# Convolutional layers are equivariant to discrete shifts but not continuous translations

arXiv: https://arxiv.org/abs/2206.04979 · HTML: https://mlsanity.com/paper/2206.04979/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 12 references that are in the map.
- First posted on arXiv 10 Jun 2022.

## Builds on (most cited first, 6 of 6)

- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Making Convolutional Networks Shift-Invariant Again](https://mlsanity.com/paper/1904.11486/) (2019)
- [Why do deep convolutional networks generalize so poorly to small image transformations?](https://mlsanity.com/paper/1805.12177/) (2018)
- [On Translation Invariance in CNNs: Convolutional Layers can Exploit Absolute Spatial Location](https://mlsanity.com/paper/2003.07064/) (2020)
- [Truly shift-invariant convolutional neural networks](https://mlsanity.com/paper/2011.14214/) (2020)

## Built on by (most cited first, 1 of 1)

- [What Do You Need for Compositional Generalization in Diffusion Planning?](https://mlsanity.com/paper/2505.18083/) (2025)

## Abstract (arXiv)

The purpose of this short and simple note is to clarify a common misconception about convolutional neural networks (CNNs). CNNs are made up of convolutional layers which are shift equivariant due to weight sharing. However, convolutional layers are not translation equivariant, even when boundary effects are ignored and when pooling and subsampling are absent. This is because shift equivariance is a discrete symmetry while translation equivariance is a continuous symmetry. This fact is well known among researchers in equivariant machine learning, but is usually overlooked among non-experts. To minimize confusion, we suggest using the term `shift equivariance' to refer to discrete shifts in pixels and `translation equivariance' to refer to continuous translations.