# Moment kernels: a simple and scalable approach for equivariance to rotations and reflections in deep convolutional networks

arXiv: https://arxiv.org/abs/2505.21736 · HTML: https://mlsanity.com/paper/2505.21736/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 37 references that are in the map.
- First posted on arXiv 27 May 2025.

## Builds on (most cited first, 20 of 20)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [You Only Look Once: Unified, Real-Time Object Detection](https://mlsanity.com/paper/1506.02640/) (2015)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [MedMNIST v2 -- A large-scale lightweight benchmark for 2D and 3D biomedical image classification](https://mlsanity.com/paper/2110.14795/) (2021)
- [VoxelMorph: A Learning Framework for Deformable Medical Image Registration](https://mlsanity.com/paper/1809.05231/) (2018)
- [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)
- [SynthMorph: learning contrast-invariant registration without acquired images](https://mlsanity.com/paper/2004.10282/) (2020)
- [Quicksilver: Fast Predictive Image Registration - a Deep Learning Approach](https://mlsanity.com/paper/1703.10908/) (2017)
- [Roto-Translation Covariant Convolutional Networks for Medical Image Analysis](https://mlsanity.com/paper/1804.03393/) (2018)
- [A Wigner-Eckart Theorem for Group Equivariant Convolution Kernels](https://mlsanity.com/paper/2010.10952/) (2020)
- [Enhancing Spatiotemporal Disease Progression Models via Latent Diffusion and Prior Knowledge](https://mlsanity.com/paper/2405.03328/) (2024)
- [A case for using rotation invariant features in state of the art feature matchers](https://mlsanity.com/paper/2204.10144/) (2022)
- [SE(3)-Equivariant and Noise-Invariant 3D Rigid Motion Tracking in Brain MRI](https://mlsanity.com/paper/2312.13534/) (2023)
- [RoTIR: Rotation-Equivariant Network and Transformers for Fish Scale Image Registration](https://mlsanity.com/paper/2401.11270/) (2024)
- [SRE-Conv: Symmetric Rotation Equivariant Convolution for Biomedical Image Classification](https://mlsanity.com/paper/2501.09753/) (2025)

## Abstract (arXiv)

Translation equivariance is a central reason convolutional neural networks have been successful in computer vision. Other symmetries, such as rotations and reflections, are similarly important in fields such as biomedical image analysis, but equivariant methods for these symmetries remain less widely adopted, especially in 3D. Existing approaches often rely on group convolutions, harmonic bases, irreducible representations, or specialized libraries, which can obscure the explicit form of admissible kernels for practitioners. We introduce moment kernels, a simple Cartesian parameterization of convolution kernels equivariant to orthogonal transformations, O(d), between tensor-valued feature fields. We prove that every such O(d)-equivariant kernel can be represented as a sum of radial functions of |x| multiplied by products of coordinate components xi and Kronecker deltas. This gives a complete, dimension-agnostic kernel family complementary to harmonic-basis approaches and implementable using standard convolution modules. We implement a discrete version of moment-kernel networks and evaluate on biomedical tasks with different transformation laws: invariant 2D image classification and equivariant 3D affine-transform regression for brain MRI. Across these tasks, moment kernels improve worst-case orientation consistency and remain trainable in 3D, while avoiding the orientation-channel expansion required by group convolutions, which reaches 48 orientations for 90-degree rotations and reflections in 3D. The resulting models provide exact consistency under grid-preserving rotations and reflections, and remain practical for standard CNN workflows.