mlsanityData to 7 Oct 2026

Universal approximations of permutation invariant/equivariant functions by deep neural networks

arXiv 1903.01939 ↗First posted 5 Mar 2019cs.LG stat.MLcited by 29

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Abstract (arXiv)

In this paper, we develop a theory about the relationship between G-invariant/equivariant functions and deep neural networks for finite group G. Especially, for a given G-invariant/equivariant function, we construct its universal approximator by deep neural network whose layers equip G-actions and each affine transformations are G-equivariant/invariant. Due to representation theory, we can show that this approximator has exponentially fewer free parameters than usual models.

Key facts. Cited by 29 papers in the map (data through 7 Oct 2026). First posted on arXiv 5 Mar 2019.

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Citing papers per year: 2021: 6; 2022: 7; 2023: 9; 2024: 2; 2025: 3; 2026: 2.

29 papers.

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  1. 2021Statistically Meaningful Approximation: a Case Study on Approximating Turing Machines with Transformerscited by 101
  2. 2022Equivariant Hypergraph Diffusion Neural Operatorscited by 45
  3. 2022Low Dimensional Invariant Embeddings for Universal Geometric Learningcited by 38
  4. 2023The Exact Sample Complexity Gain from Invariances for Kernel Regressioncited by 27
  5. 2023Explainable Equivariant Neural Networks for Particle Physics: PELICANcited by 22
  6. 2022Permutation Invariant Representations with Applications to Graph Deep Learningcited by 20
  7. 2021Explicitly antisymmetrized neural network layers for variational Monte Carlo simulationcited by 14
  8. 2021Neural Trees for Learning on Graphscited by 13
  9. 2022Universality of Group Convolutional Neural Networks Based on Ridgelet Analysis on Groupscited by 11
  10. 2021Symmetry reduction for deep reinforcement learning active control of chaotic spatiotemporal dynamicscited by 10
  11. 2022Deep Neural Network Approximation of Invariant Functions through Dynamical Systemscited by 8
  12. 2023Postprocessing of Ensemble Weather Forecasts Using Permutation-invariant Neural Networkscited by 7
17 more
  1. 2023Polynomial Width is Sufficient for Set Representation with High-dimensional Featurescited by 6
  2. 2022KASAM: Spline Additive Models for Function Approximationcited by 4
  3. 2023Learning multi-modal generative models with permutation-invariant encoders and tighter variational objectivescited by 4
  4. 2023Addressing imperfect symmetry: A novel symmetry-learning actor-critic extensioncited by 4
  5. 2025Approximation of Permutation Invariant Polynomials by Transformers: Efficient Construction in Column-Sizecited by 3
  6. 2026Rethinking Zero-Shot Time Series Classification: From Task-specific Classifiers to In-Context Inferencecited by 3
  7. 2021Capacity of Group-invariant Linear Readouts from Equivariant Representations: How Many Objects can be Linearly Classified Under All Possible Views?cited by 2
  8. 2023P-Tensors: a General Formalism for Constructing Higher Order Message Passing Networkscited by 2
  9. 2024Current Symmetry Group Equivariant Convolution Frameworks for Representation Learningcited by 2
  10. 2026Any-Dimensional Invariant Universalitycited by 1
  11. 2021Pointspectrum: Equivariance Meets Laplacian Filtering for Graph Representation Learningcited by 0
  12. 2022On the Approximation and Complexity of Deep Neural Networks to Invariant Functionscited by 0
  13. 2023A Structural Approach to the Design of Domain Specific Neural Network Architecturescited by 0
  14. 2023PolyGET: Accelerating Polymer Simulations by Accurate and Generalizable Forcefield with Equivariant Transformercited by 0
  15. 2024Decomposition of Equivariant Maps via Invariant Maps: Application to Universal Approximation under Symmetrycited by 0
  16. 2025Transformers Are Universally Consistentcited by 0
  17. 2025On the Dimension-Free Approximation of Deep Neural Networks for Symmetric Korobov Functionscited by 0