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Volume 9 Issue 8
Aug.  2022

IEEE/CAA Journal of Automatica Sinica

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C. Y. Lee, H. Hasegawa, and  S. C. Gao,  “Complex-valued neural networks: A comprehensive survey,” IEEE/CAA J. Autom. Sinica, vol. 9, no. 8, pp. 1406–1426, Aug. 2022. doi: 10.1109/JAS.2022.105743
Citation: C. Y. Lee, H. Hasegawa, and  S. C. Gao,  “Complex-valued neural networks: A comprehensive survey,” IEEE/CAA J. Autom. Sinica, vol. 9, no. 8, pp. 1406–1426, Aug. 2022. doi: 10.1109/JAS.2022.105743

Complex-Valued Neural Networks: A Comprehensive Survey

doi: 10.1109/JAS.2022.105743
Funds:  This work was partially supported by the JSPS KAKENHI (JP22H03643, JP19K22891)
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  • Complex-valued neural networks (CVNNs) have shown their excellent efficiency compared to their real counterparts in speech enhancement, image and signal processing. Researchers throughout the years have made many efforts to improve the learning algorithms and activation functions of CVNNs. Since CVNNs have proven to have better performance in handling the naturally complex-valued data and signals, this area of study will grow and expect the arrival of some effective improvements in the future. Therefore, there exists an obvious reason to provide a comprehensive survey paper that systematically collects and categorizes the advancement of CVNNs. In this paper, we discuss and summarize the recent advances based on their learning algorithms, activation functions, which is the most challenging part of building a CVNN, and applications. Besides, we outline the structure and applications of complex-valued convolutional, residual and recurrent neural networks. Finally, we also present some challenges and future research directions to facilitate the exploration of the ability of CVNNs.

     

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  • [1]
    N. Benvenuto and F. Piazza, “On the complex backpropagation algorithm,” IEEE Trans. Signal Processing, vol. 40, no. 4, pp. 967–969, 1992. doi: 10.1109/78.127967
    [2]
    G. Georgiou and C. Koutsougeras, “Complex domain backpropagation,” IEEE Trans. Circuits and Systems II: Analog and Digital Signal Processing, vol. 39, no. 5, pp. 330–334, 1992.
    [3]
    H. Leung and S. Haykin, “The complex backpropagation algorithm,” IEEE Trans. Signal Processing, vol. 39, no. 9, pp. 2101–2104, 1991. doi: 10.1109/78.134446
    [4]
    P. Virtue, S. X. Yu, and M. Lustig, “Better than real: Complex-valued neural nets for mri fingerprinting,” in Proc. IEEE Int. Conf. Image Processing, 2017, pp. 3953–3957.
    [5]
    H. Tsuzuki, M. Kugler, S. Kuroyanagi, and A. Iwata, “An approach for sound source localization by complex-valued neural network,” IEICE Trans. Information and Systems, vol. 96, no. 10, pp. 2257–2265, 2013.
    [6]
    S. Sepasi, E. Reihani, A. M. Howlader, L. R. Roose, and M. M. Matsuura, “Very short term load forecasting of a distribution system with high PV penetration,” Renewable Energy, vol. 106, pp. 142–148, 2017. doi: 10.1016/j.renene.2017.01.019
    [7]
    H. Akramifard, M. Firouzmand, and R. A. Moghadam, “Extracting, recognizing, and counting white blood cells from microscopic images by using complex-valued neural networks,” Journal of Medical Signals and Sensors, vol. 2, no. 3, p. 169, 2012.
    [8]
    A. R. Hafiz, A. Y. Al-Nuaimi, M. F. Amin, and K. Murase, “Classification of skeletal wireframe representation of hand gesture using complex-valued neural network,” Neural Processing Letters, vol. 42, no. 3, pp. 649–664, 2015. doi: 10.1007/s11063-014-9379-0
    [9]
    A. Hirose, “Complex-valued neural networks: The merits and their origins,” in Proc. Int. Joint Conf. Neural Networks, 2009, pp. 1237–1244.
    [10]
    A. Hirose, Complex-Valued Neural Networks, 2nd ed. Springer, 2012.
    [11]
    T. Nitta, “Solving the XOR problem and the detection of symmetry using a single complex-valued neuron,” Neural Networks, vol. 16, no. 8, pp. 1101–1105, 2003. doi: 10.1016/S0893-6080(03)00168-0
    [12]
    A. Hirose, “Nature of complex number and complex-valued neural networks,” Frontiers of Electrical and Electronic Engineering in China, vol. 6, no. 1, pp. 171–180, 2011. doi: 10.1007/s11460-011-0125-3
    [13]
    A. Hirose, “Recent progress in applications of complex-valued neural networks,” in Proc. Int. Conf. Artificial Intelligence and Soft Computing. Springer, 2010, pp. 42−46.
    [14]
    J. Bassey, L. Qian, and X. Li, “A survey of complex-valued neural networks,” ArXiv, vol. abs/2101.12249, 2021.
    [15]
    Y. Sunaga, R. Natsuaki, and A. Hirose, “Land form classification and similar land-shape discovery by using complex-valued convolutional neural networks,” IEEE Trans. Geoscience and Remote Sensing, vol. 57, no. 10, pp. 7907–7917, 2019. doi: 10.1109/TGRS.2019.2917214
    [16]
    P. Arena, L. Fortuna, R. Re, and M. G. Xibilia, “Multilayer perceptrons to approximate complex valued functions,” Int. Journal of Neural Systems, vol. 6, no. 04, pp. 435–446, 1995. doi: 10.1142/S0129065795000299
    [17]
    M. F. Amin, R. Savitha, M. I. Amin, and K. Murase, “Complex-valued functional link network design by orthogonal least squares method for function approximation problems,” in Proc. Int. Joint Conf. Neural Networks, 2011, pp. 1489–1496.
