Research map / Architectures and efficiency / CNN design
Lorentz Local Canonicalization: How to make any Network Lorentz-Equivariant
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.
Proposes Lorentz Local Canonicalization (LLoCa), which predicts equivariant local Lorentz frames per particle so any backbone (transformer, GNN, ParticleNet, ParT) becomes exactly Lorentz-equivariant, with tensorial message passing and Minkowski attention.
Problem
Lorentz-equivariant networks for high-energy physics rely on specialized layers that limit architecture choice and are computationally expensive; a general way to make arbitrary backbones exactly Lorentz-equivariant is needed.
Approach
Predict equivariant local reference frames (vierbeins) per particle via a softmax-weighted combination of momenta and a polar-decomposition/Minkowski Gram-Schmidt construction, express features in these frames to obtain Lorentz-invariant inputs for any backbone, and pass tensorial (scalar+vector) messages between frames with frame-transformation matrices L_i L_j^{-1}, extending Lippmann/Gerhartz e
Key results
- LLoCa-Transformer reaches 0.864 accuracy and 0.9882 AUC on JetClass, matching LLoCa-ParT and above ParT (0.861), with L-GATr at 0.866 accuracy but ~166h training vs 31h.
- On Z+4g amplitude regression, LLoCa-Transformer with 8 scalars and 2 vectors achieves MSE 1.0e-5, versus 8.3e-5 for the non-equivariant transformer.
- LLoCa models use 10-50% more FLOPs and 60-100% more training time than non-equivariant baselines, and train about 4x faster with 10x fewer FLOPs than L-GATr.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add LLoCa Lorentz equivariance | ParticleNet · JetClass | accuracy: 0.844 -> 0.845 | Helped |
| Add LLoCa Lorentz equivariance | ParticleNet · JetClass | AUC: 0.9849 -> 0.9852 | Helped |
| Add LLoCa Lorentz equivariance | ParT · JetClass | accuracy: 0.861 -> 0.864 | Helped |
| Add LLoCa Lorentz equivariance | ParT · JetClass | AUC: 0.9877 -> 0.9882 | Helped |
| Add LLoCa Lorentz equivariance | Transformer · JetClass | accuracy: 0.855 -> 0.864 | Helped |
| Add LLoCa Lorentz equivariance | Transformer · JetClass | AUC: 0.9867 -> 0.9882 | Helped |
| Use 8 scalar and 2 vector message representations rather than 16 scalars (ablation) | LLoCa-Transformer · Z+4g amplitude regression | MSE (× 10^-5): 40 ± 4 -> 1.0 ± 0.1 | Helped |
| Use 8 scalar and 2 vector message representations rather than a single 2-tensor (ablation) | LLoCa-Transformer · Z+4g amplitude regression | MSE (× 10^-5): 2.0 ± 0.4 -> 1.0 ± 0.1 | Helped |
| Use 8 scalar and 2 vector message representations rather than 4 vectors (ablation) | LLoCa-Transformer · Z+4g amplitude regression | MSE (× 10^-5): 1.4 ± 0.2 -> 1.0 ± 0.1 | Helped |
| Use tensorial message representations instead of scalar-only messages (ablation) | LLoCa-Transformer · Z+4g amplitude regression | MSE (× 10^-5): 40 ± 4 -> 2.0 ± 0.4 | Helped |
| Use data augmentation rather than exact Lorentz equivariance | LLoCa graph networks and transformers · Z+4g amplitude regression, small training datasets | MSE | Mixed |
| Use exact Lorentz equivariance rather than data augmentation | LLoCa graph networks and transformers · Z+4g amplitude regression, large training datasets | MSE | Helped |
Surprising findings
- Data augmentation outperforms exact Lorentz equivariance for small training datasets in amplitude regression.
- Tensorial (vector-containing) messages substantially outperform scalar-only messages in local canonicalization.
Limitations
- Gram-Schmidt orthogonalization needs at least three particles; for N<3 frames are sampled randomly, formally breaking equivariance to SO(3) or SO(2).
- Only covers the proper orthochronous subgroup SO+(1,3); extension to O(1,3) and Poincare group is discussed but not implemented.
- Numerical stability depends on large boosts and double precision for frame construction.
Experimental setup
Datasets: JetClass Z+ng QFT amplitude dataset (MadGraph-generated)
Benchmarks: JetClass Z+ng QFT amplitude regression
Metrics: accuracy AUC MSE of log amplitudes training time FLOPs
Baselines: PFN P-CNN LorentzNet MIParT-L L-GATr ParticleNet ParT MLP-I Global canonicalization Data augmentation
Builds on: ParticleNet ParT L-GATr Lippmann-Gerhartz local canonicalization (tensorial message passing) scaled dot-product attention
Closest papers by meaning
- Lorentz-Equivariant Geometric Algebra Transformers for High-Energy Physics (NeurIPS 2024 · similarity 0.89)
- Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groups (ICLR 2025 · similarity 0.85)
- Beyond Canonicalization: How Tensorial Messages Improve Equivariant Message Passing (ICLR 2025 · similarity 0.84)
- Equivariance via Minimal Frame Averaging for More Symmetries and Efficiency (ICML 2024 · similarity 0.84)
- Learning Chern Numbers of Multiband Topological Insulators with Gauge Equivariant Neural Networks (NeurIPS 2025 · similarity 0.84)
- Riemann Tensor Neural Networks: Learning Conservative Systems with Physics-Constrained Networks (ICML 2025 · similarity 0.84)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.