# Complex-valued Neural Networks -- Theory and Analysis

arXiv: https://arxiv.org/abs/2312.06087 · HTML: https://mlsanity.com/paper/2312.06087/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 30 references that are in the map.
- First posted on arXiv 11 Dec 2023.

## Builds on (most cited first, 11 of 11)

- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [Phase-aware Speech Enhancement with Deep Complex U-Net](https://mlsanity.com/paper/1903.03107/) (2019)
- [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)
- [Neuronal Synchrony in Complex-Valued Deep Networks](https://mlsanity.com/paper/1312.6115/) (2013)
- [On Complex Valued Convolutional Neural Networks](https://mlsanity.com/paper/1602.09046/) (2016)
- [Theory and Implementation of Complex-Valued Neural Networks](https://mlsanity.com/paper/2302.08286/) (2023)
- [Complex-valued Convolutional Neural Networks for Enhanced Radar Signal Denoising and Interference Mitigation](https://mlsanity.com/paper/2105.00929/) (2021)
- [Pixel-Wise PolSAR Image Classification via a Novel Complex-Valued Deep Fully Convolutional Network](https://mlsanity.com/paper/1909.13299/) (2019)

## Built on by (most cited first, 2 of 2)

- [EuleroDec: A Complex-Valued RVQ-VAE for Efficient and Robust Audio Coding](https://mlsanity.com/paper/2601.17517/) (2026)
- [K"ahler landscapes for complex neural network descents and guarantees including a search and destroy of the Calabi-Yau manifold](https://mlsanity.com/paper/2608.19584/) (2026)

## Abstract (arXiv)

Complex-valued neural networks (CVNNs) have recently been successful in various pioneering areas which involve wave-typed information and frequency-domain processing. This work addresses different structures and classification of CVNNs. The theory behind complex activation functions, implications related to complex differentiability and special activations for CVNN output layers are presented. The work also discusses CVNN learning and optimization using gradient and non-gradient based algorithms. Complex Backpropagation utilizing complex chain rule is also explained in terms of Wirtinger calculus. Moreover, special modules for building CVNN models, such as complex batch normalization and complex random initialization are also discussed. The work also highlights libraries and software blocks proposed for CVNN implementations and discusses future directions. The objective of this work is to understand the dynamics and most recent developments of CVNNs.