Lie Neurons: Adjoint-Equivariant Neural Networks for Semisimple Lie Algebras

TL;DR

This paper introduces an equivariant neural network framework for semi-simple Lie algebras, extending previous models to more complex algebraic structures with applications in geometry, dynamics, and shape analysis.

Contribution

It generalizes vector neuron models to Lie algebra spaces using adjoint actions, introducing novel Lie bracket and channel mixing layers for enhanced modeling capacity.

Findings

Effective on multiple Lie algebras including so(3), sl(3), sp(4)

Performs well in tasks like function fitting, dynamics learning, and shape classification

Demonstrates wide applicability and competitive results across domains

Abstract

This paper proposes an equivariant neural network that takes data in any semi-simple Lie algebra as input. The corresponding group acts on the Lie algebra as adjoint operations, making our proposed network adjoint-equivariant. Our framework generalizes the Vector Neurons, a simple $\mathrm{SO}(3)$-equivariant network, from 3-D Euclidean space to Lie algebra spaces, building upon the invariance property of the Killing form. Furthermore, we propose novel Lie bracket layers and geometric channel mixing layers that extend the modeling capacity. Experiments are conducted for the $\mathfrak{so}(3)$, $\mathfrak{sl}(3)$, and $\mathfrak{sp}(4)$ Lie algebras on various tasks, including fitting equivariant and invariant functions, learning system dynamics, point cloud registration, and homography-based shape classification. Our proposed equivariant network shows wide applicability and competitive performance in various domains.