# A General Framework for Hypercomplex-valued Extreme Learning Machines

arXiv: https://arxiv.org/abs/2101.06166 · HTML: https://mlsanity.com/paper/2101.06166/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 75 references that are in the map.
- First posted on arXiv 15 Jan 2021.

## Builds on (most cited first, 4 of 4)

- [Deep TEN: Texture Encoding Network](https://mlsanity.com/paper/1612.02844/) (2016)
- [Quaternion Gradient and Hessian](https://mlsanity.com/paper/1406.3587/) (2014)
- [A Broad Class of Discrete-Time Hypercomplex-Valued Hopfield Neural Networks](https://mlsanity.com/paper/1902.05478/) (2019)
- [A study on effectiveness of extreme learning machine](https://mlsanity.com/paper/1409.3924/) (2014)

## Built on by (most cited first, 9 of 9)

- [Universal Approximation Theorem for Vector- and Hypercomplex-Valued Neural Networks](https://mlsanity.com/paper/2401.02277/) (2024)
- [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)
- [Understanding Vector-Valued Neural Networks and Their Relationship with Real and Hypercomplex-Valued Neural Networks](https://mlsanity.com/paper/2309.07716/) (2023)
- [Current Symmetry Group Equivariant Convolution Frameworks for Representation Learning](https://mlsanity.com/paper/2409.07327/) (2024)
- [Dynamics of Structured Complex-Valued Hopfield Neural Networks](https://mlsanity.com/paper/2503.19885/) (2025)
- [Selected aspects of complex, hypercomplex and fuzzy neural networks](https://mlsanity.com/paper/2301.00007/) (2022)
- [Training-Free Color-Aware Adversarial Diffusion Sanitization for Diffusion Stegomalware Defense at Security Gateways](https://mlsanity.com/paper/2512.24499/) (2025)
- [Ghost Features and Spooky Transfer Learning for Hypercomplex-Valued Neural Networks](https://mlsanity.com/paper/2608.07735/) (2026)

## Abstract (arXiv)

This paper aims to establish a framework for extreme learning machines (ELMs) on general hypercomplex algebras. Hypercomplex neural networks are machine learning models that feature higher-dimension numbers as parameters, inputs, and outputs. Firstly, we review broad hypercomplex algebras and show a framework to operate in these algebras through real-valued linear algebra operations in a robust manner. We proceed to explore a handful of well-known four-dimensional examples. Then, we propose the hypercomplex-valued ELMs and derive their learning using a hypercomplex-valued least-squares problem. Finally, we compare real and hypercomplex-valued ELM models' performance in an experiment on time-series prediction and another on color image auto-encoding. The computational experiments highlight the excellent performance of hypercomplex-valued ELMs to treat high-dimensional data, including models based on unusual hypercomplex algebras.