# Understanding Vector-Valued Neural Networks and Their Relationship with Real and Hypercomplex-Valued Neural Networks

arXiv: https://arxiv.org/abs/2309.07716 · HTML: https://mlsanity.com/paper/2309.07716/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 38 references that are in the map.
- First posted on arXiv 14 Sep 2023.

## Builds on (most cited first, 12 of 12)

- [Squeeze-and-Excitation Networks](https://mlsanity.com/paper/1709.01507/) (2017)
- [Deep Learning in Neural Networks: An Overview](https://mlsanity.com/paper/1404.7828/) (2014)
- [Optimal approximation of piecewise smooth functions using deep ReLU neural networks](https://mlsanity.com/paper/1709.05289/) (2017)
- [Beyond Fully-Connected Layers with Quaternions: Parameterization of Hypercomplex Multiplications with 1/n Parameters](https://mlsanity.com/paper/2102.08597/) (2021)
- [Deep Quaternion Networks](https://mlsanity.com/paper/1712.04604/) (2017)
- [Geometric Clifford Algebra Networks](https://mlsanity.com/paper/2302.06594/) (2023)
- [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)
- [A General Framework for Hypercomplex-valued Extreme Learning Machines](https://mlsanity.com/paper/2101.06166/) (2021)
- [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)
- [Extending the Universal Approximation Theorem for a Broad Class of Hypercomplex-Valued Neural Networks](https://mlsanity.com/paper/2209.02456/) (2022)

## Built on by (most cited first, 4 of 4)

- [Universal Approximation Theorem for Vector- and Hypercomplex-Valued Neural Networks](https://mlsanity.com/paper/2401.02277/) (2024)
- [Time Series Compression using Quaternion Valued Neural Networks and Quaternion Backpropagation](https://mlsanity.com/paper/2403.11722/) (2024)
- [V-EfficientNets: Vector-Valued Efficiently Scaled Convolutional Neural Network Models](https://mlsanity.com/paper/2505.05659/) (2025)
- [Ghost Features and Spooky Transfer Learning for Hypercomplex-Valued Neural Networks](https://mlsanity.com/paper/2608.07735/) (2026)

## Abstract (arXiv)

Despite the many successful applications of deep learning models for multidimensional signal and image processing, most traditional neural networks process data represented by (multidimensional) arrays of real numbers. The intercorrelation between feature channels is usually expected to be learned from the training data, requiring numerous parameters and careful training. In contrast, vector-valued neural networks are conceived to process arrays of vectors and naturally consider the intercorrelation between feature channels. Consequently, they usually have fewer parameters and often undergo more robust training than traditional neural networks. This paper aims to present a broad framework for vector-valued neural networks, referred to as V-nets. In this context, hypercomplex-valued neural networks are regarded as vector-valued models with additional algebraic properties. Furthermore, this paper explains the relationship between vector-valued and traditional neural networks. Precisely, a vector-valued neural network can be obtained by placing restrictions on a real-valued model to consider the intercorrelation between feature channels. Finally, we show how V-nets, including hypercomplex-valued neural networks, can be implemented in current deep-learning libraries as real-valued networks.