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A Survey of Complex-Valued Neural Networks

arXiv 2101.12249 ↗First posted 28 Jan 2021stat.ML cs.LGcited by 61

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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.

Key facts. Cited by 61 papers in the map (data through 7 Oct 2026). Builds on 15 of its 166 references that are in the map. First posted on arXiv 28 Jan 2021.

Builds on

15 of 166 references.

References · 15

  1. 2015Deep Residual Learning for Image Recognitioncited by 51,809
  2. 2014On the Properties of Neural Machine Translation: Encoder-Decoder Approachescited by 1,303
  3. 2015Unitary Evolution Recurrent Neural Networkscited by 291
  4. 2017Deep Complex Networkscited by 215
  5. 2015Training Very Deep Networkscited by 172
  6. 2016Full-Capacity Unitary Recurrent Neural Networkscited by 90
  7. 2016Associative Long Short-Term Memorycited by 37
  8. 2018DeepNIS: Deep Neural Network for Nonlinear Electromagnetic Inverse Scatteringcited by 31
  9. 2017Better than Real: Complex-valued Neural Nets for MRI Fingerprintingcited by 30
  10. 2013Neuronal Synchrony in Complex-Valued Deep Networkscited by 28
  11. 2018Complex-valued Neural Networks with Non-parametric Activation Functionscited by 19
  12. 2017Enhanced Radar Imaging Using a Complex-valued Convolutional Neural Networkcited by 18
3 more
  1. 2018Evaluation of Complex-Valued Neural Networks on Real-Valued Classification Taskscited by 9
  2. 2019Complex Evolution Recurrent Neural Networks (ceRNNs)cited by 6
  3. 2019Semi-supervised Complex-valued GAN for Polarimetric SAR Image Classificationcited by 4

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 1; 2022: 17; 2023: 12; 2024: 11; 2025: 13; 2026: 7.

61 papers.

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  1. 2022Clifford Neural Layers for PDE Modelingcited by 91
  2. 2022Complex-Valued Autoencoders for Object Discoverycited by 40
  3. 2025Symmetry in Neural Network Parameter Spacescited by 31
  4. 2023Harmonics of Learning: Universal Fourier Features Emerge in Invariant Networkscited by 23
  5. 2023Building Blocks for a Complex-Valued Transformer Architecturecited by 16
  6. 2024FAN: Fourier Analysis Networkscited by 12
  7. 2023Theory and Implementation of Complex-Valued Neural Networkscited by 11
  8. 2022Exponential Separations in Symmetric Neural Networkscited by 10
  9. 2022C^*-algebra Net: A New Approach Generalizing Neural Network Parameters to C^*-algebracited by 9
  10. 2024ATFNet: Adaptive Time-Frequency Ensembled Network for Long-term Time Series Forecastingcited by 8
  11. 2024Unitary convolutions for learning on graphs and groupscited by 8
  12. 2022k-strip: A novel segmentation algorithm in k-space for the application of skull strippingcited by 7
38 more
  1. 2023Towards NeuroAI: Introducing Neuronal Diversity into Artificial Neural Networkscited by 6
  2. 2023Complex-valued neural networks for voice anti-spoofingcited by 6
  3. 2021SignalNet: A Low Resolution Sinusoid Decomposition and Estimation Networkcited by 5
  4. 2025CVKAN: Complex-Valued Kolmogorov-Arnold Networkscited by 5
  5. 2025iFairy: the First 2-bit Complex LLM with All Parameters in ±1, ± icited by 5
  6. 2022Complex-Value Spatio-temporal Graph Convolutional Neural Networks and its Applications to Electric Power Systems AIcited by 4
  7. 2023Non-contact Sensing for Anomaly Detection in Wind Turbine Blades: A focus-SVDD with Complex-Valued Auto-Encoder Approachcited by 4
  8. 2023SISMIK for brain MRI: Deep-learning-based motion estimation and model-based motion correction in k-spacecited by 4
  9. 2022Adversarial Audio Synthesis with Complex-valued Polynomial Networkscited by 3
  10. 2025Enhancing deep neural networks through complex-valued representations and Kuramoto synchronization dynamicscited by 3
  11. 2022A Hybrid Complex-valued Neural Network Framework with Applications to Electroencephalogram (EEG)cited by 2
  12. 2023Spectral Batch Normalization: Normalization in the Frequency Domaincited by 2
  13. 2023Complex-Valued Neural Networks for Data-Driven Signal Processing and Signal Understandingcited by 2
  14. 2024Steinmetz Neural Networks for Complex-Valued Datacited by 2
  15. 2024PHemoNet: A Multimodal Network for Physiological Signalscited by 2
  16. 2024Towards a Categorical Foundation of Deep Learning: A Surveycited by 2
  17. 2025DDRM-PR: Fourier Phase Retrieval using Denoising Diffusion Restoration Modelscited by 2
  18. 2025The neural networks with tensor weights and emergent fermionic Wick rules in the large-width limitcited by 2
  19. 2025Holographic Transformers for Complex-Valued Signal Processing: Integrating Phase Interference into Self-Attentioncited by 2
  20. 2025Fairy2i: Training Complex LLMs from Real LLMs with All Parameters in ± 1, ± icited by 2
  21. 2022Machine Learning Decoder for 5G NR PUCCH Format 0cited by 1
  22. 2022Learning the Propagation of Worms in Wireless Sensor Networkscited by 1
  23. 2022Impact of PolSAR pre-processing and balancing methods on complex-valued neural networks segmentation taskscited by 1
  24. 2022Fully complex-valued deep learning model for visual perceptioncited by 1
  25. 2022Harmonic (Quantum) Neural Networkscited by 1
  26. 2023Complex-valued Neural Networks -- Theory and Analysiscited by 1
  27. 2024An Intrinsic Vector Heat Networkcited by 1
  28. 2024Improving Quaternion Neural Networks with Quaternionic Activation Functionscited by 1
  29. 2025RotaTouille: Rotation Equivariant Deep Learning for Contourscited by 1
  30. 2025CoPE: A Lightweight Complex Positional Encodingcited by 1
  31. 2026ToTMNet: FFT-Accelerated Toeplitz Temporal Mixing Network for Lightweight Remote Photoplethysmographycited by 1
  32. 2022SUTD-PRCM Dataset and Neural Architecture Search Approach for Complex Metasurface Designcited by 0
  33. 2022A photonic chip-based machine learning approach for the prediction of molecular propertiescited by 0
  34. 2022On the Approximation and Complexity of Deep Neural Networks to Invariant Functionscited by 0
  35. 2022Characterizing 4-string contact interaction using machine learningcited by 0
  36. 2023Emerging Approaches for THz Array Imaging: A Tutorial Review and Software Toolcited by 0
  37. 2023Generalized Activation via Multivariate Projectioncited by 0
  38. 2024DeepCSHAP: Utilizing Shapley Values to Explain Deep Complex-Valued Neural Networkscited by 0

50 most cited of 61 All 61 (JSON)