# Novel Complex-Valued Hopfield Neural Networks with Phase and Magnitude Quantization

arXiv: https://arxiv.org/abs/2507.00461 · HTML: https://mlsanity.com/paper/2507.00461/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 48 references that are in the map.
- First posted on arXiv 1 Jul 2025.

## Builds on (most cited first, 10 of 10)

- [Hopfield Networks is All You Need](https://mlsanity.com/paper/2008.02217/) (2020)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [Large Associative Memory Problem in Neurobiology and Machine Learning](https://mlsanity.com/paper/2008.06996/) (2020)
- [The universal approximation theorem for complex-valued neural networks](https://mlsanity.com/paper/2012.03351/) (2020)
- [Complex-valued Neural Networks with Non-parametric Activation Functions](https://mlsanity.com/paper/1802.08026/) (2018)
- [A Broad Class of Discrete-Time Hypercomplex-Valued Hopfield Neural Networks](https://mlsanity.com/paper/1902.05478/) (2019)
- [New Insights on Learning Rules for Hopfield Networks: Memory and Objective Function Minimisation](https://mlsanity.com/paper/2010.01472/) (2020)
- [Hierarchical Associative Memory, Parallelized MLP-Mixer, and Symmetry Breaking](https://mlsanity.com/paper/2406.12220/) (2024)
- [CVKAN: Complex-Valued Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2502.02417/) (2025)
- [Dynamics of Structured Complex-Valued Hopfield Neural Networks](https://mlsanity.com/paper/2503.19885/) (2025)

## Abstract (arXiv)

This research paper introduces two novel complex-valued Hopfield neural networks (CvHNNs) that incorporate phase and magnitude quantization. The first CvHNN employs a ceiling-type activation function that operates on the rectangular coordinate representation of the complex net contribution. The second CvHNN similarly incorporates phase and magnitude quantization but utilizes a ceiling-type activation function based on the polar coordinate representation of the complex net contribution. The proposed CvHNNs, with their phase and magnitude quantization, significantly increase the number of states compared to existing models in the literature, thereby expanding the range of potential applications for CvHNNs.