[Paper Review] Lorentz Group Equivariant Neural Network for Particle Physics

Alexander Bogatskiy, Brandon Anderson+4 more|arXiv (Cornell University)|Jun 8, 2020

Computational Physics and Python ApplicationsComputer Science46 references68 citations

TL;DR

A Lorentz group equivariant neural network (LGN) is built using finite-dimensional Lorentz representations and Clebsch-Gordan decompositions, applied to jet constituent energy-momenta for top quark tagging. The approach yields compact, interpretable models with competitive performance on a public dataset.

ABSTRACT

We present a neural network architecture that is fully equivariant with respect to transformations under the Lorentz group, a fundamental symmetry of space and time in physics. The architecture is based on the theory of the finite-dimensional representations of the Lorentz group and the equivariant nonlinearity involves the tensor product. For classification tasks in particle physics, we demonstrate that such an equivariant architecture leads to drastically simpler models that have relatively few learnable parameters and are much more physically interpretable than leading approaches that use CNNs and point cloud approaches. The competitive performance of the network is demonstrated on a public classification dataset [27] for tagging top quark decays given energy-momenta of jet constituents produced in proton-proton collisions.

Motivation & Objective

• Motivate incorporating fundamental Lorentz symmetry into neural networks for high-energy physics data.
• Develop an architecture that is fully equivariant under Lorentz transformations using finite-dimensional representations.
• Demonstrate parameter efficiency and interpretability compared with CNNs and point-cloud methods.
• Apply the model to a public jet-energy-momentum classification dataset to tag top quark decays.

Proposed method

• Use finite-dimensional representations of the Lorentz group to ensure model equivariance.
• Employ tensor products and Clebsch-Gordan decomposition to build Lorentz-equivariant nonlinearities.
• Parameterize equivariant linear maps via isotypic component (Clebsch-Gordan) blocks with per-block matrices.
• Train an architecture where activations live in Lorentz representations and invariants drive final outputs.
• Ground the approach in an equivariant universal approximation framework applicable to SL(2,C) and SO+(1,3) representations.

Experimental results

Research questions

• RQ1 Can a Lorentz group equivariant network achieve competitive top-quark tagging performance using only Lorentz-invariant inputs and equivariant processing?
• RQ2 How does enforcing Lorentz equivariance affect model size, interpretability, and learning efficiency relative to non-equivariant baselines?
• RQ3 What are the practical gains in exploiting the Clebsch-Gordan decomposition for learning Lorentz-invariant and equivariant quantities from jet energy-momentum data?

Key findings

• The LGN architecture is fully equivariant under the Lorentz group by construction.
• Tensor product and Clebsch-Gordan decomposition provide the nonlinear and linear building blocks for Lorentz-equivariant learning.
• The representation-based design yields models with relatively few learnable parameters and enhanced physical interpretability compared with CNNs or point-cloud approaches.
• The method is demonstrated on a public KasPleThRu19 dataset for top quark tagging using jet constituent energy-momenta, showing competitive performance.