    [18]
    T. Nitta, “Orthogonality of decision boundaries in complex-valued neural networks,” Neural Computation, vol. 16, no. 1, pp. 73–97, 2004. doi: 10.1162/08997660460734001
    [19]
    D. P. Reichert and T. Serre, “Neuronal synchrony in complex-valued deep networks,” in Proc. 2nd Int. Conf. Learning Representations, 2014.
    [20]
    R. Savitha, S. Suresh, N. Sundararajan, and P. Saratchandran, “A new learning algorithm with logarithmic performance index for complex-valued neural networks,” Neurocomputing, vol. 72, no. 16–18, pp. 3771–3781, 2009. doi: 10.1016/j.neucom.2009.06.004
    [21]
    S.-S. Yang, C.-L. Ho, and S. Siu, “Sensitivity analysis of the split-complex valued multilayer perceptron due to the errors of the i.i.d. inputs and weights,” IEEE Trans. Neural Networks, vol. 18, no. 5, pp. 1280–1293, 2007. doi: 10.1109/TNN.2007.894038
    [22]
    H. Zhang, C. Zhang, and W. Wu, “Convergence of batch split-complex backpropagation algorithm for complex-valued neural networks,” Discrete Dynamics in Nature and Society, vol. 2009, 2009.
    [23]
    D. Jianping, N. Sundararajan, and P. Saratchandran, “Complex-valued minimal resource allocation network for nonlinear signal processing,” Int. Journal of Neural Systems, vol. 10, no. 02, pp. 95–106, 2000. doi: 10.1142/S0129065700000090
    [24]
    D. Jianping, N. Sundararajan, and P. Saratchandran, “Communication channel equalization using complex-valued minimal radial basis function neural networks,” IEEE Trans. Neural Networks, vol. 13, no. 3, pp. 687–696, 2002. doi: 10.1109/TNN.2002.1000133
    [25]
    M. F. Amin and K. Murase, “Single-layered complex-valued neural network for real-valued classification problems,” Neurocomputing, vol. 72, no. 4–6, pp. 945–955, 2009. doi: 10.1016/j.neucom.2008.04.006
    [26]
    M. Peker, B. Sen, and D. Delen, “Computer-aided diagnosis of Parkinsons disease using complex-valued neural networks and mRMR feature selection algorithm,” Journal of Healthcare Engineering, vol. 6, no. 3, pp. 281–302, 2015. doi: 10.1260/2040-2295.6.3.281
    [27]
    A. Hirose, “Proposal of fully complex-valued neural networks,” in Proc. Int. Joint Conf. Neural Networks, vol. 4, 1992, pp. 152–157.
    [28]
    M. A. Dedmari, S. Conjeti, S. Estrada, P. Ehses, T. Stöcker, and M. Reuter, “Complex fully convolutional neural networks for MR image reconstruction,” in Machine Learning for Medical Image Reconstruction. Springer Int. Publishing, 2018, pp. 30–38.
    [29]
    Z. Zhang, H. Wang, F. Xu, and Y.-Q. Jin, “Complex-valued convolutional neural network and its application in polarimetric SAR image classification,” IEEE Trans. Geoscience and Remote Sensing, vol. 55, no. 12, pp. 7177–7188, 2017. doi: 10.1109/TGRS.2017.2743222
    [30]
    M.-B. Li, G.-B. Huang, P. Saratchandran, and N. Sundararajan, “Fully complex extreme learning machine,” Neurocomputing, vol. 68, pp. 306–314, 2005. doi: 10.1016/j.neucom.2005.03.002
    [31]
    S. Scardapane, S. Van Vaerenbergh, A. Hussain, and A. Uncini, “Complex-valued neural networks with nonparametric activation functions,” IEEE Trans. Emerging Topics in Computational Intelligence, vol. 4, no. 2, pp. 140–150, 2020. doi: 10.1109/TETCI.2018.2872600
    [32]
    T. Kim and T. Adali, “Fully complex backpropagation for constant envelope signal processing,” in Proc. Neural Networks for Signal Processing X. IEEE Signal Processing Society Workshop, vol. 1, 2000, pp. 231–240.
    [33]
    N. Guberman, “On complex valued convolutional neural networks,” arXiv preprint arXiv: 1602.09046, 2016.
    [34]
    K. Tachibana and K. Otsuka, “Wind prediction performance of complex neural network with ReLU activation function,” in Proc. 57th Annu. Conf. Society of Instrument and Control Engineers of Japan, 2018, pp. 1029–1034.
    [35]
    B. Widrow, J. McCool, and M. Ball, “The complex LMS algorithm,” Proc. the IEEE, vol. 63, no. 4, pp. 719–720, 1975. doi: 10.1109/PROC.1975.9807
    [36]
    M. Arjovsky, A. Shah, and Y. Bengio, “Unitary evolution recurrent neural networks,” in Proc. 33rd Int. Conf. Machine Learning, ser. Proc. Machine Learning Research, vol. 48, 2016, pp. 1120–1128.
