# Riesz networks: scale invariant neural networks in a single forward pass

arXiv: https://arxiv.org/abs/2305.04665 · HTML: https://mlsanity.com/paper/2305.04665/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 57 references that are in the map.
- First posted on arXiv 8 May 2023.

## Builds on (most cited first, 22 of 22)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Feature Pyramid Networks for Object Detection](https://mlsanity.com/paper/1612.03144/) (2016)
- [Spatial Transformer Networks](https://mlsanity.com/paper/1506.02025/) (2015)
- [Multi-Scale Context Aggregation by Dilated Convolutions](https://mlsanity.com/paper/1511.07122/) (2015)
- [Deep Scale-spaces: Equivariance Over Scale](https://mlsanity.com/paper/1905.11697/) (2019)
- [Scale-Equivariant Steerable Networks](https://mlsanity.com/paper/1910.11093/) (2019)
- [A Unified Multi-scale Deep Convolutional Neural Network for Fast Object Detection](https://mlsanity.com/paper/1607.07155/) (2016)
- [Gabor Convolutional Networks](https://mlsanity.com/paper/1705.01450/) (2017)
- [Scale-Invariant Convolutional Neural Networks](https://mlsanity.com/paper/1411.6369/) (2014)
- [Locally Scale-Invariant Convolutional Neural Networks](https://mlsanity.com/paper/1412.5104/) (2014)
- [Structured Receptive Fields in CNNs](https://mlsanity.com/paper/1605.02971/) (2016)
- [Scale-covariant and scale-invariant Gaussian derivative networks](https://mlsanity.com/paper/2011.14759/) (2020)
- [Scale equivariance in CNNs with vector fields](https://mlsanity.com/paper/1807.11783/) (2018)
- [Scale-invariant scale-channel networks: Deep networks that generalise to previously unseen scales](https://mlsanity.com/paper/2106.06418/) (2021)
- [Fully trainable Gaussian derivative convolutional layer](https://mlsanity.com/paper/2207.08424/) (2022)
- [Provably scale-covariant continuous hierarchical networks based on scale-normalized differential expressions coupled in cascade](https://mlsanity.com/paper/1905.13555/) (2019)
- [Scale Equivariant Neural Networks with Morphological Scale-Spaces](https://mlsanity.com/paper/2105.01335/) (2021)
- [Understanding when spatial transformer networks do not support invariance, and what to do about it](https://mlsanity.com/paper/2004.11678/) (2020)
- [Crack Modeling via Minimum-Weight Surfaces in 3d Voronoi Diagrams](https://mlsanity.com/paper/2210.05093/) (2022)
- [Methods for segmenting cracks in 3d images of concrete: A comparison based on semi-synthetic images](https://mlsanity.com/paper/2112.09493/) (2021)

## Built on by (most cited first, 8 of 8)

- [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)
- [TO-GATE: Clarifying Questions and Summarizing Responses with Trajectory Optimization for Eliciting Human Preference](https://mlsanity.com/paper/2506.02827/) (2025)
- [Riesz feature representation: scale equivariant scattering network for classification tasks](https://mlsanity.com/paper/2307.08467/) (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)
- [Learning and extrapolating scale-invariant processes](https://mlsanity.com/paper/2601.14810/) (2026)
- [Approach to Finding a Robust Deep Learning Model](https://mlsanity.com/paper/2505.17254/) (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)

Scale invariance of an algorithm refers to its ability to treat objects equally independently of their size. For neural networks, scale invariance is typically achieved by data augmentation. However, when presented with a scale far outside the range covered by the training set, neural networks may fail to generalize. Here, we introduce the Riesz network, a novel scale invariant neural network. Instead of standard 2d or 3d convolutions for combining spatial information, the Riesz network is based on the Riesz transform which is a scale equivariant operation. As a consequence, this network naturally generalizes to unseen or even arbitrary scales in a single forward pass. As an application example, we consider detecting and segmenting cracks in tomographic images of concrete. In this context, 'scale' refers to the crack thickness which may vary strongly even within the same sample. To prove its scale invariance, the Riesz network is trained on one fixed crack width. We then validate its performance in segmenting simulated and real tomographic images featuring a wide range of crack widths. An additional experiment is carried out on the MNIST Large Scale data set.