# Theory and Implementation of Complex-Valued Neural Networks

arXiv: https://arxiv.org/abs/2302.08286 · HTML: https://mlsanity.com/paper/2302.08286/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 61 references that are in the map.
- First posted on arXiv 16 Feb 2023.

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

- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [A Survey of Complex-Valued Neural Networks](https://mlsanity.com/paper/2101.12249/) (2021)
- [Better than Real: Complex-valued Neural Nets for MRI Fingerprinting](https://mlsanity.com/paper/1707.00070/) (2017)
- [On Complex Valued Convolutional Neural Networks](https://mlsanity.com/paper/1602.09046/) (2016)
- [Complex-valued Neural Networks with Non-parametric Activation Functions](https://mlsanity.com/paper/1802.08026/) (2018)
- [Evaluation of Complex-Valued Neural Networks on Real-Valued Classification Tasks](https://mlsanity.com/paper/1811.12351/) (2018)
- [Pixel-Wise PolSAR Image Classification via a Novel Complex-Valued Deep Fully Convolutional Network](https://mlsanity.com/paper/1909.13299/) (2019)
- [On the Computation of Complex-valued Gradients with Application to Statistically Optimum Beamforming](https://mlsanity.com/paper/1701.00392/) (2017)

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

- [Cauchy activation function and XNet](https://mlsanity.com/paper/2409.19221/) (2024)
- [CoNO: Complex Neural Operator for Continous Dynamical Physical Systems](https://mlsanity.com/paper/2406.02597/) (2024)
- [Complex-Valued Neural Networks for Data-Driven Signal Processing and Signal Understanding](https://mlsanity.com/paper/2309.07948/) (2023)
- [Complex-valued Neural Networks -- Theory and Analysis](https://mlsanity.com/paper/2312.06087/) (2023)
- [The neural networks with tensor weights and emergent fermionic Wick rules in the large-width limit](https://mlsanity.com/paper/2507.05303/) (2025)
- [Emerging Approaches for THz Array Imaging: A Tutorial Review and Software Tool](https://mlsanity.com/paper/2309.08844/) (2023)
- [Neural Controlled Differential Equations with Quantum Hidden Evolutions](https://mlsanity.com/paper/2404.19673/) (2024)
- [A Complex UNet Approach for Non-Invasive Fetal ECG Extraction Using Single-Channel Dry Textile Electrodes](https://mlsanity.com/paper/2506.22457/) (2025)
- [Exploring Polarimetric Properties Preservation during Reconstruction of PolSAR images using Complex-valued Convolutional Neural Networks](https://mlsanity.com/paper/2602.07094/) (2026)
- [Batch Normalization for Neural Networks on Complex Domains](https://mlsanity.com/paper/2605.00467/) (2026)
- [Entanglement is Half the Story: Post-Selection vs. Partial Traces](https://mlsanity.com/paper/2605.02385/) (2026)

## Abstract (arXiv)

This work explains in detail the theory behind Complex-Valued Neural Network (CVNN), including Wirtinger calculus, complex backpropagation, and basic modules such as complex layers, complex activation functions, or complex weight initialization. We also show the impact of not adapting the weight initialization correctly to the complex domain. This work presents a strong focus on the implementation of such modules on Python using cvnn toolbox. We also perform simulations on real-valued data, casting to the complex domain by means of the Hilbert Transform, and verifying the potential interest of CVNN even for non-complex data.