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

TL;DR

The paper tackles learning on data endowed with Lie algebra symmetry by introducing Lie Neurons, an adjoint‑equivariant network operating on elements of finite‑dimensional semisimple Lie algebras. It builds linear and nonlinear layers around the adjoint action and Killing form, plus a geometric channel mixing module, to enable expressive Lie‑algebraic representations and invariants. The approach is demonstrated across $\mathfrak{so}(3)$, $\mathfrak{sl}(3)$, and $\mathfrak{sp}(4)$ on tasks including BCH regression, rigid‑body dynamics, point‑cloud registration, and homography‑based classification, achieving strong performance and data efficiency relative to non‑equivariant baselines and prior Lie‑equivariant methods. By extending vector‑space based equivariance to adjoint actions on semisimple Lie algebras, the work broadens the applicability of symmetry‑aware learning to a wider class of geometric data and dynamical systems, with potential impact in robotics, vision, and physics‑informed ML.

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.

Figures (6)

• Figure 1: Lie Neurons can be applied in learning dynamics. In this example, we learn the dynamics of a simulated free-rotation International Space Station, which we pose as an initial value problem in the Neural ODE framework. This figure shows the estimated trajectories and the learned vector field from the Lie Neurons. Detail descriptions can be found in Section \ref{['sec:modeling_dynamics']}.
• Figure 2: The framework used in modeling the Euler-Poincaré equation. Lie neurons learn the dynamic equation, and an off-the-shelf ODE solver is employed to solve the ODE. Here, $m$ is a set of learnable equivariant weights.
• Figure 3: A visualization of the three Platonic solids in our classification task. The yellow and blue colors highlight a neighboring pair of faces, between which the homography transforms in the image plane are taken as input to our models.
• Figure 4: Comparison between existing equivariant networks and our work. The existing equivariant networks take in vectors in $\mathbb{R}^n$ and are equivariant to the left action of a group. Our adjoint equivariant network takes elements in the Lie algebra as inputs and is equivariant to the adjoint action, which corresponds to a change of basis operation.
• Figure 5: The network architecture used in each experiment.

Theorems & Definitions (3)

• Definition 3.1
• Theorem 3.2
• Theorem 3.3