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

arXiv: https://arxiv.org/abs/2401.02277 · HTML: https://mlsanity.com/paper/2401.02277/

## 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 (most cited first, 15 of 15)

- [ResNet with one-neuron hidden layers is a Universal Approximator](https://mlsanity.com/paper/1806.10909/) (2018)
- [The universal approximation theorem for complex-valued neural networks](https://mlsanity.com/paper/2012.03351/) (2020)
- [PHNNs: Lightweight Neural Networks via Parameterized Hypercomplex Convolutions](https://mlsanity.com/paper/2110.04176/) (2021)
- [On the approximation by single hidden layer feedforward neural networks with fixed weights](https://mlsanity.com/paper/1708.06219/) (2017)
- [On the Kolmogorov neural networks](https://mlsanity.com/paper/2311.00049/) (2023)
- [A General Framework for Hypercomplex-valued Extreme Learning Machines](https://mlsanity.com/paper/2101.06166/) (2021)
- [Deep Octonion Networks](https://mlsanity.com/paper/1903.08478/) (2019)
- [Dual Quaternion Ambisonics Array for Six-Degree-of-Freedom Acoustic Representation](https://mlsanity.com/paper/2204.01851/) (2022)
- [Acute Lymphoblastic Leukemia Detection Using Hypercomplex-Valued Convolutional Neural Networks](https://mlsanity.com/paper/2205.13273/) (2022)
- [Learning Speech Emotion Representations in the Quaternion Domain](https://mlsanity.com/paper/2204.02385/) (2022)
- [Extending the Universal Approximation Theorem for a Broad Class of Hypercomplex-Valued Neural Networks](https://mlsanity.com/paper/2209.02456/) (2022)
- [Understanding Vector-Valued Neural Networks and Their Relationship with Real and Hypercomplex-Valued Neural Networks](https://mlsanity.com/paper/2309.07716/) (2023)
- [Backpropagation with N-D Vector-Valued Neurons Using Arbitrary Bilinear Products](https://mlsanity.com/paper/1805.09621/) (2018)
- [Approximation error of single hidden layer neural networks with fixed weights](https://mlsanity.com/paper/2202.03289/) (2022)
- [Online regularization of complex-valued neural networks for structure optimization in wireless-communication channel prediction](https://mlsanity.com/paper/1901.10121/) (2019)

## Built on by (most cited first, 11 of 11)

- [Universal approximation theorem for neural networks with inputs from a topological vector space](https://mlsanity.com/paper/2409.12913/) (2024)
- [Towards Explaining Hypercomplex Neural Networks](https://mlsanity.com/paper/2403.17929/) (2024)
- [On shallow feedforward neural networks with inputs from a topological space](https://mlsanity.com/paper/2504.02321/) (2025)
- [Topological DeepONets and a generalization of the Chen-Chen operator approximation theorem](https://mlsanity.com/paper/2603.11972/) (2026)
- [Fully tensorial approach to hypercomplex-valued neural networks](https://mlsanity.com/paper/2407.00449/) (2024)
- [Dynamics of Structured Complex-Valued Hopfield Neural Networks](https://mlsanity.com/paper/2503.19885/) (2025)
- [V-EfficientNets: Vector-Valued Efficiently Scaled Convolutional Neural Network Models](https://mlsanity.com/paper/2505.05659/) (2025)
- [A Cycle-Consistency Constrained Framework for Dynamic Solution Space Reduction in Noninjective Regression](https://mlsanity.com/paper/2507.04659/) (2025)
- [Exploring Polarimetric Properties Preservation during Reconstruction of PolSAR images using Complex-valued Convolutional Neural Networks](https://mlsanity.com/paper/2602.07094/) (2026)
- [Universality of shallow and deep neural networks on non-Euclidean spaces](https://mlsanity.com/paper/2602.23381/) (2026)
- [Ghost Features and Spooky Transfer Learning for Hypercomplex-Valued Neural Networks](https://mlsanity.com/paper/2608.07735/) (2026)

## 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.