# A Broad Class of Discrete-Time Hypercomplex-Valued Hopfield Neural Networks

arXiv: https://arxiv.org/abs/1902.05478 · HTML: https://mlsanity.com/paper/1902.05478/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Feb 2019.

## Built on by (most cited first, 5 of 5)

- [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)
- [Novel computational workflows for natural and biomedical image processing based on hypercomplex algebras](https://mlsanity.com/paper/2502.07758/) (2025)
- [Dynamics of Structured Complex-Valued Hopfield Neural Networks](https://mlsanity.com/paper/2503.19885/) (2025)
- [Novel Complex-Valued Hopfield Neural Networks with Phase and Magnitude Quantization](https://mlsanity.com/paper/2507.00461/) (2025)

## Abstract (arXiv)

In this paper, we address the stability of a broad class of discrete-time hypercomplex-valued Hopfield-type neural networks. To ensure the neural networks belonging to this class always settle down at a stationary state, we introduce novel hypercomplex number systems referred to as real-part associative hypercomplex number systems. Real-part associative hypercomplex number systems generalize the well-known Cayley-Dickson algebras and real Clifford algebras and include the systems of real numbers, complex numbers, dual numbers, hyperbolic numbers, quaternions, tessarines, and octonions as particular instances. Apart from the novel hypercomplex number systems, we introduce a family of hypercomplex-valued activation functions called B-projection functions. Broadly speaking, a B-projection function projects the activation potential onto the set of all possible states of a hypercomplex-valued neuron. Using the theory presented in this paper, we confirm the stability analysis of several discrete-time hypercomplex-valued Hopfield-type neural networks from the literature. Moreover, we introduce and provide the stability analysis of a general class of Hopfield-type neural networks on Cayley-Dickson algebras.