Computing Representations for Lie Algebraic Networks
This work addresses the problem of building equivariant networks for complex symmetry groups in physics and other domains, representing a novel method for a known bottleneck.
The paper tackles the challenge of constructing neural networks equivariant to arbitrary continuous symmetry groups by introducing an algorithm that computes Lie group representations from Lie algebra structure constants, enabling the creation of Poincaré-equivariant models for relativistic point cloud object tracking.
Recent work has constructed neural networks that are equivariant to continuous symmetry groups such as 2D and 3D rotations. This is accomplished using explicit Lie group representations to derive the equivariant kernels and nonlinearities. We present three contributions motivated by frontier applications of equivariance beyond rotations and translations. First, we relax the requirement for explicit Lie group representations with a novel algorithm that finds representations of arbitrary Lie groups given only the structure constants of the associated Lie algebra. Second, we provide a self-contained method and software for building Lie group-equivariant neural networks using these representations. Third, we contribute a novel benchmark dataset for classifying objects from relativistic point clouds, and apply our methods to construct the first object-tracking model equivariant to the Poincaré group.