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arXiv · 2610.17424

On Making $\mathrm{SE}(2)$-Invariant Networks Optimal

Tomas Karella·Emily Shinkle·Alice Elisabeth Anastasia Allen·Pieter J. Swart·Nicholas Lubbers·Roxana Bujack
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Abstract

Respecting a-priori known symmetries of underlying data is a key principle in designing efficient neural architectures. Convolutional neural networks are a canonical example, encoding translation equivariance. Extending this inductive bias to richer symmetry groups such as $\mathrm{SE}(2)$ in the most effective way possible is an ongoing architectural and computational challenge. Building on classical invariant theory, we derive invariant feature maps that are provably optimal: minimal and information-preserving with respect to the symmetry group. Our construction is designed to integrate seamlessly into standard architectures, requiring no modification to nonlinearities or normalization layers. As a concrete implementation, we introduce a plug-and-play module for convolutional neural networks that enforces $\mathrm{SE}(2)$ invariance, capturing both translations and rotations within standard architectures. The resulting models are both efficient and expressive. Experiments show competitive or improved accuracy compared to state-of-the-art equivariant methods at significantly lower computational cost. Finally, we provide a theoretical and empirical analysis of the relationship between equivariance and invariance, offering insight into their roles in controlling robustness and flexibility.

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Tomas Karella, Emily Shinkle, Alice Elisabeth Anastasia Allen, Pieter J. Swart, Nicholas Lubbers, Roxana Bujack. 2026-10-12. On Making $\mathrm{SE}(2)$-Invariant Networks Optimal. https://arxiv.org/abs/2610.17424

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