LEARNING IRREDUCIBLE REPRESENTATIONS OF NONCOMMUTATIVE LIE GROUPS

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 Poincaré group.

1. INTRODUCTION

Many tasks in machine learning exactly or approximately obey a continuous symmetry such as 2D rotations. An ML model is said to be equivariant to such a symmetry if the model respects it automatically (without training). Equivariant models have been applied to tasks ranging from computer vision to molecular chemistry, leading to a generalization of equivariance techniques beyond 2D rotations to other symmetries such as 3D rotations. This is enabled by known mathematical results about each new set of symmetries. Specifically, explicit group representation matrices for each new symmetry group are required. For many important symmetries, formulae are readily available to produce these representations. For other symmetries we are not so lucky, and the representations may be difficult to find explicitly. In the worst cases, the classification of the group representations is an open problem in mathematics. For example, in the important case of the homogeneous Galilean group, which we define in section 2, the classification of the finite dimensional representations is a so-called "wild algebraic problem" for which we have only partial solutions (De Montigny et al., 2006; Niederle & Nikitin, 2006; Levy-Leblond, 1971 ). To construct an equivariant network without prior knowledge of the group representations, novel approaches are needed. In this work, we propose an algorithm LearnRep that finds the representation matrices with high precision. We validate that LearnRep succeeds for the Poincaré group, a set of symmetries governing phenomena from particle physics to object tracking. We further validate LearnRep on two additional sets of symmetries where formulae are known. We apply the Poincaré group representations obtained by LearnRep to construct SpacetimeNet, a Poincaré-equivariant object-tracking model. As far as we are aware, LearnRep is the first automated solver which can find explicit representation matrices for sets of symmetries which form noncompact, noncommutative Lie groups Further, SpacetimeNet is the first object-tracking model with a rigorous guarantee of Poincaré group equivariance.

1.1. GROUP REPRESENTATIONS AND EQUIVARIANT MACHINE LEARNING

Group theory provides the mathematical framework for describing symmetries and building equivariant ML models. Informally, a symmetry group G is a set of invertible transformations α, β ∈ G which can be composed together using a product operation αβ. We are interested in continuous symmetries for which G is a Lie group. In prior constructions of Lie group-equivariant models, group representations are required. For a group G, an n-dimensional (real) group representation ρ : G → R n×n is a mapping from each element α ∈ G to an n × n-dimensional matrix ρ(α), such that for any two elements α, β ∈ G, we have ρ(α)ρ(β) = ρ(αβ).

