# RotDCF: Decomposition of Convolutional Filters for Rotation-Equivariant Deep Networks

arXiv: https://arxiv.org/abs/1805.06846 · HTML: https://mlsanity.com/paper/1805.06846/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 May 2018.

## Built on by (most cited first, 17 of 17)

- [Grounding inductive biases in natural images:invariance stems from variations in data](https://mlsanity.com/paper/2106.05121/) (2021)
- [Self-Interpretable Model with TransformationEquivariant Interpretation](https://mlsanity.com/paper/2111.04927/) (2021)
- [Deformation Robust Roto-Scale-Translation Equivariant CNNs](https://mlsanity.com/paper/2111.10978/) (2021)
- [On the Implicit Bias of Linear Equivariant Steerable Networks](https://mlsanity.com/paper/2303.04198/) (2023)
- [Tilt your Head: Activating the Hidden Spatial-Invariance of Classifiers](https://mlsanity.com/paper/2405.03730/) (2024)
- [Empowering Networks With Scale and Rotation Equivariance Using A Similarity Convolution](https://mlsanity.com/paper/2303.00326/) (2023)
- [ChaRRNets: Channel Robust Representation Networks for RF Fingerprinting](https://mlsanity.com/paper/2105.03568/) (2021)
- [Rotation-Scale Equivariant Steerable Filters](https://mlsanity.com/paper/2304.04600/) (2023)
- [Current Symmetry Group Equivariant Convolution Frameworks for Representation Learning](https://mlsanity.com/paper/2409.07327/) (2024)
- [FILTRA: Rethinking Steerable CNN by Filter Transform](https://mlsanity.com/paper/2105.11636/) (2021)
- [Similarity Equivariant Linear Transformation of Joint Orientation-Scale Space Representations](https://mlsanity.com/paper/2203.06786/) (2022)
- [Fast expansion into harmonics on the disk: a steerable basis with fast radial convolutions](https://mlsanity.com/paper/2207.13674/) (2022)
- [ADA-GNN: Atom-Distance-Angle Graph Neural Network for Crystal Material Property Prediction](https://mlsanity.com/paper/2401.11768/) (2024)
- [Generative Quanta Color Imaging](https://mlsanity.com/paper/2403.19066/) (2024)
- [A Rotation Meanout Network with Invariance for Dermoscopy Image Classification and Retrieval](https://mlsanity.com/paper/2208.00627/) (2022)
- [Efficient 3D affinely equivariant CNNs with adaptive fusion of augmented spherical Fourier-Bessel bases](https://mlsanity.com/paper/2402.16825/) (2024)
- [Accelerated Rotation-Invariant Convolution for UAV Image Segmentation](https://mlsanity.com/paper/2512.08888/) (2025)

## Abstract (arXiv)

Explicit encoding of group actions in deep features makes it possible for convolutional neural networks (CNNs) to handle global deformations of images, which is critical to success in many vision tasks. This paper proposes to decompose the convolutional filters over joint steerable bases across the space and the group geometry simultaneously, namely a rotation-equivariant CNN with decomposed convolutional filters (RotDCF). This decomposition facilitates computing the joint convolution, which is proved to be necessary for the group equivariance. It significantly reduces the model size and computational complexity while preserving performance, and truncation of the bases expansion serves implicitly to regularize the filters. On datasets involving in-plane and out-of-plane object rotations, RotDCF deep features demonstrate greater robustness and interpretability than regular CNNs. The stability of the equivariant representation to input variations is also proved theoretically under generic assumptions on the filters in the decomposed form. The RotDCF framework can be extended to groups other than rotations, providing a general approach which achieves both group equivariance and representation stability at a reduced model size.