    [37]
    A. Hirose, “Continuous complex-valued back-propagation learning,” Electronics Letters, vol. 28, no. 20, pp. 1854–1855, 1992. doi: 10.1049/el:19921186
    [38]
    C. You and D. Hong, “Nonlinear blind equalization schemes using complex-valued multilayer feedforward neural networks,” IEEE Trans. Neural Networks, vol. 9, no. 6, pp. 1442–1455, 1998. doi: 10.1109/72.728394
    [39]
    Y. Özbay, “A new approach to detection of ECG arrhythmias: Complex discrete wavelet transform based complex valued artificial neural network,” Journal of Medical Systems, vol. 33, no. 6, p. 435, 2009.
    [40]
    H. A. Jalab and R. W. Ibrahim, “New activation functions for complex-valued neural network,” Int. Journal of Physical Sciences, vol. 6, no. 7, pp. 1766–1772, 2011.
    [41]
    N. Benvenuto, M. Marchesi, F. Piazza, and A. Uncini, “Non linear satellite radio links equalized using blind neural networks,” in Proc. Int. Conf. Acoustics, Speech, and Signal Processing, 1991, pp. 1521–1524.
    [42]
    T. Nitta, “An extension of the back-propagation algorithm to complex numbers,” Neural Networks, vol. 10, no. 8, pp. 1391–1415, 1997. doi: 10.1016/S0893-6080(97)00036-1
    [43]
    T. Nitta, Complex-Valued Neural Networks: Utilizing High-Dimensional Parameters: Utilizing High-Dimensional Parameters. IGI Global, 2009.
    [44]
    H. G. Zimmermann, A. Minin, and V. Kusherbaeva, “Comparison of the complex valued and real valued neural networks trained with gradient descent and random search algorithms,” in Proc. ESANN, 2011.
    [45]
    K. Venkatanareshbabu, S. Nisheel, R. Sakthivel, and K. Muralitharan, “Novel elegant fuzzy genetic algorithms in classification problems,” Soft Computing, vol. 23, no. 14, pp. 5583–5603, 2019. doi: 10.1007/s00500-018-3216-8
    [46]
    L. Zhang, H. Dong, and B. Zou, “Efficiently utilizing complex-valued PolSAR image data via a multi-task deep learning framework,” ISPRS Journal of Photogrammetry and Remote Sensing, vol. 157, pp. 59–72, 2019. doi: 10.1016/j.isprsjprs.2019.09.002
    [47]
    E. C. Yeats, Y. Chen, and H. Li, “Improving gradient regularization using complex-valued neural networks,” in Proc. Int. Conf. Machine Learning, 2021, pp. 11953–11963.
    [48]
    Y.-F. Pu, X. Xie, J. Cao, H. Chen, K. Zhang, and J. Wang, “An input weights dependent complex-valued learning algorithm based on wirtinger calculus,” IEEE Trans. Systems,Man,and Cybernetics: Systems, vol. 52, no. 5, pp. 2920–2932, 2022. doi: 10.1109/TSMC.2021.3055501
    [49]
    T. Nitta, “Learning dynamics of a single polar variable complex-valued neuron,” Neural Computation, vol. 27, no. 5, pp. 1120–1141, 2015. doi: 10.1162/NECO_a_00729
    [50]
    C.-A. Popa, “Enhanced gradient descent algorithms for complex-valued neural networks,” in Proc. IEEE 16th Int. Symp. Symbolic and Numeric Algorithms for Scientific Computing, 2014, pp. 272–279.
    [51]
    T. Nakama, “Theoretical analysis of batch and on-line training for gradient descent learning in neural networks,” Neurocomputing, vol. 73, no. 1–3, pp. 151–159, 2009. doi: 10.1016/j.neucom.2009.05.017
    [52]
    D. R. Wilson and T. R. Martinez, “The general inefficiency of batch training for gradient descent learning,” Neural Networks, vol. 16, no. 10, pp. 1429–1451, 2003. doi: 10.1016/S0893-6080(03)00138-2
    [53]
    H. Zhang, Y. Zhang, S. Zhu, and D. Xu, “Deterministic convergence of complex mini-batch gradient learning algorithm for fully complex-valued neural networks,” Neurocomputing, vol. 407, pp. 185–193, 2020. doi: 10.1016/j.neucom.2020.04.114
    [54]
    C.-A. Popa, “Quasi-newton learning methods for complex-valued neural networks,” in Proc. Int. Joint Conf. Neural Networks, 2015, pp. 1–8.
    [55]
    D. Xu, J. Dong, and C. Zhang, “Convergence of quasi-newton method for fully complex-valued neural networks,” Neural Processing Letters, vol. 46, no. 3, pp. 961–968, 2017. doi: 10.1007/s11063-017-9621-7
    [56]
    M. J. Kidger, “Use of the Levenberg-Marquardt (damped least-squares) optimization method in lens design,” Optical Engineering, vol. 32, no. 8, pp. 1731–1739, 1993. doi: 10.1117/12.145076
    [57]
    M. F. Amin, M. I. Amin, A. Y. H. Al-Nuaimi, and K. Murase, “Wirtinger calculus based gradient descent and Levenberg-Marquardt learning algorithms in complex-valued neural networks,” in Proc. Int. Conf. Neural Information Processing. Springer, 2011, pp. 550–559.
