# Scale Steerable Filters for Locally Scale-Invariant Convolutional Neural Networks

arXiv: https://arxiv.org/abs/1906.03861 · HTML: https://mlsanity.com/paper/1906.03861/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jun 2019.

## Built on by (most cited first, 22 of 22)

- [Approximately Equivariant Networks for Imperfectly Symmetric Dynamics](https://mlsanity.com/paper/2201.11969/) (2022)
- [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)
- [Physics-Guided Deep Learning for Dynamical Systems: A Survey](https://mlsanity.com/paper/2107.01272/) (2021)
- [DISCO: accurate Discrete Scale Convolutions](https://mlsanity.com/paper/2106.02733/) (2021)
- [Clifford-Steerable Convolutional Neural Networks](https://mlsanity.com/paper/2402.14730/) (2024)
- [Discovering Symmetry Breaking in Physical Systems with Relaxed Group Convolution](https://mlsanity.com/paper/2310.02299/) (2023)
- [Scale-invariant scale-channel networks: Deep networks that generalise to previously unseen scales](https://mlsanity.com/paper/2106.06418/) (2021)
- [Truly Scale-Equivariant Deep Nets with Fourier Layers](https://mlsanity.com/paper/2311.02922/) (2023)
- [Scale Equivariant U-Net](https://mlsanity.com/paper/2210.04508/) (2022)
- [Improving the Sample-Complexity of Deep Classification Networks with Invariant Integration](https://mlsanity.com/paper/2202.03967/) (2022)
- [Scale-Equivariant Deep Learning for 3D Data](https://mlsanity.com/paper/2304.05864/) (2023)
- [Implicit Equivariance in Convolutional Networks](https://mlsanity.com/paper/2111.14157/) (2021)
- [Learning Geometry-aware Representations by Sketching](https://mlsanity.com/paper/2304.08204/) (2023)
- [Scale generalisation properties of extended scale-covariant and scale-invariant Gaussian derivative networks on image datasets with spatial scaling variations](https://mlsanity.com/paper/2409.11140/) (2024)
- [Just a Matter of Scale? Reevaluating Scale Equivariance in Convolutional Neural Networks](https://mlsanity.com/paper/2211.10288/) (2022)
- [Deep Neural Networks with Efficient Guaranteed Invariances](https://mlsanity.com/paper/2303.01567/) (2023)
- [Scalar Invariant Networks with Zero Bias](https://mlsanity.com/paper/2211.08486/) (2022)
- [SO(2) and O(2) Equivariance in Image Recognition with Bessel-Convolutional Neural Networks](https://mlsanity.com/paper/2304.09214/) (2023)
- [Evaluating Machine Learning Models with NERO: Non-Equivariance Revealed on Orbits](https://mlsanity.com/paper/2305.19889/) (2023)
- [Restore Translation Using Equivariant Neural Networks](https://mlsanity.com/paper/2306.16938/) (2023)
- [Scale-invariant Gaussian derivative residual networks](https://mlsanity.com/paper/2603.02843/) (2026)

## Abstract (arXiv)

Augmenting transformation knowledge onto a convolutional neural network's weights has often yielded significant improvements in performance. For rotational transformation augmentation, an important element to recent approaches has been the use of a steerable basis i.e. the circular harmonics. Here, we propose a scale-steerable filter basis for the locally scale-invariant CNN, denoted as log-radial harmonics. By replacing the kernels in the locally scale-invariant CNN citelsicnn with scale-steered kernels, significant improvements in performance can be observed on the MNIST-Scale and FMNIST-Scale datasets. Training with a scale-steerable basis results in filters which show meaningful structure, and feature maps demonstrate which demonstrate visibly higher spatial-structure preservation of input. Furthermore, the proposed scale-steerable CNN shows on-par generalization to global affine transformation estimation methods such as Spatial Transformers, in response to test-time data distortions.