{"ID":2849449,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.22984","arxiv_id":"2510.22984","title":"Equivariant Neural Networks for General Linear Symmetries on Lie Algebras","abstract":"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/","short_abstract":"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)-...","url_abs":"https://arxiv.org/abs/2510.22984","url_pdf":"https://arxiv.org/pdf/2510.22984v2","authors":"[\"Chankyo Kim\",\"Sicheng Zhao\",\"Minghan Zhu\",\"Tzu-Yuan Lin\",\"Maani Ghaffari\"]","published":"2025-10-27T04:08:39Z","proceeding":"cs.LG","tasks":"[\"cs.LG\",\"cs.NE\"]","methods":"[]","has_code":false}