Diffeomorphism-Equivariant Neural Networks

2026-02-06

AI Summary

This work proposes a novel approach to extend neural network equivariance to the infinite-dimensional group of diffeomorphisms, thereby enhancing model generalization under arbitrary smooth geometric deformations. By formulating diffeomorphic equivariance as a differentiable optimization problem within an energy-based normalization framework and integrating differentiable image registration techniques, the method achieves approximate equivariant responses to unseen continuous deformations without requiring data augmentation or retraining. This is the first effort to generalize equivariance to the infinite-dimensional diffeomorphism group, significantly improving robustness and generalization performance on complex geometric transformations in both image segmentation and classification tasks.

Technology Category

• Computer Vision: Diffusion Models for Vision
• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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.

Problem

• diffeomorphism
• equivariance
• infinite-dimensional groups
• neural networks
• symmetry

Innovation

• diffeomorphism-equivariance
• energy-based canonicalisation
• infinite-dimensional groups
• differentiable image registration
• equivariant neural networks