# On Complex Valued Convolutional Neural Networks

arXiv: https://arxiv.org/abs/1602.09046 · HTML: https://mlsanity.com/paper/1602.09046/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Feb 2016.

## Built on by (most cited first, 26 of 26)

- [How important are activation functions in regression and classification? A survey, performance comparison, and future directions](https://mlsanity.com/paper/2209.02681/) (2022)
- [CubeLearn: End-to-end Learning for Human Motion Recognition from Raw mmWave Radar Signals](https://mlsanity.com/paper/2111.03976/) (2021)
- [Building Blocks for a Complex-Valued Transformer Architecture](https://mlsanity.com/paper/2306.09827/) (2023)
- [Theory and Implementation of Complex-Valued Neural Networks](https://mlsanity.com/paper/2302.08286/) (2023)
- [Theoretical Exploration of Flexible Transmitter Model](https://mlsanity.com/paper/2111.06027/) (2021)
- [Quaternion Convolutional Neural Networks: Current Advances and Future Directions](https://mlsanity.com/paper/2307.08663/) (2023)
- [A Survey of Deep Learning for Complex Speech Spectrograms](https://mlsanity.com/paper/2505.08694/) (2025)
- [A Hybrid Complex-valued Neural Network Framework with Applications to Electroencephalogram (EEG)](https://mlsanity.com/paper/2207.14799/) (2022)
- [Adapting Pre-trained Language Models for Quantum Natural Language Processing](https://mlsanity.com/paper/2302.13812/) (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)
- [Deep Learning-based MRI Reconstruction with Artificial Fourier Transform Network (AFTNet)](https://mlsanity.com/paper/2312.10892/) (2023)
- [Steinmetz Neural Networks for Complex-Valued Data](https://mlsanity.com/paper/2409.10075/) (2024)
- [Learning Convolutional Neural Networks in the Frequency Domain](https://mlsanity.com/paper/2204.06718/) (2022)
- [Towards White-Box Deep Wireless Sensing](https://mlsanity.com/paper/2507.21799/) (2025)
- [RotaTouille: Rotation Equivariant Deep Learning for Contours](https://mlsanity.com/paper/2508.16359/) (2025)
- [When do complex-valued neural networks help? A study of representation, geometry, and optimization](https://mlsanity.com/paper/2605.27673/) (2026)
- [Neural Networks with Complex-Valued Weights Have No Spurious Local Minima](https://mlsanity.com/paper/2103.07287/) (2021)
- [Spectral Complexity-scaled Generalization Bound of Complex-valued Neural Networks](https://mlsanity.com/paper/2112.03467/) (2021)
- [Emerging Approaches for THz Array Imaging: A Tutorial Review and Software Tool](https://mlsanity.com/paper/2309.08844/) (2023)
- [TFDMNet: A Novel Network Structure Combines the Time Domain and Frequency Domain Features](https://mlsanity.com/paper/2401.15949/) (2024)
- [GASPnet: Global Agreement to Synchronize Phases](https://mlsanity.com/paper/2507.16674/) (2025)
- [Phase-Aware Deep Learning with Complex-Valued CNNs for Audio Signal Applications](https://mlsanity.com/paper/2510.09926/) (2025)
- [Quantum Decision Transformers (QDT): Synergistic Entanglement and Interference for Offline Reinforcement Learning](https://mlsanity.com/paper/2512.14726/) (2025)
- [Exploring Polarimetric Properties Preservation during Reconstruction of PolSAR images using Complex-valued Convolutional Neural Networks](https://mlsanity.com/paper/2602.07094/) (2026)
- [Complex-Valued Unitary Representations as Classification Heads for Improved Uncertainty Quantification in Deep Neural Networks](https://mlsanity.com/paper/2602.15283/) (2026)

## Abstract (arXiv)

Convolutional neural networks (CNNs) are the cutting edge model for supervised machine learning in computer vision. In recent years CNNs have outperformed traditional approaches in many computer vision tasks such as object detection, image classification and face recognition. CNNs are vulnerable to overfitting, and a lot of research focuses on finding regularization methods to overcome it. One approach is designing task specific models based on prior knowledge. Several works have shown that properties of natural images can be easily captured using complex numbers. Motivated by these works, we present a variation of the CNN model with complex valued input and weights. We construct the complex model as a generalization of the real model. Lack of order over the complex field raises several difficulties both in the definition and in the training of the network. We address these issues and suggest possible solutions. The resulting model is shown to be a restricted form of a real valued CNN with twice the parameters. It is sensitive to phase structure, and we suggest it serves as a regularized model for problems where such structure is important. This suggestion is verified empirically by comparing the performance of a complex and a real network in the problem of cell detection. The two networks achieve comparable results, and although the complex model is hard to train, it is significantly less vulnerable to overfitting. We also demonstrate that the complex network detects meaningful phase structure in the data.