# Scale-covariant and scale-invariant Gaussian derivative networks

arXiv: https://arxiv.org/abs/2011.14759 · HTML: https://mlsanity.com/paper/2011.14759/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Nov 2020.

## Built on by (most cited first, 25 of 25)

- [FlexConv: Continuous Kernel Convolutions with Differentiable Kernel Sizes](https://mlsanity.com/paper/2110.08059/) (2021)
- [Resolution learning in deep convolutional networks using scale-space theory](https://mlsanity.com/paper/2106.03412/) (2021)
- [Scale-invariant scale-channel networks: Deep networks that generalise to previously unseen scales](https://mlsanity.com/paper/2106.06418/) (2021)
- [Scale Equivariant U-Net](https://mlsanity.com/paper/2210.04508/) (2022)
- [Scale-Equivariant UNet for Histopathology Image Segmentation](https://mlsanity.com/paper/2304.04595/) (2023)
- [Covariance properties under natural image transformations for the generalized Gaussian derivative model for visual receptive fields](https://mlsanity.com/paper/2303.09803/) (2023)
- [Riesz networks: scale invariant neural networks in a single forward pass](https://mlsanity.com/paper/2305.04665/) (2023)
- [Foveation in the Era of Deep Learning](https://mlsanity.com/paper/2312.01450/) (2023)
- [Discrete approximations of Gaussian smoothing and Gaussian derivatives](https://mlsanity.com/paper/2311.11317/) (2023)
- [Frequency learning for structured CNN filters with Gaussian fractional derivatives](https://mlsanity.com/paper/2111.06660/) (2021)
- [Scale-Equivariant Deep Learning for 3D Data](https://mlsanity.com/paper/2304.05864/) (2023)
- [Unified theory for joint covariance properties under geometric image transformations for spatio-temporal receptive fields according to the generalized Gaussian derivative model for visual receptive fields](https://mlsanity.com/paper/2311.10543/) (2023)
- [Riesz feature representation: scale equivariant scattering network for classification tasks](https://mlsanity.com/paper/2307.08467/) (2023)
- [Approximation properties relative to continuous scale space for hybrid discretizations of Gaussian derivative operators](https://mlsanity.com/paper/2405.05095/) (2024)
- [Rotation-Scale Equivariant Steerable Filters](https://mlsanity.com/paper/2304.04600/) (2023)
- [Covariant spatio-temporal receptive fields for spiking neural networks](https://mlsanity.com/paper/2405.00318/) (2024)
- [Learning Images Across Scales Using Adversarial Training](https://mlsanity.com/paper/2406.08924/) (2024)
- [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)
- [PDE-CNNs: Axiomatic Derivations and Applications](https://mlsanity.com/paper/2403.15182/) (2024)
- [Symmetry-aware Neural Architecture for Embodied Visual Navigation](https://mlsanity.com/paper/2112.09515/) (2021)
- [Feature-Centered First Order Structure Tensor Scale-Space in 2D and 3D](https://mlsanity.com/paper/2409.13389/) (2024)
- [Learning and extrapolating scale-invariant processes](https://mlsanity.com/paper/2601.14810/) (2026)
- [Modelling and analysis of the 8 filters from the "master key filters hypothesis" for depthwise-separable deep networks in relation to idealized receptive fields based on scale-space theory](https://mlsanity.com/paper/2509.12746/) (2025)
- [Hybrid Lie semi-group and cascade structures for the generalized Gaussian derivative model for visual receptive fields](https://mlsanity.com/paper/2509.15748/) (2025)
- [Scale-invariant Gaussian derivative residual networks](https://mlsanity.com/paper/2603.02843/) (2026)

## Abstract (arXiv)

This paper presents a hybrid approach between scale-space theory and deep learning, where a deep learning architecture is constructed by coupling parameterized scale-space operations in cascade. By sharing the learnt parameters between multiple scale channels, and by using the transformation properties of the scale-space primitives under scaling transformations, the resulting network becomes provably scale covariant. By in addition performing max pooling over the multiple scale channels, a resulting network architecture for image classification also becomes provably scale invariant. We investigate the performance of such networks on the MNISTLargeScale dataset, which contains rescaled images from original MNIST over a factor of 4 concerning training data and over a factor of 16 concerning testing data. It is demonstrated that the resulting approach allows for scale generalization, enabling good performance for classifying patterns at scales not present in the training data.