# Equivariant Neural Networks for General Linear Symmetries on Lie Algebras

arXiv: https://arxiv.org/abs/2510.22984 · HTML: https://mlsanity.com/paper/2510.22984/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 49 of its 59 references that are in the map.
- First posted on arXiv 27 Oct 2025.

## Builds on (most cited first, 30 of 49)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Gaussian Grouping: Segment and Edit Anything in 3D Scenes](https://mlsanity.com/paper/2312.00732/) (2023)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs](https://mlsanity.com/paper/2206.11990/) (2022)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [Frame Averaging for Invariant and Equivariant Network Design](https://mlsanity.com/paper/2110.03336/) (2021)
- [On the Universality of Invariant Networks](https://mlsanity.com/paper/1901.09342/) (2019)
- [Lorentz Group Equivariant Neural Network for Particle Physics](https://mlsanity.com/paper/2006.04780/) (2020)
- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [ManiGaussian: Dynamic Gaussian Splatting for Multi-task Robotic Manipulation](https://mlsanity.com/paper/2403.08321/) (2024)
- [LieTransformer: Equivariant self-attention for Lie Groups](https://mlsanity.com/paper/2012.10885/) (2020)
- [Particle Transformer for Jet Tagging](https://mlsanity.com/paper/2202.03772/) (2022)
- [Energy Flow Networks: Deep Sets for Particle Jets](https://mlsanity.com/paper/1810.05165/) (2018)
- [Clifford Group Equivariant Neural Networks](https://mlsanity.com/paper/2305.11141/) (2023)
- [Geometric Algebra Transformer](https://mlsanity.com/paper/2305.18415/) (2023)
- [Lattice gauge equivariant convolutional neural networks](https://mlsanity.com/paper/2012.12901/) (2020)
- [Group Equivariant Fourier Neural Operators for Partial Differential Equations](https://mlsanity.com/paper/2306.05697/) (2023)
- [Dynamic 3D Gaussian Tracking for Graph-Based Neural Dynamics Modeling](https://mlsanity.com/paper/2410.18912/) (2024)
- [PDO-eConvs: Partial Differential Operator Based Equivariant Convolutions](https://mlsanity.com/paper/2007.10408/) (2020)
- [Learning to Fly in Seconds](https://mlsanity.com/paper/2311.13081/) (2023)
- [VN-Transformer: Rotation-Equivariant Attention for Vector Neurons](https://mlsanity.com/paper/2206.04176/) (2022)

## Built on by (most cited first, 4 of 4)

- [The Token Is a Group Element: On Lie-Algebra Attention over Matrix Lie Groups](https://mlsanity.com/paper/2606.20547/) (2026)
- [E3DGS: Unified Geometric-Photometric Equivariance for 3D Gaussian Splatting via Color-as-Geometry Embedding](https://mlsanity.com/paper/2607.15536/) (2026)
- [mathbbSL(n) Representation Learning: An Intrinsic Mixed-Curvature Space with Higher Curvature Capacities and Deeper Order-Aware Composition](https://mlsanity.com/paper/2609.15083/) (2026)
- [GINIO: A Geometric SO(3)-Equivariant Interface for Neural Inertial Odometry](https://mlsanity.com/paper/2609.25338/) (2026)

## Abstract (arXiv)

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/