Diffeomorphism-Equivariant Neural Networks

TL;DR

This paper introduces a novel method to incorporate diffeomorphism equivariance into neural networks by formulating it as an optimization problem, enabling better generalization to unseen transformations without extensive data augmentation.

Contribution

It extends equivariance to infinite-dimensional groups using energy-based canonicalisation and differentiable image registration techniques.

Findings

Achieves approximate equivariance in segmentation and classification tasks.

Generalizes to unseen transformations without retraining.

Reduces reliance on data augmentation.

Abstract

Incorporating group symmetries via equivariance into neural networks has emerged as a robust approach for overcoming the efficiency and data demands 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 canonicalisation. Formulating equivariance as an optimisation 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 generalises to unseen transformations without relying on extensive data augmentation or retraining.