mlsanityData to 7 Oct 2026

Equivalence of approximation by convolutional neural networks and fully-connected networks

arXiv 1809.00973 ↗First posted 4 Sep 2018math.FA cs.LGcited by 38

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Abstract (arXiv)

Convolutional neural networks are the most widely used type of neural networks in applications. In mathematical analysis, however, mostly fully-connected networks are studied. In this paper, we establish a connection between both network architectures. Using this connection, we show that all upper and lower bounds concerning approximation rates of fully-connected neural networks for functions f ∈ C -- for an arbitrary function class C -- translate to essentially the same bounds concerning approximation rates of convolutional neural networks for functions f ∈ Cequi, with the class Cequi consisting of all translation equivariant functions whose first coordinate belongs to C. All presented results consider exclusively the case of convolutional neural networks without any pooling operation and with circular convolutions, i.e., not based on zero-padding.

Key facts. Cited by 38 papers in the map (data through 7 Oct 2026). First posted on arXiv 4 Sep 2018.

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Citing papers per year: 2021: 6; 2022: 11; 2023: 7; 2024: 6; 2025: 4; 2026: 4.

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  1. 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 207
  2. 2022Anti-Oversmoothing in Deep Vision Transformers via the Fourier Domain Analysis: From Theory to Practicecited by 136
  3. 2021Besov Function Approximation and Binary Classification on Low-Dimensional Manifolds Using Convolutional Residual Networkscited by 32
  4. 2021Universal Approximation Under Constraints is Possible with Transformerscited by 32
  5. 2021Theory of Deep Convolutional Neural Networks III: Approximating Radial Functionscited by 22
  6. 2021Approximation Properties of Deep ReLU CNNscited by 15
  7. 2023A Brief Survey on the Approximation Theory for Sequence Modellingcited by 15
  8. 2023Optimal rates of approximation by shallow ReLUk neural networks and applications to nonparametric regressioncited by 15
  9. 2022Fully-Connected Network on Noncompact Symmetric Space and Ridgelet Transform based on Helgason-Fourier Analysiscited by 14
  10. 2021Convergence of Deep Convolutional Neural Networkscited by 13
  11. 2024Generalization Error Guaranteed Auto-Encoder-Based Nonlinear Model Reduction for Operator Learningcited by 12
  12. 2022Universality of Group Convolutional Neural Networks Based on Ridgelet Analysis on Groupscited by 11
26 more
  1. 2022Learning Ability of Interpolating Deep Convolutional Neural Networkscited by 9
  2. 2024A practical existence theorem for reduced order models based on convolutional autoencoderscited by 9
  3. 2022Deep Neural Network Approximation of Invariant Functions through Dynamical Systemscited by 8
  4. 2026Identifiable Equivariant Networks are Layerwise Equivariantcited by 7
  5. 2022Benefits of Overparameterized Convolutional Residual Networks: Function Approximation under Smoothness Constraintcited by 6
  6. 2022On the Universal Approximation Property of Deep Fully Convolutional Neural Networkscited by 6
  7. 2023Tailor: Altering Skip Connections for Resource-Efficient Inferencecited by 6
  8. 2021Universal Regular Conditional Distributionscited by 4
  9. 2022Analysis of convolutional neural network image classifiers in a rotationally symmetric modelcited by 4
  10. 2023Deep Convolutional Neural Networks with Zero-Padding: Feature Extraction and Learningcited by 4
  11. 2025Higher Order Approximation Rates for ReLU CNNs in Korobov Spacescited by 4
  12. 2025On Universality Classes of Equivariant Networkscited by 4
  13. 2023Universal approximation with complex-valued deep narrow neural networkscited by 3
  14. 2024Spectral GNN via Two-dimensional (2-D) Graph Convolutioncited by 3
  15. 2024Deep Neural Networks are Adaptive to Function Regularity and Data Distribution in Approximation and Estimationcited by 3
  16. 2025New universal operator approximation theorem for encoder-decoder architecturescited by 3
  17. 2023Equivariant and Steerable Neural Networks: A review with special emphasis on the symmetric groupcited by 2
  18. 2024Approximation Rates and VC-Dimension Bounds for (P)ReLU MLP Mixture of Expertscited by 2
  19. 2025Two-Dimensional Deep ReLU CNN Approximation for Korobov Functions: A Constructive Approachcited by 2
  20. 2022Optimal Convergence Rates of Deep Convolutional Neural Networks: Additive Ridge Functionscited by 1
  21. 2022Neural Network Approximation of Continuous Functions in High Dimensions with Applications to Inverse Problemscited by 1
  22. 2026Equivalence of approximation by networks of single- and multi-spike neuronscited by 1
  23. 2022Design equivariant neural networks for 3D point cloudcited by 0
  24. 2024Physics-Informed Neural Network Based Digital Image Correlation Methodcited by 0
  25. 2026Adaptivity Under Realizability Constraints: Comparing In-Context and Agentic Learningcited by 0
  26. 2026Neural Flow Operators can Approximate any Operator: Abstract Frameworks and Universal Approximationscited by 0