% vs 84.02% for E2-CNN with about 230k vs 365k parameters.
• Per-epoch time on NFFA (k=3, A100) is 98 s vs 227 s for e2cnn; speedups reach up to 14x for higher moment orders.

Surprising findings

• Adding an explicit equivariant (type-1) branch did not improve accuracy over the purely invariant model on RotMNIST.
• Replacing the optimal basis with its flexible variant gave no measurable accuracy gain in dense CNN features, as reference-moment degeneracies were found to be extremely rare.

Limitations

• Invariance is only approximate in discretized implementations since the theory assumes continuous transformations.
• Invariant map depends on non-zero moment responses; features may be unreliable when magnitudes are very small.
• Benefit is mainly for tasks with strong rotational symmetry; in high-data regimes augmentation and capacity can close the gap.

Experimental setup

Datasets: MNIST-Rot-Test RotMNIST RESISC45 EuroSAT DTD Lost Oil Wells Galaxy Zoo Colorectal Histology NFFA-Europe

Benchmarks: MNIST-Rot-Test RotMNIST RESISC45 EuroSAT DTD NFFA Galaxy Zoo Colorectal Histology

Metrics: accuracy top-5 accuracy rotation consistency index (RCI) parameter count forward-pass runtime

Baselines: E(2)-CNN H-Nets ResNet18 Flusser basis flexible basis

Builds on: Flusser moment invariants steerable CNNs E(2)-CNN Bezout map Equiformer

Closest papers by meaning

• Affine Equivariant Networks Based on Differential Invariants (CVPR 2024 · similarity 0.89)
• A Tale of Two Symmetries: Exploring the Loss Landscape of Equivariant Models (NeurIPS 2025 · similarity 0.89)
• Improving Equivariant Networks with Probabilistic Symmetry Breaking (ICLR 2025 · similarity 0.89)
• Bridging Symmetry and Robustness: On the Role of Equivariance in Enhancing Adversarial Robustness (NeurIPS 2025 · similarity 0.89)
• Probing Equivariance and Symmetry Breaking in Convolutional Networks (NeurIPS 2025 · similarity 0.89)
• Lie Group Decompositions for Equivariant Neural Networks (ICLR 2024 · similarity 0.88)