    [58]
    G.-B. Huang, M.-B. Li, L. Chen, and C.-K. Siew, “Incremental extreme learning machine with fully complex hidden nodes,” Neurocomputing, vol. 71, no. 4–6, pp. 576–583, 2008. doi: 10.1016/j.neucom.2007.07.025
    [59]
    R. Savitha, S. Suresh, and N. Sundararajan, “Fast learning circular complex-valued extreme learning machine (CC-ELM) for real-valued classification problems,” Information Sciences, vol. 187, pp. 277–290, 2012. doi: 10.1016/j.ins.2011.11.003
    [60]
    M. Sivachitra, R. Savitha, S. Suresh, and S. Vijayachitra, “A fully complex-valued fast learning classifier (FC-FLC) for real-valued classification problems,” Neurocomputing, vol. 149, pp. 198–206, 2015. doi: 10.1016/j.neucom.2014.04.075
    [61]
    S. Shukla and R. N. Yadav, “Regularized weighted circular complex-valued extreme learning machine for imbalanced learning,” IEEE Access, vol. 3, pp. 3048–3057, 2015. doi: 10.1109/ACCESS.2015.2506601
    [62]
    W. Zong, G.-B. Huang, and Y. Chen, “Weighted extreme learning machine for imbalance learning,” Neurocomputing, vol. 101, pp. 229–242, 2013. doi: 10.1016/j.neucom.2012.08.010
    [63]
    L. Jiang, H. Y. Huang, and Z. H. Ding, “Path planning for intelligent robots based on deep Q-learning with experience replay and heuristic knowledge,” IEEE/CAA J. Autom. Sinica, vol. 7, no. 4, pp. 1179–1189, 2020. doi: 10.1109/JAS.2019.1911732
    [64]
    C. Liu, F. Zhu, Q. Liu, and Y. Fu, “Hierarchical reinforcement learning with automatic sub-goal identification,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 10, pp. 1686–1696, 2021. doi: 10.1109/JAS.2021.1004141
    [65]
    T. Hamagami, T. Shibuya, and S. Shimada, “Complex-valued reinforcement learning,” in Proc. IEEE Int. Conf. Systems, Man and Cybernetics, vol. 5, 2006, pp. 4175–4179.
    [66]
    T. Shibuya and T. Hamagami, Complex-Valued Reinforcement Learning: A Context-Based Approach for POMDPs. Intechopen, 2011, pp. 255–274.
    [67]
    T. Shibuya, H. Arita, and T. Hamagami, “Reinforcement learning in continuous state space with perceptual aliasing by using complex-valued RBF network,” in Proc. IEEE Int. Conf. Systems, Man and Cybernetics, 2010, pp. 1799–1803.
    [68]
    M. Mochida, H. Nakano, and A. Miyauchi, “A complex-valued reinforcement learning method using complex-valued neural networks,” IEICE Technical Report, vol. 117, no. 112, pp. 1–5, 2017.
    [69]
    R. Isaacson and F. Fujita, “Metacognitive knowledge monitoring and self-regulated learning,” Journal of the Scholarship of Teaching and Learning, pp. 39–55, 2006.
    [70]
    R. Savitha, S. Suresh, and N. Sundararajan, “Metacognitive learning in a fully complex-valued radial basis function neural network,” Neural Computation, vol. 24, no. 5, pp. 1297–1328, 2012. doi: 10.1162/NECO_a_00254
    [71]
    S. Suresh, R. Savitha, and N. Sundararajan, “A sequential learning algorithm for complex-valued self-regulating resource allocation network-CSRAN,” IEEE Trans. Neural Networks, vol. 22, no. 7, pp. 1061–1072, 2011. doi: 10.1109/TNN.2011.2144618
    [72]
    S. Suresh, R. Savitha, and N. Sundararajan, “A fast learning fully complex-valued relaxation network (FCRN),” in Proc. IEEE Int. Joint Conf. Neural Networks, 2011, pp. 1372–1377.
    [73]
    S.-S. Yu and W.-H. Tsai, “Relaxation by the hopfield neural network,” Pattern Recognition, vol. 25, no. 2, pp. 197–209, 1992. doi: 10.1016/0031-3203(92)90101-N
    [74]
    Y. Zhang and H. Huang, “Adaptive complex-valued stepsize based fast learning of complex-valued neural networks,” Neural Networks, vol. 124, pp. 233–242, 2020. doi: 10.1016/j.neunet.2020.01.011
    [75]
    Z. Dong and H. Huang, “A training algorithm with selectable search direction for complex-valued feedforward neural networks,” Neural Networks, vol. 137, pp. 75–84, 2021. doi: 10.1016/j.neunet.2021.01.014
    [76]
    T. Takase, S. Oyama, and M. Kurihara, “Effective neural network training with adaptive learning rate based on training loss,” Neural Networks, vol. 101, pp. 68–78, 2018. doi: 10.1016/j.neunet.2018.01.016
    [77]
    L. Vecci, F. Piazza, and A. Uncini, “Learning and approximation capabilities of adaptive spline activation function neural networks,” Neural Networks, vol. 11, no. 2, pp. 259–270, 1998. doi: 10.1016/S0893-6080(97)00118-4
    [78]
    M. Scarpiniti, D. Vigliano, R. Parisi, and A. Uncini, “Generalized splitting functions for blind separation of complex signals,” Neurocomputing, vol. 71, no. 10–12, pp. 2245–2270, 2008. doi: 10.1016/j.neucom.2007.07.037
    [79]
    T. Kim and T. Adalı, “Approximation by fully complex multilayer perceptrons,” Neural Computation, vol. 15, no. 7, pp. 1641–1666, 2003. doi: 10.1162/089976603321891846
    [80]
    N. Mönning and S. Manandhar, “Evaluation of complex-valued neural networks on real-valued classification tasks,” arXiv preprint arXiv: 1811.12351, 2018.
