Equivariant Neural Networks for General Linear Symmetries on Lie Algebras

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/

💡 Research Summary

The paper introduces Reductive Lie Neurons (ReLNs), a novel neural architecture that is exactly equivariant to the adjoint action of the general linear group GL(n) on its Lie algebra gl(n). Existing equivariant networks largely target compact groups (e.g., SO(3), SE(3)) and simple vector features, leaving matrix‑valued data such as covariances, inertias, and shape tensors under‑served. A core difficulty for GL(n) is that its Lie algebra is reductive but not semisimple, so the canonical Killing form is degenerate on the center, preventing the construction of invariant scalars needed for gating, normalization, and nonlinearities.

To overcome this, the authors construct a non‑degenerate Ad‑invariant bilinear form eB on any reductive Lie algebra. For gl(n) = ℝ·I ⊕ sl(n), the form is given explicitly by eB(X,Y)=2n·tr(XY)−tr(X)·tr(Y). This form combines the Killing form on the semisimple part with an arbitrary inner product on the center, yielding a full‑rank, Ad‑invariant metric. eB enables stable invariant contractions, which the authors exploit to design two equivariant nonlinear primitives: ReLN‑ReLU (an invariant‑gate built from eB) and ReLN‑Bracket (the Lie bracket