Lie Neurons: Adjoint-Equivariant Neural Networks for Semisimple Lie Algebras

This paper proposes an equivariant neural network that takes data in any semi-simple Lie algebra as input. The corresponding group acts on the Lie algebra as adjoint operations, making our proposed network adjoint-equivariant. Our framework generalizes the Vector Neurons, a simple $\mathrm{SO}(3)$-equivariant network, from 3-D Euclidean space to Lie algebra spaces, building upon the invariance property of the Killing form. Furthermore, we propose novel Lie bracket layers and geometric channel mixing layers that extend the modeling capacity. Experiments are conducted …

3-d algebra cs.ai cs.lg data framework making network networks neural network neural networks neurons operations paper simple space spaces vector