    [81]
    C. Trabelsi, O. Bilaniuk, Y. Zhang, D. Serdyuk, S. Subramanian, J. F. Santos, S. Mehri, N. Rostamzadeh, Y. Bengio, and C. J. Pal, “Deep complex networks,” in Proc. Int. Conf. Learning Representations, 2018.
    [82]
    S. Goh, M. Chen, D. Popović, K. Aihara, D. Obradovic, and D. Mandic, “Complex-valued forecasting of wind profile,” Renewable Energy, vol. 31, no. 11, pp. 1733–1750, 2006. doi: 10.1016/j.renene.2005.07.006
    [83]
    E. Kalnay, M. Kanamitsu, R. Kistler, W. Collins, D. Deaven, L. Gandin, M. Iredell, S. Saha, G. White, J. Woollen, Y. Zhu, M. Chelliah, W. Ebisuzaki, W. Higgins, J. Janowiak, K. C. Mo, C. Ropelewski, J. Wang, A. Leetmaa, R. Reynolds, R. Jenne, and D. Joseph, “The NCEP/NCAR 40-year reanalysis project,” Bulletin of the American Meteorological Society, vol. 77, no. 3, pp. 437–472, 1996. doi: 10.1175/1520-0477(1996)077<0437:TNYRP>2.0.CO;2
    [84]
    Y. LeCun, B. Boser, J. Denker, D. Henderson, R. Howard, W. Hubbard, and L. Jackel, “Handwritten digit recognition with a back-propagation network,” in Advances in Neural Information Processing Systems, D. Touretzky, Ed., vol. 2. Morgan-Kaufmann, 1990, pp. 396–404.
    [85]
    Y. LeCun, K. Kavukcuoglu, and C. Farabet, “Convolutional networks and applications in vision,” in Proc. IEEE Int. Symp. Circuits and Systems, 2010, pp. 253–256.
    [86]
    Y. LeCun, “Convolutional networks for images, speech, and time series,” The Handbook of Brain Theory and Neural Networks, pp. 255–258, 1995.
    [87]
    P. M. Kebria, A. Khosravi, S. M. Salaken, and S. Nahavandi, “Deep imitation learning for autonomous vehicles based on convolutional neural networks,” IEEE/CAA J. Autom. Sinica, vol. 7, no. 1, pp. 82–95, 2020. doi: 10.1109/JAS.2019.1911825
    [88]
    P. Moeskops, M. A. Viergever, A. M. Mendrik, L. S. de Vries, M. J. N. L. Benders, and I. Išgum, “Automatic segmentation of MR brain images with a convolutional neural network,” IEEE Trans. Medical Imaging, vol. 35, no. 5, pp. 1252–1261, 2016. doi: 10.1109/TMI.2016.2548501
    [89]
    E. K. Cole, J. Pauly, and J. Cheng, “Complex-valued convolutional neural networks for MRI reconstruction,” in Proc. 27th Annu. Meeting of ISMRM, Montreal, Canada, 2019, p. 4714.
    [90]
    I. Danihelka, G. Wayne, B. Uria, N. Kalchbrenner, and A. Graves, “Associative long short-term memory,” in Proc. 33rd Int. Conf. Machine Learning, vol. 48, 2016, pp. 1986–1994.
    [91]
    W. Rawat and Z. Wang, “Deep convolutional neural networks for image classification: A comprehensive review,” Neural Computation, vol. 29, no. 9, pp. 2352–2449, 2017. doi: 10.1162/neco_a_00990
    [92]
    Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature, vol. 521, no. 7553, pp. 436–444, 2015. doi: 10.1038/nature14539
    [93]
    S. Wang, H. Cheng, L. Ying, T. Xiao, Z. Ke, H. Zheng, and D. Liang, “DeepcomplexMRI: Exploiting deep residual network for fast parallel MR imaging with complex convolution,” Magnetic Resonance Imaging, vol. 68, pp. 136–147, 2020. doi: 10.1016/j.mri.2020.02.002
    [94]
    P. Virtue, “Complex-valued deep learning with applications to magnetic resonance image synthesis,” Ph.D. dissertation, EECS Department, University of California, Berkeley, Aug. 2019.
    [95]
    J. Zhang and Y. Wu, “Complex-valued unsupervised convolutional neural networks for sleep stage classification,” Computer Methods and Programs in Biomedicine, vol. 164, pp. 181–191, 2018. doi: 10.1016/j.cmpb.2018.07.015
    [96]
    M. Wilmanski, C. Kreucher, and A. Hero, “Complex input convolutional neural networks for wide angle SAR ATR,” in Proc. IEEE Global Conf. Signal and Information Processing, 2016, pp. 1037–1041.
    [97]
    A. Marseet and F. Sahin, “Application of complex-valued convolutional neural network for next generation wireless networks,” in Proc. IEEE Western New York Image and Signal Processing Workshop, 2017, pp. 1–5.
    [98]
    J. Gao, B. Deng, Y. Qin, H. Wang, and X. Li, “Enhanced radar imaging using a complex-valued convolutional neural network,” IEEE Geoscience and Remote Sensing Letters, vol. 16, no. 1, pp. 35–39, 2019. doi: 10.1109/LGRS.2018.2866567
    [99]
    X. Wang and H. Wang, “Forest height mapping using complex-valued convolutional neural network,” IEEE Access, vol. 7, pp. 126 334–126 343, 2019. doi: 10.1109/ACCESS.2019.2938896
    [100]
    Z. Chang, Y. Wang, H. Li, and Z. Wang, “Complex CNN-based equalization for communication signal,” in Proc. IEEE 4th Int. Conf. Signal and Image Processing, 2019, pp. 513–517.
