# Truly shift-equivariant convolutional neural networks with adaptive polyphase upsampling

arXiv: https://arxiv.org/abs/2105.04040 · HTML: https://mlsanity.com/paper/2105.04040/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 25 references that are in the map.
- First posted on arXiv 9 May 2021.

## Builds on (most cited first, 17 of 17)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [fastMRI: An Open Dataset and Benchmarks for Accelerated MRI](https://mlsanity.com/paper/1811.08839/) (2018)
- [Spherical CNNs](https://mlsanity.com/paper/1801.10130/) (2018)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Making Convolutional Networks Shift-Invariant Again](https://mlsanity.com/paper/1904.11486/) (2019)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Why do deep convolutional networks generalize so poorly to small image transformations?](https://mlsanity.com/paper/1805.12177/) (2018)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [Equivariance Through Parameter-Sharing](https://mlsanity.com/paper/1702.08389/) (2017)
- [Scale-Equivariant Steerable Networks](https://mlsanity.com/paper/1910.11093/) (2019)
- [Exploring the Landscape of Spatial Robustness](https://mlsanity.com/paper/1712.02779/) (2017)
- [Delving Deeper into Anti-aliasing in ConvNets](https://mlsanity.com/paper/2008.09604/) (2020)
- [Truly shift-invariant convolutional neural networks](https://mlsanity.com/paper/2011.14214/) (2020)
- [Shift Equivariance in Object Detection](https://mlsanity.com/paper/2008.05787/) (2020)
- [The LoDoPaB-CT Dataset: A Benchmark Dataset for Low-Dose CT Reconstruction Methods](https://mlsanity.com/paper/1910.01113/) (2019)

## Built on by (most cited first, 10 of 10)

- [Imaging with Equivariant Deep Learning](https://mlsanity.com/paper/2209.01725/) (2022)
- [Learnable Polyphase Sampling for Shift Invariant and Equivariant Convolutional Networks](https://mlsanity.com/paper/2210.08001/) (2022)
- [Robust Perception through Equivariance](https://mlsanity.com/paper/2212.06079/) (2022)
- [LoFi: Neural Local Fields for Scalable Image Reconstruction](https://mlsanity.com/paper/2411.04995/) (2024)
- [Evaluating Machine Learning Models with NERO: Non-Equivariance Revealed on Orbits](https://mlsanity.com/paper/2305.19889/) (2023)
- [Translation-Equivariance of Normalization Layers and Aliasing in Convolutional Neural Networks](https://mlsanity.com/paper/2505.19805/) (2025)
- [Equivariant Splitting: Self-supervised learning from incomplete data](https://mlsanity.com/paper/2510.00929/) (2025)
- [Interpretable time series neural representation for classification purposes](https://mlsanity.com/paper/2310.16696/) (2023)
- [Fast Equivariant Imaging: Accelerating Unsupervised Learning and Model Adaptation via Inexact Splitting](https://mlsanity.com/paper/2507.06764/) (2025)
- [UNet-AF: Alias-free UNet architectures](https://mlsanity.com/paper/2603.11323/) (2026)

## Abstract (arXiv)

Convolutional neural networks lack shift equivariance due to the presence of downsampling layers. In image classification, adaptive polyphase downsampling (APS-D) was recently proposed to make CNNs perfectly shift invariant. However, in networks used for image reconstruction tasks, it can not by itself restore shift equivariance. We address this problem by proposing adaptive polyphase upsampling (APS-U), a non-linear extension of conventional upsampling, which allows CNNs with symmetric encoder-decoder architecture (for example U-Net) to exhibit perfect shift equivariance. With MRI and CT reconstruction experiments, we show that networks containing APS-D/U layers exhibit state of the art equivariance performance without sacrificing on image reconstruction quality. In addition, unlike prior methods like data augmentation and anti-aliasing, the gains in equivariance obtained from APS-D/U also extend to images outside the training distribution.