# Deep Complex Networks

arXiv: https://arxiv.org/abs/1705.09792 · HTML: https://mlsanity.com/paper/1705.09792/

## Key facts

- Cited by 217 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 May 2017.

## Built on by (most cited first, 30 of 217)

- [SpeechBrain: A General-Purpose Speech Toolkit](https://mlsanity.com/paper/2106.04624/) (2021)
- [R2RNet: Low-light Image Enhancement via Real-low to Real-normal Network](https://mlsanity.com/paper/2106.14501/) (2021)
- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [A Survey of Complex-Valued Neural Networks](https://mlsanity.com/paper/2101.12249/) (2021)
- [Physics-Driven Deep Learning for Computational Magnetic Resonance Imaging](https://mlsanity.com/paper/2203.12215/) (2022)
- [RF-Diffusion: Radio Signal Generation via Time-Frequency Diffusion](https://mlsanity.com/paper/2404.09140/) (2024)
- [Geometric Clifford Algebra Networks](https://mlsanity.com/paper/2302.06594/) (2023)
- [Cold Diffusion for Speech Enhancement](https://mlsanity.com/paper/2211.02527/) (2022)
- [Looking into Your Speech: Learning Cross-modal Affinity for Audio-visual Speech Separation](https://mlsanity.com/paper/2104.02775/) (2021)
- [High-Order Pooling for Graph Neural Networks with Tensor Decomposition](https://mlsanity.com/paper/2205.11691/) (2022)
- [Harmonics of Learning: Universal Fourier Features Emerge in Invariant Networks](https://mlsanity.com/paper/2312.08550/) (2023)
- [Fourier-Net: Fast Image Registration with Band-limited Deformation](https://mlsanity.com/paper/2211.16342/) (2022)
- [Atmospheric Turbulence Removal with Complex-Valued Convolutional Neural Network](https://mlsanity.com/paper/2204.06989/) (2022)
- [CubeLearn: End-to-end Learning for Human Motion Recognition from Raw mmWave Radar Signals](https://mlsanity.com/paper/2111.03976/) (2021)
- [Bispectral Neural Networks](https://mlsanity.com/paper/2209.03416/) (2022)
- [Speech Denoising Without Clean Training Data: A Noise2Noise Approach](https://mlsanity.com/paper/2104.03838/) (2021)
- [RPMixer: Shaking Up Time Series Forecasting with Random Projections for Large Spatial-Temporal Data](https://mlsanity.com/paper/2402.10487/) (2024)
- [Building Blocks for a Complex-Valued Transformer Architecture](https://mlsanity.com/paper/2306.09827/) (2023)
- [FilterTS: Comprehensive Frequency Filtering for Multivariate Time Series Forecasting](https://mlsanity.com/paper/2505.04158/) (2025)
- [CLNet: Complex Input Lightweight Neural Network designed for Massive MIMO CSI Feedback](https://mlsanity.com/paper/2102.07507/) (2021)
- [Noise-aware Physics-informed Machine Learning for Robust PDE Discovery](https://mlsanity.com/paper/2206.12901/) (2022)
- [Dual-Octave Convolution for Accelerated Parallel MR Image Reconstruction](https://mlsanity.com/paper/2104.05345/) (2021)
- [Learning ground states of quantum Hamiltonians with graph networks](https://mlsanity.com/paper/2110.06390/) (2021)
- [MoRe-Fi: Motion-robust and Fine-grained Respiration Monitoring via Deep-Learning UWB Radar](https://mlsanity.com/paper/2111.08195/) (2021)
- [Theory and Implementation of Complex-Valued Neural Networks](https://mlsanity.com/paper/2302.08286/) (2023)
- [FFCNet: Fourier Transform-Based Frequency Learning and Complex Convolutional Network for Colon Disease Classification](https://mlsanity.com/paper/2207.01287/) (2022)
- [Investigating the Design Space of Diffusion Models for Speech Enhancement](https://mlsanity.com/paper/2312.04370/) (2023)
- [Detecting quantum entanglement with unsupervised learning](https://mlsanity.com/paper/2103.04804/) (2021)
- [DONet: Dual-Octave Network for Fast MR Image Reconstruction](https://mlsanity.com/paper/2105.05980/) (2021)
- [Dual adaptive training of photonic neural networks](https://mlsanity.com/paper/2212.06141/) (2022)

## Abstract (arXiv)

At present, the vast majority of building blocks, techniques, and architectures for deep learning are based on real-valued operations and representations. However, recent work on recurrent neural networks and older fundamental theoretical analysis suggests that complex numbers could have a richer representational capacity and could also facilitate noise-robust memory retrieval mechanisms. Despite their attractive properties and potential for opening up entirely new neural architectures, complex-valued deep neural networks have been marginalized due to the absence of the building blocks required to design such models. In this work, we provide the key atomic components for complex-valued deep neural networks and apply them to convolutional feed-forward networks and convolutional LSTMs. More precisely, we rely on complex convolutions and present algorithms for complex batch-normalization, complex weight initialization strategies for complex-valued neural nets and we use them in experiments with end-to-end training schemes. We demonstrate that such complex-valued models are competitive with their real-valued counterparts. We test deep complex models on several computer vision tasks, on music transcription using the MusicNet dataset and on Speech Spectrum Prediction using the TIMIT dataset. We achieve state-of-the-art performance on these audio-related tasks.