# Riesz feature representation: scale equivariant scattering network for classification tasks

arXiv: https://arxiv.org/abs/2307.08467 · HTML: https://mlsanity.com/paper/2307.08467/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 47 references that are in the map.
- First posted on arXiv 17 Jul 2023.

## Builds on (most cited first, 19 of 19)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [Understanding Deep Convolutional Networks](https://mlsanity.com/paper/1601.04920/) (2016)
- [Kymatio: Scattering Transforms in Python](https://mlsanity.com/paper/1812.11214/) (2018)
- [Deep Roto-Translation Scattering for Object Classification](https://mlsanity.com/paper/1412.8659/) (2014)
- [A Mathematical Theory of Deep Convolutional Neural Networks for Feature Extraction](https://mlsanity.com/paper/1512.06293/) (2015)
- [Scaling the Scattering Transform: Deep Hybrid Networks](https://mlsanity.com/paper/1703.08961/) (2017)
- [Scattering Networks for Hybrid Representation Learning](https://mlsanity.com/paper/1809.06367/) (2018)
- [Scale-covariant and scale-invariant Gaussian derivative networks](https://mlsanity.com/paper/2011.14759/) (2020)
- [Scale-invariant scale-channel networks: Deep networks that generalise to previously unseen scales](https://mlsanity.com/paper/2106.06418/) (2021)
- [Parametric Scattering Networks](https://mlsanity.com/paper/2107.09539/) (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)
- [A Learnable ScatterNet: Locally Invariant Convolutional Layers](https://mlsanity.com/paper/1903.03137/) (2019)
- [Exploring the ability of CNNs to generalise to previously unseen scales over wide scale ranges](https://mlsanity.com/paper/2004.01536/) (2020)
- [Provably scale-covariant continuous hierarchical networks based on scale-normalized differential expressions coupled in cascade](https://mlsanity.com/paper/1905.13555/) (2019)
- [Riesz networks: scale invariant neural networks in a single forward pass](https://mlsanity.com/paper/2305.04665/) (2023)
- [Visualizing and Improving Scattering Networks](https://mlsanity.com/paper/1709.01355/) (2017)
- [Monogenic Wavelet Scattering Network for Texture Image Classification](https://mlsanity.com/paper/2202.12491/) (2022)

## Built on by (most cited first, 4 of 4)

- [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)
- [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)
- [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)

Scattering networks yield powerful and robust hierarchical image descriptors which do not require lengthy training and which work well with very few training data. However, they rely on sampling the scale dimension. Hence, they become sensitive to scale variations and are unable to generalize to unseen scales. In this work, we define an alternative feature representation based on the Riesz transform. We detail and analyze the mathematical foundations behind this representation. In particular, it inherits scale equivariance from the Riesz transform and completely avoids sampling of the scale dimension. Additionally, the number of features in the representation is reduced by a factor four compared to scattering networks. Nevertheless, our representation performs comparably well for texture classification with an interesting addition: scale equivariance. Our method yields superior performance when dealing with scales outside of those covered by the training dataset. The usefulness of the equivariance property is demonstrated on the digit classification task, where accuracy remains stable even for scales four times larger than the one chosen for training. As a second example, we consider classification of textures.