Towards Diffeomorphism-Equivariant Neural Networks via Canonicalization

Incorporating group symmetries via equivariance into neural networks has emerged as a robust approach for improving efficiency and overcoming the data requirements of modern deep learning. While most existing approaches, such as group convolutions and averaging-based methods, focus on compact, finite, or low-dimensional groups with linear actions, this work explores how equivariance can be extended to infinite-dimensional groups. We propose a strategy designed to induce diffeomorphism equivariance in pre-trained neural networks via energy-based canonicalization. Formulating equivariance as an optimization problem allows us to access the rich toolbox of already established differentiable image registration methods. Empirical results on segmentation and classification tasks confirm that our approach achieves approximate equivariance and generalizes to unseen transformations without relying on extensive data augmentation or retraining.