# Equivariant neural networks for inverse problems

arXiv: https://arxiv.org/abs/2102.11504 · HTML: https://mlsanity.com/paper/2102.11504/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 56 references that are in the map.
- First posted on arXiv 23 Feb 2021.

## Builds on (most cited first, 22 of 22)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [fastMRI: An Open Dataset and Benchmarks for Accelerated MRI](https://mlsanity.com/paper/1811.08839/) (2018)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Harmonic Networks: Deep Translation and Rotation Equivariance](https://mlsanity.com/paper/1612.04642/) (2016)
- [Learned Primal-dual Reconstruction](https://mlsanity.com/paper/1707.06474/) (2017)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [Deep Scale-spaces: Equivariance Over Scale](https://mlsanity.com/paper/1905.11697/) (2019)
- [Scale-Equivariant Steerable Networks](https://mlsanity.com/paper/1910.11093/) (2019)
- [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018)
- [Roto-Translation Covariant Convolutional Networks for Medical Image Analysis](https://mlsanity.com/paper/1804.03393/) (2018)
- [Exploiting Cyclic Symmetry in Convolutional Neural Networks](https://mlsanity.com/paper/1602.02660/) (2016)
- [Recurrent Inference Machines for Solving Inverse Problems](https://mlsanity.com/paper/1706.04008/) (2017)
- [Manifold Learning for Knowledge Discovery and Intelligent Inverse Design of Photonic Nanostructures: Breaking the Geometric Complexity](https://mlsanity.com/paper/2102.04454/) (2021)

## Built on by (most cited first, 29 of 29)

- [Equivariance with Learned Canonicalization Functions](https://mlsanity.com/paper/2211.06489/) (2022)
- [Robust Equivariant Imaging: a fully unsupervised framework for learning to image from noisy and partial measurements](https://mlsanity.com/paper/2111.12855/) (2021)
- [Imaging with Equivariant Deep Learning](https://mlsanity.com/paper/2209.01725/) (2022)
- [Equivariant plug-and-play image reconstruction](https://mlsanity.com/paper/2312.01831/) (2023)
- [Lie Algebra Canonicalization: Equivariant Neural Operators under arbitrary Lie Groups](https://mlsanity.com/paper/2410.02698/) (2024)
- [Equivariant Bootstrapping for Uncertainty Quantification in Imaging Inverse Problems](https://mlsanity.com/paper/2310.11838/) (2023)
- [Rotation Equivariant Proximal Operator for Deep Unfolding Methods in Image Restoration](https://mlsanity.com/paper/2312.15701/) (2023)
- [Improving Equivariant Networks with Probabilistic Symmetry Breaking](https://mlsanity.com/paper/2503.21985/) (2025)
- [Learning Hamiltonians of constrained mechanical systems](https://mlsanity.com/paper/2201.13254/) (2022)
- [MEPNet: A Model-Driven Equivariant Proximal Network for Joint Sparse-View Reconstruction and Metal Artifact Reduction in CT Images](https://mlsanity.com/paper/2306.14274/) (2023)
- [Scale-Equivariant Unrolled Neural Networks for Data-Efficient Accelerated MRI Reconstruction](https://mlsanity.com/paper/2204.10436/) (2022)
- [Rotation Equivariant Arbitrary-scale Image Super-Resolution](https://mlsanity.com/paper/2508.05160/) (2025)
- [A Deep Learning Localization Method for Measuring Abdominal Muscle Dimensions in Ultrasound Images](https://mlsanity.com/paper/2109.14919/) (2021)
- [Stochastic Primal-Dual Deep Unrolling](https://mlsanity.com/paper/2110.10093/) (2021)
- [Equivariant Neural Tangent Kernels](https://mlsanity.com/paper/2406.06504/) (2024)
- [SRE-CNN: A Spatiotemporal Rotation-Equivariant CNN for Cardiac Cine MR Imaging](https://mlsanity.com/paper/2409.08537/) (2024)
- [An analytic theory of convolutional neural network inverse problems solvers](https://mlsanity.com/paper/2601.10334/) (2026)
- [Inverting Data Transformations via Diffusion Sampling](https://mlsanity.com/paper/2602.08267/) (2026)
- [Equivariance Regularization for Image Reconstruction](https://mlsanity.com/paper/2202.05062/) (2022)
- [Equivariant Denoisers for Image Restoration](https://mlsanity.com/paper/2412.05343/) (2024)
- [Tensor-Tensor Products, Group Representations, and Semidefinite Programming](https://mlsanity.com/paper/2507.12729/) (2025)
- [Equivariant Splitting: Self-supervised learning from incomplete data](https://mlsanity.com/paper/2510.00929/) (2025)
- [Equivariant Deep Equilibrium Models for Imaging Inverse Problems](https://mlsanity.com/paper/2511.18667/) (2025)
- [Image-to-Image Translation Framework Embedded with Rotation Symmetry Priors](https://mlsanity.com/paper/2604.12805/) (2026)
- [Learning end-to-end inversion of circular Radon transforms in the partial radial setup](https://mlsanity.com/paper/2308.14144/) (2023)
- [Equivariant Multiscale Learned Invertible Reconstruction for Cone Beam CT](https://mlsanity.com/paper/2401.11256/) (2024)
- [DUN-SRE: Deep Unrolling Network with Spatiotemporal Rotation Equivariance for Dynamic MRI Reconstruction](https://mlsanity.com/paper/2506.10309/) (2025)
- [Equivariant Multiscale Learned Invertible Reconstruction for Cone Beam CT: From Simulated to Real Data](https://mlsanity.com/paper/2512.21180/) (2025)
- [Aligning Network Equivariance with Data Symmetry: A Theoretical Framework and Adaptive Approach for Image Restoration](https://mlsanity.com/paper/2605.13744/) (2026)

## Abstract (arXiv)

In recent years the use of convolutional layers to encode an inductive bias (translational equivariance) in neural networks has proven to be a very fruitful idea. The successes of this approach have motivated a line of research into incorporating other symmetries into deep learning methods, in the form of group equivariant convolutional neural networks. Much of this work has been focused on roto-translational symmetry of mathbf Rd, but other examples are the scaling symmetry of mathbf Rd and rotational symmetry of the sphere. In this work, we demonstrate that group equivariant convolutional operations can naturally be incorporated into learned reconstruction methods for inverse problems that are motivated by the variational regularisation approach. Indeed, if the regularisation functional is invariant under a group symmetry, the corresponding proximal operator will satisfy an equivariance property with respect to the same group symmetry. As a result of this observation, we design learned iterative methods in which the proximal operators are modelled as group equivariant convolutional neural networks. We use roto-translationally equivariant operations in the proposed methodology and apply it to the problems of low-dose computerised tomography reconstruction and subsampled magnetic resonance imaging reconstruction. The proposed methodology is demonstrated to improve the reconstruction quality of a learned reconstruction method with a little extra computational cost at training time but without any extra cost at test time.