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). Two parallel techniques have been developed for implementing Lie group equivariant neural networks. The first approach was described in general by Cohen et al. (2019) . For the latter approach taken by Thomas et al. ( 2018 2020), convolutions and nonlinearities are performed directly on the irreducible representations of the group, which we define in section 2.4. A common thread in these works has been to utilize existing formulas derived for the matrix elements of these irreducible representations. However, these formulas are only available for specific Lie groups where the representation theory is well-understood. A more convenient approach for extending equivariance to novel Lie groups would utilize an automated computational technique to obtain the required representations. The primary contribution of this work is such a technique.

1.2. CONTRIBUTIONS

In this work, we automate the generation of explicit group representation matrices of Lie groups using an algorithm called LearnRep. LearnRep poses an optimization problem defined by the Lie algebra associated with a Lie group, whose solutions are the representations of the algebra. A penalty term is used to prevent the formation of trivial representations. Gradient descent of the resulting loss function produces nontrivial representations upon convergence. We apply LearnRep to three noncommutative Lie groups for which the finite-dimensional representations are well-understood, allowing us to verify that the representations produced are irreducible by computing their Clebsch-Gordan coefficients and applying Schur's Lemma. One of the Lie groups where LearnRep performs well is the Lorentz group of special relativity. Prior work has applied Lorentz-equivariant models to particle physics. In this work we explain that the Lorentz group along with the larger Poincaré group also governs everyday object-tracking tasks. We construct a Poincaré-equivariant neural network architecture called SpacetimeNet and demonstrate that it can learn to solve a 3D object-tracking task subject to "motion equivariance," where the inputs are a time series of points in space. In summary, our contributions are: • LearnRep, an algorithm which can find irreducible representations of a noncompact and noncommutative Lie group. • SpacetimeNet, a Poincaré group-equivariant neural network applied to object-tracking tasks. Our work contributes towards a general framework and toolset for building neural networks equivariant to novel Lie groups, and motivates further study of Lorentz equivariance for object tracking.

1.3. ORGANIZATION

We summarize all necessary background and terminology in section 2. We describe the LearnRep algorithm in section 3.1 and SpacetimeNet in section 3.2. We summarize related work in section 4. We present our experimental results in section 5: our experiments in learning irreducible Lie group representations with LearnRep in section 5.1 and the performance of our Poincaré-equivariant SpacetimeNet model on a 3D object tracking task in section 5.2.

2. TECHNICAL BACKGROUND

We explain the most crucial concepts here and defer to Appendix A.1 for a derivation of the representation theory of the Lorentz group. 2.1 SYMMETRY GROUPS SO(n) AND SO(m, n) A 3D rotation may be defined as a matrix A :∈ R 3×3 which satisfies the following properties, in which u, v = 3 i=1 u i v i : (i) det A = 1 (ii) ∀ u, v ∈ R 3 , A u, A v = u, v ; these imply the set of 3D rotations forms a group under matrix multiplication and this group is denoted SO(3). This definition directly generalizes to the n-dimensional rotation group SO(n). For



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