Lie Neurons: Adjoint-Equivariant Neural Networks for Semisimple Lie Algebras (2310.04521v3)

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.

• The paper introduces Lie Neurons, leveraging the adjoint action of Lie groups to maintain symmetry in semisimple Lie algebra inputs.
• It employs innovative layers like LN-ReLU and LN-Bracket along with geometric channel mixing to boost network expressivity and robustness.
• Experimental results in BCH approximation, rigid body dynamics, and point cloud registration validate improved accuracy and flexible performance.

Equivariant Neural Networks on Lie Algebras: A Study of Lie Neurons

The paper "Lie Neurons: Adjoint-Equivariant Neural Networks for Semisimple Lie Algebras" extends the framework of equivariant neural networks to handle inputs from semisimple Lie algebras. This research advances the application of geometric learning into domains where data are naturally gauged by continuous symmetry transformations represented by Lie groups and algebras. The proposed architecture, termed Lie Neurons, promises a robust framework for neural networks that must retain their structure under the adjoint action of associated Lie groups.

Theoretical Framework

The network architecture leverages the intrinsic properties of Lie algebras, notably the adjoint representation and the Killing form. By effectively translating these mathematical constructs into processes within neural network layers, the architecture achieves equivariance. This design allows the model to maintain the symmetry structure of its inputs, offering a mechanism to preserve functional relations invariantly under transformations by a Lie group.

Key components of the architecture include:

• Linear Layers: Operating on the feature dimension, they avoid disrupting the adjoint equivariance by functioning orthogonally to transformations in the geometric dimension.
• Nonlinear Activation Functions: Two novel layers are presented—LN-ReLU, which relies on the adjoint-invariant property of the Killing form, and LN-Bracket, exploiting the structure of the Lie bracket.
• Geometric Channel Mixing: A unique component that facilitates dimensional mixing for Lie algebras, crucially enhancing the expressivity of the models in tasks sensitive to such operations.

Experimental Analysis

The applicability and performance robustness of Lie Neurons were assessed through a diverse set of experiments, focusing primarily on the $\mathfrak{so}(3)$ and $\mathfrak{sl}(3)$ algebras:

1. Baker–Campbell–Hausdorff (BCH) Formula Approximation: The network outperformed baselines in regressing this formula, using $\mathfrak{so}(3)$ elements, demonstrating superior accuracy owing to the bracket layer's inherent design.
2. Dynamic Modeling of Rigid Body Rotations: Implementing Lie Neurons within a Neural ODE framework, the network effectively learned the dynamics of the free-rotating International Space Station, confirming its efficacy in tasks requiring exact adherence to geometric properties after transformations.
3. Point Cloud Registration: The results showed commendable performance parity with existing networks, underscoring the flexibility of Lie Neurons in standard geometric deep learning problems.
4. Platonic Solids Classification via Homography: Using $\mathfrak{sl}(3)$, the network demonstrated robust classification capabilities of 3D structures under varied viewing transformations, maintaining accuracy across both original and transformed perspectives.

Implications and Future Directions

The implications for both theoretical exploration and applied machine learning are multi-faceted. The ability of Lie Neurons to tightly integrate group theoretic concepts such as Lie brackets into the core computational graph of neural networks highlights a promising approach to leveraging symmetry and invariance more deeply within AI systems. This capability can be particularly beneficial to domains such as robotics, physics-based simulations, and any application where the underlying data distribution respects continuous group symmetries.

Future research might explore extending these concepts beyond semisimple algebras, exploring alternative algebraic structures where similar properties can be exploited. Moreover, enhancing practical methods for basis discovery in arbitrary datasets would expand the architecture's usability across a broader spectrum of problem domains.

In summary, the Lie Neurons present a significant step forward in embedding group symmetry into the core functionality of neural networks, enriching the toolbox for researchers and practitioners aiming to integrate deep learning robustly with geometric and algebraic insights.