mlsanityData to 14 Oct 2026

Universal Approximation Theorem for Vector- and Hypercomplex-Valued Neural Networks

arXiv 2401.02277 ↗First posted 4 Jan 2024cs.LG cs.NEcited by 11

Not on the map. Why?

Abstract (arXiv)

The universal approximation theorem states that a neural network with one hidden layer can approximate continuous functions on compact sets with any desired precision. This theorem supports using neural networks for various applications, including regression and classification tasks. Furthermore, it is valid for real-valued neural networks and some hypercomplex-valued neural networks such as complex-, quaternion-, tessarine-, and Clifford-valued neural networks. However, hypercomplex-valued neural networks are a type of vector-valued neural network defined on an algebra with additional algebraic or geometric properties. This paper extends the universal approximation theorem for a wide range of vector-valued neural networks, including hypercomplex-valued models as particular instances. Precisely, we introduce the concept of non-degenerate algebra and state the universal approximation theorem for neural networks defined on such algebras.

Key facts. Cited by 11 papers in the map (data through 14 Oct 2026). Builds on 15 of its 63 references that are in the map. First posted on arXiv 4 Jan 2024.

Builds on

15 of 63 references.

References · 15

  1. 2018ResNet with one-neuron hidden layers is a Universal Approximatorcited by 79
  2. 2020The universal approximation theorem for complex-valued neural networkscited by 28
  3. 2021PHNNs: Lightweight Neural Networks via Parameterized Hypercomplex Convolutionscited by 26
  4. 2017On the approximation by single hidden layer feedforward neural networks with fixed weightscited by 19
  5. 2023On the Kolmogorov neural networkscited by 19
  6. 2021A General Framework for Hypercomplex-valued Extreme Learning Machinescited by 9
  7. 2019Deep Octonion Networkscited by 8
  8. 2022Dual Quaternion Ambisonics Array for Six-Degree-of-Freedom Acoustic Representationcited by 8
  9. 2022Acute Lymphoblastic Leukemia Detection Using Hypercomplex-Valued Convolutional Neural Networkscited by 8
  10. 2022Learning Speech Emotion Representations in the Quaternion Domaincited by 7
  11. 2022Extending the Universal Approximation Theorem for a Broad Class of Hypercomplex-Valued Neural Networkscited by 5
  12. 2023Understanding Vector-Valued Neural Networks and Their Relationship with Real and Hypercomplex-Valued Neural Networkscited by 4
3 more
  1. 2018Backpropagation with N-D Vector-Valued Neurons Using Arbitrary Bilinear Productscited by 2
  2. 2022Approximation error of single hidden layer neural networks with fixed weightscited by 2
  3. 2019Online regularization of complex-valued neural networks for structure optimization in wireless-communication channel predictioncited by 1

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 3; 2025: 4; 2026: 4.

11 papers.

  1. 2024Universal approximation theorem for neural networks with inputs from a topological vector spacecited by 4
  2. 2024Towards Explaining Hypercomplex Neural Networkscited by 3
  3. 2025On shallow feedforward neural networks with inputs from a topological spacecited by 2
  4. 2026Topological DeepONets and a generalization of the Chen-Chen operator approximation theoremcited by 2
  5. 2024Fully tensorial approach to hypercomplex-valued neural networkscited by 1
  6. 2025Dynamics of Structured Complex-Valued Hopfield Neural Networkscited by 1
  7. 2025V-EfficientNets: Vector-Valued Efficiently Scaled Convolutional Neural Network Modelscited by 0
  8. 2025A Cycle-Consistency Constrained Framework for Dynamic Solution Space Reduction in Noninjective Regressioncited by 0
  9. 2026Exploring Polarimetric Properties Preservation during Reconstruction of PolSAR images using Complex-valued Convolutional Neural Networkscited by 0
  10. 2026Universality of shallow and deep neural networks on non-Euclidean spacescited by 0
  11. 2026Ghost Features and Spooky Transfer Learning for Hypercomplex-Valued Neural Networkscited by 0