# A Survey of Complex-Valued Neural Networks

arXiv: https://arxiv.org/abs/2101.12249 · HTML: https://mlsanity.com/paper/2101.12249/

## Key facts

- Cited by 61 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 166 references that are in the map.
- First posted on arXiv 28 Jan 2021.

## Builds on (most cited first, 15 of 15)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [On the Properties of Neural Machine Translation: Encoder-Decoder Approaches](https://mlsanity.com/paper/1409.1259/) (2014)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [Training Very Deep Networks](https://mlsanity.com/paper/1507.06228/) (2015)
- [Full-Capacity Unitary Recurrent Neural Networks](https://mlsanity.com/paper/1611.00035/) (2016)
- [Associative Long Short-Term Memory](https://mlsanity.com/paper/1602.03032/) (2016)
- [DeepNIS: Deep Neural Network for Nonlinear Electromagnetic Inverse Scattering](https://mlsanity.com/paper/1810.03990/) (2018)
- [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)
- [Complex-valued Neural Networks with Non-parametric Activation Functions](https://mlsanity.com/paper/1802.08026/) (2018)
- [Enhanced Radar Imaging Using a Complex-valued Convolutional Neural Network](https://mlsanity.com/paper/1712.10096/) (2017)
- [Evaluation of Complex-Valued Neural Networks on Real-Valued Classification Tasks](https://mlsanity.com/paper/1811.12351/) (2018)
- [Complex Evolution Recurrent Neural Networks (ceRNNs)](https://mlsanity.com/paper/1906.02246/) (2019)
- [Semi-supervised Complex-valued GAN for Polarimetric SAR Image Classification](https://mlsanity.com/paper/1906.03605/) (2019)

## Built on by (most cited first, 30 of 61)

- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [Complex-Valued Autoencoders for Object Discovery](https://mlsanity.com/paper/2204.02075/) (2022)
- [Symmetry in Neural Network Parameter Spaces](https://mlsanity.com/paper/2506.13018/) (2025)
- [Harmonics of Learning: Universal Fourier Features Emerge in Invariant Networks](https://mlsanity.com/paper/2312.08550/) (2023)
- [Building Blocks for a Complex-Valued Transformer Architecture](https://mlsanity.com/paper/2306.09827/) (2023)
- [FAN: Fourier Analysis Networks](https://mlsanity.com/paper/2410.02675/) (2024)
- [Theory and Implementation of Complex-Valued Neural Networks](https://mlsanity.com/paper/2302.08286/) (2023)
- [Exponential Separations in Symmetric Neural Networks](https://mlsanity.com/paper/2206.01266/) (2022)
- [C^*-algebra Net: A New Approach Generalizing Neural Network Parameters to C^*-algebra](https://mlsanity.com/paper/2206.09513/) (2022)
- [ATFNet: Adaptive Time-Frequency Ensembled Network for Long-term Time Series Forecasting](https://mlsanity.com/paper/2404.05192/) (2024)
- [Unitary convolutions for learning on graphs and groups](https://mlsanity.com/paper/2410.05499/) (2024)
- [k-strip: A novel segmentation algorithm in k-space for the application of skull stripping](https://mlsanity.com/paper/2205.09706/) (2022)
- [Complex-valued neural networks for voice anti-spoofing](https://mlsanity.com/paper/2308.11800/) (2023)
- [Towards NeuroAI: Introducing Neuronal Diversity into Artificial Neural Networks](https://mlsanity.com/paper/2301.09245/) (2023)
- [SignalNet: A Low Resolution Sinusoid Decomposition and Estimation Network](https://mlsanity.com/paper/2106.05490/) (2021)
- [CVKAN: Complex-Valued Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2502.02417/) (2025)
- [iFairy: the First 2-bit Complex LLM with All Parameters in ±1, ± i](https://mlsanity.com/paper/2508.05571/) (2025)
- [Complex-Value Spatio-temporal Graph Convolutional Neural Networks and its Applications to Electric Power Systems AI](https://mlsanity.com/paper/2208.08485/) (2022)
- [Non-contact Sensing for Anomaly Detection in Wind Turbine Blades: A focus-SVDD with Complex-Valued Auto-Encoder Approach](https://mlsanity.com/paper/2306.10808/) (2023)
- [SISMIK for brain MRI: Deep-learning-based motion estimation and model-based motion correction in k-space](https://mlsanity.com/paper/2312.13220/) (2023)
- [Adversarial Audio Synthesis with Complex-valued Polynomial Networks](https://mlsanity.com/paper/2206.06811/) (2022)
- [Enhancing deep neural networks through complex-valued representations and Kuramoto synchronization dynamics](https://mlsanity.com/paper/2502.21077/) (2025)
- [A Hybrid Complex-valued Neural Network Framework with Applications to Electroencephalogram (EEG)](https://mlsanity.com/paper/2207.14799/) (2022)
- [Spectral Batch Normalization: Normalization in the Frequency Domain](https://mlsanity.com/paper/2306.16999/) (2023)
- [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)
- [Steinmetz Neural Networks for Complex-Valued Data](https://mlsanity.com/paper/2409.10075/) (2024)
- [PHemoNet: A Multimodal Network for Physiological Signals](https://mlsanity.com/paper/2410.00010/) (2024)
- [Towards a Categorical Foundation of Deep Learning: A Survey](https://mlsanity.com/paper/2410.05353/) (2024)
- [DDRM-PR: Fourier Phase Retrieval using Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2501.03030/) (2025)

## Abstract (arXiv)

Artificial neural networks (ANNs) based machine learning models and especially deep learning models have been widely applied in computer vision, signal processing, wireless communications, and many other domains, where complex numbers occur either naturally or by design. However, most of the current implementations of ANNs and machine learning frameworks are using real numbers rather than complex numbers. There are growing interests in building ANNs using complex numbers, and exploring the potential advantages of the so-called complex-valued neural networks (CVNNs) over their real-valued counterparts. In this paper, we discuss the recent development of CVNNs by performing a survey of the works on CVNNs in the literature. Specifically, a detailed review of various CVNNs in terms of activation function, learning and optimization, input and output representations, and their applications in tasks such as signal processing and computer vision are provided, followed by a discussion on some pertinent challenges and future research directions.