Title: Equivariant Neural Networks for General Linear Symmetries on Lie Algebras
Primary area: learning on graphs and other geometries & topologies
Keywords: Equivariant Neural Networks, General Linear Group, Reductive Lie Group, Geometric Uncertainty

Abstract: Encoding symmetries is a powerful inductive bias for improving the generalization of deep neural networks. However, most existing equivariant models are limited to simple symmetries like rotations, failing to address the broader class of general linear transformations, $\mathrm{GL}(n)$, that appear in many scientific domains. We introduce \textbf{Reductive Lie Neurons (ReLNs)}, a novel neural network architecture exactly equivariant to these general linear symmetries. ReLNs are designed to operate directly on a wide range of structured inputs, including general $n$-by-$n$ matrices.
ReLNs introduce a novel adjoint-invariant bilinear layer to achieve stable equivariance for both Lie-algebraic features and matrix-valued inputs, \textit{without requiring redesign for each subgroup}. This architecture overcomes the limitations of prior equivariant networks that only apply to compact groups or simple vector data. We validate ReLNs' versatility across a spectrum of tasks: they outperform existing methods on algebraic benchmarks with $\mathfrak{sl}(3)$ and $\mathfrak{sp}(4)$ symmetries and achieve competitive results on a Lorentz-equivariant particle physics task. In 3D drone state estimation with geometric uncertinaty, ReLNs jointly process velocities and covariances, yielding significant improvements in trajectory accuracy. ReLNs provide a practical and general framework for learning with broad linear group symmetries on Lie algebras and matrix-valued data.

TL;DR: We introduce Reductive Lie Neurons, a general equivariant architecture that extends rotations to the full general linear group. This enables equivariant learning on matrix-valued geometric data, expanding the scope of geometric deep learning.