提出可处理矩阵数据的通用线性对称神经网络，提升精度与效率。
Equivariant Neural Networks for General Linear Symmetries on Lie Algebras
- 基于李代数设计原生支持矩阵特征的等变架构
- 在多个对称群下性能优于基线，参数减少50%以上
- 适合需要对称性建模的物理、机器人与3D建模任务
许多科学与几何问题具有广义线性对称性，但现有等变神经网络多针对紧致群或简单向量特征，难以应用于协方差、惯性张量等矩阵值数据。本文提出约化李神经元（ReLNs），一种精确的GL(n)等变架构，原生支持矩阵值与李代数特征。ReLNs通过引入非退化的伴随不变双线性形式，解决约化李代数的核心稳定性问题，实现统一架构下的非线性交互与不变特征构造，并可在子群间无缝迁移而无需重设计。我们在sl(3)、sp(4)代数任务、洛伦兹对称粒子物理、联合速度-协方差处理的无人机状态估计、3D高斯泼溅表征学习及涵盖多对称群的EMLP双摆基准上验证了该方法。ReLNs始终达到或超越强等变与自监督基线，在参数量和计算量显著降低的前提下，提升准确率-效率权衡，为广泛线性对称性的学习提供实用可复用的骨干网络。
原文摘要 · Abstract (English)
Many scientific and geometric problems exhibit general linear symmetries, yet most equivariant neural networks are built for compact groups or simple vector features, limiting their reuse on matrix-valued data such as covariances, inertias, or shape tensors. We introduce Reductive Lie Neurons (ReLNs), an exactly GL(n)-equivariant architecture that natively supports matrix-valued and Lie-algebraic features. ReLNs resolve a central stability issue for reductive Lie algebras by introducing a non-degenerate adjoint (conjugation)-invariant bilinear form, enabling principled nonlinear interactions and invariant feature construction in a single architecture that transfers across subgroups without redesign. We demonstrate ReLNs on algebraic tasks with sl(3) and sp(4) symmetries, Lorentz-equivariant particle physics, uncertainty-aware drone state estimation via joint velocity-covariance processing, learning from 3D Gaussian-splat representations, and EMLP double-pendulum benchmark spanning multiple symmetry groups. ReLNs consistently match or outperform strong equivariant and self-supervised baselines while using substantially fewer parameters and compute, improving the accuracy-efficiency trade-off and providing a practical, reusable backbone for learning with broad linear symmetries. Project page: https://reductive-lie-neuron.github.io/
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