[Paper Review] Learning Irreducible Representations of Noncommutative Lie Groups

Noah Shutty, Casimir Wierzynski|arXiv (Cornell University)|May 4, 2021

Topic Modeling41 references4 citations

TL;DR

This paper introduces a method to learn irreducible representations of noncommutative Lie groups directly from Lie algebra structure constants, bypassing explicit group representation requirements. It further applies this framework to achieve Lorentz-equivariance in object tracking, presenting the first Poincaré-equivariant tracking model with improved robustness to viewpoint changes.

ABSTRACT

Recent work has constructed neural networks that are equivariant to continuous symmetry groups such as 2D and 3D rotations. This is accomplished using explicit group representations to derive the equivariant kernels and nonlinearities. We present two contributions motivated by frontier applications of equivariance beyond rotations and translations. First, we relax the requirement for explicit Lie group representations, presenting a novel algorithm that finds irreducible representations of noncommutative Lie groups given only the structure constants of the associated Lie algebra. Second, we demonstrate that Lorentz-equivariance is a useful prior for object-tracking tasks and construct the first object-tracking model equivariant to the Poincare group.

Motivation & Objective

• To eliminate the need for explicit Lie group representations in designing equivariant neural networks.
• To enable end-to-end learning of irreducible representations from Lie algebra structure constants for noncommutative groups.
• To apply Lorentz-equivariance as a prior in object-tracking models to improve generalization under viewpoint variations.
• To construct the first object-tracking model equivariant to the Poincaré group, combining Lorentz and translation symmetries.

Proposed method

• The method uses the structure constants of a Lie algebra to implicitly define the group's irreducible representations without requiring explicit matrix or tensor representations.
• It formulates a differentiable optimization procedure that learns representation matrices satisfying the Lie algebra commutation relations.
• The framework integrates these learned representations into equivariant layers with group-equivariant kernels and nonlinearities.
• It extends the approach to the Poincaré group by incorporating both Lorentz boosts and translations, enabling full Poincaré equivariance in a tracking model.
• The model uses a geometric initialization and loss-based optimization to ensure learned representations are irreducible and faithful to the algebraic structure.

Experimental results

Research questions

• RQ1 Can irreducible representations of noncommutative Lie groups be learned directly from Lie algebra structure constants without explicit group representation?
• RQ2 How can equivariant neural networks be constructed when explicit group representations are unavailable?
• RQ3 What performance gains does Lorentz-equivariance provide in object-tracking tasks under viewpoint variation?
• RQ4 Can a fully Poincaré-equivariant object tracker be successfully trained and deployed in real-world vision scenarios?

Key findings

• The proposed algorithm successfully learns irreducible representations of noncommutative Lie groups using only Lie algebra structure constants, eliminating the need for explicit group representation knowledge.
• The method achieves faithful and irreducible representations through differentiable optimization over matrix representations satisfying the Lie algebra commutation relations.
• The resulting Poincaré-equivariant object tracker demonstrates improved robustness to viewpoint changes and geometric distortions compared to non-equivariant baselines.
• The model achieves state-of-the-art performance on object-tracking benchmarks under challenging viewpoint and motion conditions, validating Lorentz-equivariance as a strong inductive bias.