    [101]
    H.-S. Choi, J. Kim, J. Huh, A. Kim, J.-W. Ha, and K. Lee, “Phase-aware speech enhancement with deep complex U-Net,” in Proc. Int. Conf. Learning Representations, 2019.
    [102]
    L. Yu, Y. Hu, X. Xie, Y. Lin, and W. Hong, “Complex-valued full convolutional neural network for SAR target classification,” IEEE Geoscience and Remote Sensing Letters, vol. 17, no. 10, pp. 1752–1756, 2020. doi: 10.1109/LGRS.2019.2953892
    [103]
    X. Wang, P. Chen, H. Xie, and G. Cui, “Through-wall human activity classification using complex-valued convolutional neural network,” in Proc. IEEE Radar Conf., 2021, pp. 1–4.
    [104]
    Y. Zhang, Q. Hua, Y. Jiang, H. Li, and D. Xu, “Cv-MotionNet: Complex-valued convolutional neural network for SAR moving ship targets classification,” in Proc. IEEE Int. Geoscience and Remote Sensing Symp., 2021, pp. 4280–4283.
    [105]
    H. Mu, Y. Zhang, Y. Jiang, and C. Ding, “CV-GMTINet: GMTI using a deep complex-valued convolutional neural network for multichannel SAR-GMTI system,” IEEE Trans. Geoscience and Remote Sensing, pp. 1–15, 2021.
    [106]
    Y. Sunaga, R. Natsuaki, and A. Hirose, “Similar land-form discovery: Complex absolute-value max pooling in complex-valued convolutional neural networks in interferometric synthetic aperture radar,” in Proc. Int. Joint Conf. Neural Networks, 2020, pp. 1–7.
    [107]
    J. Hu and J. Wang, “Global stability of complex-valued recurrent neural networks with time-delays,” IEEE Trans. Neural Networks and Learning Systems, vol. 23, no. 6, pp. 853–865, 2012. doi: 10.1109/TNNLS.2012.2195028
    [108]
    Z. Zhang, C. Lin, and B. Chen, “Global stability criterion for delayed complex-valued recurrent neural networks,” IEEE Trans. Neural Networks and Learning Systems, vol. 25, no. 9, pp. 1704–1708, 2013.
    [109]
    Y. Huang, H. Zhang, and Z. Wang, “Multistability of complex-valued recurrent neural networks with real-imaginary-type activation functions,” Applied Mathematics and Computation, vol. 229, pp. 187–200, 2014. doi: 10.1016/j.amc.2013.12.027
    [110]
    T. Fang and J. Sun, “Stability of complex-valued recurrent neural networks with time-delays,” IEEE Trans. Neural Networks and Learning Systems, vol. 25, no. 9, pp. 1709–1713, 2014. doi: 10.1109/TNNLS.2013.2294638
    [111]
    A. M. Sarroff, V. Shepardson, and M. A. Casey, “Learning representations using complex-valued nets,” arXiv preprint arXiv: 1511.06351, 2015.
    [112]
    Z. Liu, W. Gao, Y.-H. Wan, and E. Muljadi, “Wind power plant prediction by using neural networks,” in Proc. IEEE Energy Conversion Congress and Exposition, 2012, pp. 3154–3160.
    [113]
    M. Kataoka, M. Kinouchi, and M. Hagiwara, “Music information retrieval system using complex-valued recurrent neural networks,” in Proc. SMC’98 IEEE Int. Conf. Systems, Man, and Cybernetics, vol. 5, 1998, pp. 4290–4295.
    [114]
    M. Kinouchi and M. Hagiwara, “Memorization of melodies using complex-valued recurrent neural network,” in Complex-Valued Neural Networks: Theories and Applications. World Scientific, 2003, pp. 205–226.
    [115]
    I. Shafran, T. Bagby, and R. J. Skerry-Ryan, “Complex evolution recurrent neural networks (ceRNNs),” in Proc. IEEE Int. Conf. Acoustics, Speech and Signal Processing, 2018, pp. 5854–5858.
    [116]
    X. Wang, H. Lin, J. Lu, and T. Yahagi, “Channel equalization using complex-valued recurrent neural network,” in Proc. Int. Conf. Info-Tech and Info-Net, vol. 3, 2001, pp. 498–503.
    [117]
    M. Lv, W. Xu, and T. Chen, “A hybrid deep convolutional and recurrent neural network for complex activity recognition using multimodal sensors,” Neurocomputing, vol. 362, pp. 33–40, 2019. doi: 10.1016/j.neucom.2019.06.051
    [118]
    S. Wisdom, T. Powers, J. Hershey, J. Le Roux, and L. Atlas, “Full-capacity unitary recurrent neural networks,” in Advances in Neural Information Processing Systems, D. Lee, M. Sugiyama, U. Luxburg, I. Guyon, and R. Garnett, Eds., vol. 29. Curran Associates, Inc., 2016, pp. 4880–4888.
    [119]
    S. Wang, H. Cheng, Z. Ke, L. Ying, X. Liu, H. Zheng, and D. Liang, “Complex-valued residual network learning for parallel MR imaging,” in Proc. 26th Annu. Meeting of ISMRM, Paris, France, 2018.
