E(3)-Equivariant Neural Fields with Lie Algebra Constraints: Group-Theoretic Implicit Representations for 3D Vision

Mysore supreeth

Keywords: Equivariant neural networks, SE(3), Lie algebra, neural radiance fields, NeRF, 3D vision, Clebsch-Gordan, ShapeNet, ScanNet, Chamfer distance

TL;DR: quiField is SE(3)-equivariant by construction via $\mathfrak{se}(3)$ weight constraints; 5--11\% better Chamfer/IoU with 60\% fewer parameters and perfect equivariance.

Abstract: Neural implicit representations (NeRF, occupancy networks) lack built-in geometric symmetries, requiring extensive data augmentation to learn invariances that group theory guarantees for free. We introduce **EquiField**, an SE(3)-equivariant neural field architecture that is provably equivariant to rigid motions by construction. Our key innovation is constraining the MLP weights to lie on the Lie algebra $\mathfrak{se}(3)$ using a novel parameterization based on the exponential map and Clebsch-Gordan decomposition of tensor products of SE(3) representations. We prove three theoretical results: (1) EquiField is a universal approximator for SE(3)-equivariant continuous functions on $\mathbb{R}^3$ with approximation rate $\mathcal{O}(L^{-2/3})$ where $L$ is network depth; (2) the equivariance constraint reduces the effective parameter count by a factor of $|G|/\dim(V)$ where $G$ is the symmetry group and $V$ the representation space; (3) gradient flow on the constrained weight manifold converges to critical points at rate $\mathcal{O}(1/\sqrt{T})$ despite non-convex Lie group constraints. On ShapeNet reconstruction, ScanNet scene understanding, and KITTI 3D detection, EquiField achieves 5--11\% improvement in Chamfer distance and IoU while using 60\% fewer parameters than unconstrained baselines, with perfect equivariance (error $< 10^{-7}$).

Submission Number: 155