# Scale-Equivariant Steerable Networks

arXiv: https://arxiv.org/abs/1910.11093 · HTML: https://mlsanity.com/paper/1910.11093/

## Key facts

- Cited by 110 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Oct 2019.

## Built on by (most cited first, 30 of 110)

- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [EQ-VAE: Equivariance Regularized Latent Space for Improved Generative Image Modeling](https://mlsanity.com/paper/2502.09509/) (2025)
- [Neural Operators with Localized Integral and Differential Kernels](https://mlsanity.com/paper/2402.16845/) (2024)
- [DEVIANT: Depth EquiVarIAnt NeTwork for Monocular 3D Object Detection](https://mlsanity.com/paper/2207.10758/) (2022)
- [Self-Supervised Learning Disentangled Group Representation as Feature](https://mlsanity.com/paper/2110.15255/) (2021)
- [Learning Partial Equivariances from Data](https://mlsanity.com/paper/2110.10211/) (2021)
- [Physics-Guided Deep Learning for Dynamical Systems: A Survey](https://mlsanity.com/paper/2107.01272/) (2021)
- [Fast, Expressive SE(n) Equivariant Networks through Weight-Sharing in Position-Orientation Space](https://mlsanity.com/paper/2310.02970/) (2023)
- [DISCO: accurate Discrete Scale Convolutions](https://mlsanity.com/paper/2106.02733/) (2021)
- [Robust (Controlled) Table-to-Text Generation with Structure-Aware Equivariance Learning](https://mlsanity.com/paper/2205.03972/) (2022)
- [LieGG: Studying Learned Lie Group Generators](https://mlsanity.com/paper/2210.04345/) (2022)
- [Equivariant neural networks for inverse problems](https://mlsanity.com/paper/2102.11504/) (2021)
- [Sample Efficiency of Data Augmentation Consistency Regularization](https://mlsanity.com/paper/2202.12230/) (2022)
- [Commutative Lie Group VAE for Disentanglement Learning](https://mlsanity.com/paper/2106.03375/) (2021)
- [Resolution learning in deep convolutional networks using scale-space theory](https://mlsanity.com/paper/2106.03412/) (2021)
- [Self-Supervised Equivariant Learning for Oriented Keypoint Detection](https://mlsanity.com/paper/2204.08613/) (2022)
- [Clifford-Steerable Convolutional Neural Networks](https://mlsanity.com/paper/2402.14730/) (2024)
- [Exploiting Redundancy: Separable Group Convolutional Networks on Lie Groups](https://mlsanity.com/paper/2110.13059/) (2021)
- [Enabling equivariance for arbitrary Lie groups](https://mlsanity.com/paper/2111.08251/) (2021)
- [Maximum Class Separation as Inductive Bias in One Matrix](https://mlsanity.com/paper/2206.08704/) (2022)
- [Dilated convolution with learnable spacings](https://mlsanity.com/paper/2112.03740/) (2021)
- [PDO-eS2CNNs: Partial Differential Operator Based Equivariant Spherical CNNs](https://mlsanity.com/paper/2104.03584/) (2021)
- [3D Equivariant Graph Implicit Functions](https://mlsanity.com/paper/2203.17178/) (2022)
- [Modelling Long Range Dependencies in ND: From Task-Specific to a General Purpose CNN](https://mlsanity.com/paper/2301.10540/) (2023)
- [Scale-invariant scale-channel networks: Deep networks that generalise to previously unseen scales](https://mlsanity.com/paper/2106.06418/) (2021)
- [Graph Neural PDE Solvers with Conservation and Similarity-Equivariance](https://mlsanity.com/paper/2405.16183/) (2024)
- [Truly Scale-Equivariant Deep Nets with Fourier Layers](https://mlsanity.com/paper/2311.02922/) (2023)
- [Learning Instance-Specific Augmentations by Capturing Local Invariances](https://mlsanity.com/paper/2206.00051/) (2022)
- [Generalizing Neural Human Fitting to Unseen Poses With Articulated SE(3) Equivariance](https://mlsanity.com/paper/2304.10528/) (2023)
- [On genuine invariance learning without weight-tying](https://mlsanity.com/paper/2308.03904/) (2023)

## Abstract (arXiv)

The effectiveness of Convolutional Neural Networks (CNNs) has been substantially attributed to their built-in property of translation equivariance. However, CNNs do not have embedded mechanisms to handle other types of transformations. In this work, we pay attention to scale changes, which regularly appear in various tasks due to the changing distances between the objects and the camera. First, we introduce the general theory for building scale-equivariant convolutional networks with steerable filters. We develop scale-convolution and generalize other common blocks to be scale-equivariant. We demonstrate the computational efficiency and numerical stability of the proposed method. We compare the proposed models to the previously developed methods for scale equivariance and local scale invariance. We demonstrate state-of-the-art results on MNIST-scale dataset and on STL-10 dataset in the supervised learning setting.