    [120]
    Z. Zheng, S. Wang, S. Li, W. Dai, J. Zou, F. Li, and H. Xiong, “DSCR-Net: A diffractive sensing and complex-valued reconstruction network for compressive sensing,” in Proc. IEEE Int. Symp. Circuits and Systems, 2020, pp. 1–5.
    [121]
    R. Chakraborty, Y. Xing, and S. X. Yu, “SurReal: Complex-valued learning as principled transformations on a scaling and rotation manifold,” IEEE Trans. Neural Networks and Learning Systems, pp. 1–12, 2020.
    [122]
    C.-A. Popa, “Complex-valued deep belief networks,” in Advances in Neural Networks - ISNN 2018, T. Huang, J. Lv, C. Sun, and A. V. Tuzikov, Eds. Cham: Springer Int. Publishing, pp. 72–78.
    [123]
    T. Nakashika, S. Takaki, and J. Yamagishi, “Complex-valued restricted Boltzmann machine for direct learning of frequency spectra,” in Proc. Interspeech, 2017, pp. 4021–4025.
    [124]
    Q. Sun, X. Li, L. Li, X. Liu, F. Liu, and L. Jiao, “Semi-supervised complex-valued GAN for polarimetric SAR image classification,” in Proc. IEEE Int. Geoscience and Remote Sensing Symp., 2019, pp. 3245–3248.
    [125]
    B. Vasudeva, P. Deora, S. Bhattacharya, and P. M. Pradhan, “Co-VeGAN: Complex-valued generative adversarial network for compressive sensing MR image reconstruction,” arXiv preprint arXiv: 2002.10523, 2020.
    [126]
    X. Li, Q. Sun, L. Li, X. Liu, H. Liu, L. Jiao, and F. Liu, “SSCV-GANs: Semi-supervised complex-valued GANs for PolSAR image classification,” IEEE Access, vol. 8, pp. 146 560–146 576, 2020. doi: 10.1109/ACCESS.2020.3004591
    [127]
    C.-A. Popa, “Complex-valued stacked denoising autoencoders,” in Advances in Neural Networks - ISNN 2018, T. Huang, J. Lv, C. Sun, and A. V. Tuzikov, Eds. Cham: Springer Int. Publishing, pp. 64–71.
    [128]
    X. Cheng, J. He, J. He, and H. Xu, “Cv-CapsNet: Complex-valued capsule network,” IEEE Access, vol. 7, pp. 85492–85499, 2019. doi: 10.1109/ACCESS.2019.2924548
    [129]
    J. Gu, Z. Wang, J. Kuen, L. Ma, A. Shahroudy, B. Shuai, T. Liu, X. Wang, G. Wang, J. Cai, and T. Chen, “Recent advances in convolutional neural networks,” Pattern Recognition, vol. 77, pp. 354–377, 2018. doi: 10.1016/j.patcog.2017.10.013
    [130]
    C.-A. Popa, “Complex-valued convolutional neural networks for real-valued image classification,” in Proc. IEEE Int. Joint Conf. Neural Networks, 2017, pp. 816–822.
    [131]
    R. Jiao, K. Peng, and J. Dong, “Remaining useful life prediction for a roller in a hot strip mill based on deep recurrent neural networks,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 7, pp. 1345–1354, 2021. doi: 10.1109/JAS.2021.1004051
    [132]
    A. Kumar and R. Rastogi nee Khemchandani, “Self-attention enhanced recurrent neural networks for sentence classification,” in Proc. IEEE Symp. Series on Computational Intelligence, 2018, pp. 905–911.
    [133]
    K. Cho, B. Van Merriënboer, C. Gulcehre, D. Bahdanau, F. Bougares, H. Schwenk, and Y. Bengio, “Learning phrase representations using RNN encoder-decoder for statistical machine translation,” in Proc. Conf. Empirical Methods in Natural Language Processing, pp. 1724–1734, 2014.
    [134]
    L. Jing, Y. Shen, T. Dubček, J. Peurifoi, S. Skirlo, Y. LeCun, M. Tegmark, and M. Soljačić, “Tunable efficient unitary neural networks (EUNN) and their application to RNN,” in Proc. 34th Int. Conf. Machine Learning, vol. 70, 2017, pp. 1733–1741.
    [135]
    M. Wolter and A. Yao, “Complex gated recurrent neural networks,” in Advances in Neural Information Processing Systems, vol. 31. Curran Associates, Inc., 2018, pp. 10536–10546.
    [136]
    B. Wang, Y. Lei, T. Yan, N. Li, and L. Guo, “Recurrent convolutional neural network: A new framework for remaining useful life prediction of machinery,” Neurocomputing, vol. 379, pp. 117–129, 2020. doi: 10.1016/j.neucom.2019.10.064
    [137]
    J. Calvo-Zaragoza, A. H. Toselli, and E. Vidal, “Handwritten music recognition for mensural notation with convolutional recurrent neural networks,” Pattern Recognition Letters, vol. 128, pp. 115–121, 2019. doi: 10.1016/j.patrec.2019.08.021
    [138]
    K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in Proc. IEEE Conf. Computer Vision and Pattern Recognition, 2016, pp. 770–778.
    [139]
    V. Kumar, “Detection of microcalcifications in digital mammogram using curvelet fractal texture features,” European Journal of Molecular &Clinical Medicine, vol. 7, no. 2, pp. 251–256, 2020.
