# Neural Fourier Transform: A General Approach to Equivariant Representation Learning

arXiv: https://arxiv.org/abs/2305.18484 · HTML: https://mlsanity.com/paper/2305.18484/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 51 references that are in the map.
- First posted on arXiv 29 May 2023.

## Builds on (most cited first, 29 of 29)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Optuna: A Next-generation Hyperparameter Optimization Framework](https://mlsanity.com/paper/1907.10902/) (2019)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [PlenOctrees for Real-time Rendering of Neural Radiance Fields](https://mlsanity.com/paper/2103.14024/) (2021)
- [Disentangling by Factorising](https://mlsanity.com/paper/1802.05983/) (2018)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Deep Learning for Classical Japanese Literature](https://mlsanity.com/paper/1812.01718/) (2018)
- [Kubric: A scalable dataset generator](https://mlsanity.com/paper/2203.03570/) (2022)
- [ABO: Dataset and Benchmarks for Real-World 3D Object Understanding](https://mlsanity.com/paper/2110.06199/) (2021)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups](https://mlsanity.com/paper/1802.03690/) (2018)
- [Towards a Definition of Disentangled Representations](https://mlsanity.com/paper/1812.02230/) (2018)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [A General Theory of Equivariant CNNs on Homogeneous Spaces](https://mlsanity.com/paper/1811.02017/) (2018)
- [Learning Steerable Filters for Rotation Equivariant CNNs](https://mlsanity.com/paper/1711.07289/) (2017)
- [A Practical Method for Constructing Equivariant Multilayer Perceptrons for Arbitrary Matrix Groups](https://mlsanity.com/paper/2104.09459/) (2021)
- [Equivariant Point Network for 3D Point Cloud Analysis](https://mlsanity.com/paper/2103.14147/) (2021)
- [Meta-Learning Symmetries by Reparameterization](https://mlsanity.com/paper/2007.02933/) (2020)
- [Explorations in Homeomorphic Variational Auto-Encoding](https://mlsanity.com/paper/1807.04689/) (2018)
- [Learning Symmetric Embeddings for Equivariant World Models](https://mlsanity.com/paper/2204.11371/) (2022)
- [Equivariant Neural Rendering](https://mlsanity.com/paper/2006.07630/) (2020)
- [Learning the Irreducible Representations of Commutative Lie Groups](https://mlsanity.com/paper/1402.4437/) (2014)
- [Unsupervised Learning of Group Invariant and Equivariant Representations](https://mlsanity.com/paper/2202.07559/) (2022)
- [Spherical Regression: Learning Viewpoints, Surface Normals and 3D Rotations on n-Spheres](https://mlsanity.com/paper/1904.05404/) (2019)
- [Intertwiners between Induced Representations (with Applications to the Theory of Equivariant Neural Networks)](https://mlsanity.com/paper/1803.10743/) (2018)
- [Unsupervised Learning of Equivariant Structure from Sequences](https://mlsanity.com/paper/2210.05972/) (2022)
- [Learning Structured Latent Factors from Dependent Data:A Generative Model Framework from Information-Theoretic Perspective](https://mlsanity.com/paper/2007.10623/) (2020)

## Built on by (most cited first, 12 of 12)

- [GTA: A Geometry-Aware Attention Mechanism for Multi-View Transformers](https://mlsanity.com/paper/2310.10375/) (2023)
- [Steerers: A framework for rotation equivariant keypoint descriptors](https://mlsanity.com/paper/2312.02152/) (2023)
- [DiLA: Disentangled Latent Action World Models](https://mlsanity.com/paper/2605.15725/) (2026)
- [Neural Isometries: Taming Transformations for Equivariant ML](https://mlsanity.com/paper/2405.19296/) (2024)
- [Affine steerers for structured keypoint description](https://mlsanity.com/paper/2408.14186/) (2024)
- [AtlasD: Automatic Local Symmetry Discovery](https://mlsanity.com/paper/2504.10777/) (2025)
- [Equivariance by Contrast: Identifiable Equivariant Embeddings from Unlabeled Finite Group Actions](https://mlsanity.com/paper/2510.21706/) (2025)
- [Inter-environmental world modeling for continuous and compositional dynamics](https://mlsanity.com/paper/2503.09911/) (2025)
- [KEEC: Koopman Embedded Equivariant Control](https://mlsanity.com/paper/2312.01544/) (2023)
- [Algebraic Priors for Approximately Equivariant Networks](https://mlsanity.com/paper/2506.08244/) (2025)
- [Equivariant Latent Alignment via Flow Matching under Group Symmetries](https://mlsanity.com/paper/2605.30705/) (2026)
- [Blind Recovery of Latent Domains via Unsupervised Symmetry Discovery](https://mlsanity.com/paper/2606.17782/) (2026)

## Abstract (arXiv)

Symmetry learning has proven to be an effective approach for extracting the hidden structure of data, with the concept of equivariance relation playing the central role. However, most of the current studies are built on architectural theory and corresponding assumptions on the form of data. We propose Neural Fourier Transform (NFT), a general framework of learning the latent linear action of the group without assuming explicit knowledge of how the group acts on data. We present the theoretical foundations of NFT and show that the existence of a linear equivariant feature, which has been assumed ubiquitously in equivariance learning, is equivalent to the existence of a group invariant kernel on the dataspace. We also provide experimental results to demonstrate the application of NFT in typical scenarios with varying levels of knowledge about the acting group.