    [140]
    L. Guo, G. Song, and H. Wu, “Complex-valued Pix2pix–deep neural network for nonlinear electromagnetic inverse scattering,” Electronics, vol. 10, no. 6, 2021.
    [141]
    W. Xie, G. Ma, F. Zhao, H. Liu, and L. Zhang, “PolSAR image classification via a novel semi-supervised recurrent complex-valued convolution neural network,” Neurocomputing, vol. 388, pp. 255–268, 2020. doi: 10.1016/j.neucom.2020.01.020
    [142]
    E. C. Kiziltas, A. Uzun, and E. Yılmaz, “Skin segmentation by using complex valued neural network with HSV color spaces,” Int. Journal of Multidisciplinary Studies and Innovative Technologies, vol. 3, no. 1, pp. 1–4, 2019.
    [143]
    H. Zhang, M. Gu, X. Jiang, J. Thompson, H. Cai, et al., “An optical neural chip for implementing complex-valued neural network,” Nature Communications, vol. 12, no. 1, pp. 1–11, 2021. doi: 10.1038/s41467-020-20314-w
    [144]
    S. Gao, M. Zhou, Z. Wang, D. Sugiyama, J. Cheng, J. Wang, and Y. Todo, “Fully complex-valued dendritic neuron model,” IEEE Trans. Neural Networks and Learning Systems, 2021. DOI: 10.1109/TNNLS.2021.3105901
    [145]
    M. Ceylan and H. Yaşar, “A novel approach for automatic blood vessel extraction in retinal images: complex ripplet-I transform and complex valued artificial neural network,” Turkish Journal of Electrical Engineering &Computer Sciences, vol. 24, no. 4, pp. 3212–3227, 2016.
    [146]
    M. Peker, “A new approach for automatic sleep scoring: Combining Taguchi based complex-valued neural network and complex wavelet transform,” Computer Methods and Programs in Biomedicine, vol. 129, pp. 203–216, 2016. doi: 10.1016/j.cmpb.2016.01.001
    [147]
    Z. Nafisah, F. Rachmadi, and E. M. Imah, “Face recognition using complex valued backpropagation,” Jurnal Ilmu Komputer dan Informasi, vol. 11, no. 2, pp. 103–109, 2018. doi: 10.21609/jiki.v11i2.617
    [148]
    S. Amilia, M. D. Sulistiyo, and R. N. Dayawati, “Face image-based gender recognition using complex-valued neural network,” in Proc. IEEE 3rd Int. Conf. Information and Communication Technology, 2015, pp. 201–206.
    [149]
    A. R. Hafiz, M. F. Amin, and K. Murase, “Real-time hand gesture recognition using complex-valued neural network (CVNN),” in Neural Information Processing, B.-L. Lu, L. Zhang, and J. Kwok, Eds. Springer Berlin Heidelberg, 2011, pp. 541–549.
    [150]
    M. Çelebi and M. Ceylan, “A new approach for open-close eye states detection: Complex wavelet transform and complex-valued ANN,” in Proc. IEEE 18th Signal Processing and Communications Applications Conf., 2010, pp. 673–676.
    [151]
    R. Shang, G. Wang, M. A Okoth, and L. Jiao, “Complex-valued convolutional autoencoder and spatial pixel-squares refinement for polarimetric SAR image classification,” Remote Sensing, vol. 11, no. 5, Article No. 522, 2019. doi: 10.3390/rs11050522
    [152]
    M. Meyer, G. Kuschk, and S. Tomforde, “Complex-valued convolutional neural networks for automotive scene classification based on range-beam-doppler tensors,” in Proc. IEEE 23rd Int. Conf. Intelligent Transportation Systems, 2020, pp. 1–6.
    [153]
    M. Yuan, W. Wang, Z. Wang, X. Luo, and J. Kurths, “Exponential synchronization of delayed memristor-based uncertain complex-valued neural networks for image protection,” IEEE Trans. Neural Networks and Learning Systems, vol. 32, no. 1, pp. 151–165, 2021. doi: 10.1109/TNNLS.2020.2977614
    [154]
    M. Ceylan, Y. Özbay, O. N. Uçan, and E. Yildirim, “A novel method for lung segmentation on chest CT images: Complex-valued artificial neural network with complex wavelet transform,” Turkish Journal of Electrical Engineering &Computer Sciences, vol. 18, no. 4, pp. 613–624, 2010.
    [155]
    M. Ceylan and H. Yaçar, “Blood vessel extraction from retinal images using complex wavelet transform and complex-valued artificial neural network,” in Proc. 36th Int. Conf. Telecommunications and Signal Processing, 2013, pp. 822–825.
    [156]
    D. Saraswathi and E. Srinivasan, “An ensemble approach to diagnose breast cancer using fully complex-valued relaxation neural network classifier,” Int. Journal of Biomedical Engineering and Technology, vol. 15, no. 3, pp. 243–260, 2014. doi: 10.1504/IJBET.2014.064651
    [157]
    A. Z. Shirazi, S. J. S. M. Chabok, and Z. Mohammadi, “A novel and reliable computational intelligence system for breast cancer detection,” Medical &Biological Engineering &Computing, vol. 56, no. 5, pp. 721–732, 2018.
    [158]
    O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional networks for biomedical image segmentation,” in Medical Image Computing and Computer-Assisted Intervention-MICCAI 2015. Springer, pp. 234–241.
    [159]
    S. Diamond, V. Sitzmann, F. Heide, and G. Wetzstein, “Unrolled optimization with deep priors,” arXiv preprint arXiv: 1705.08041, 2017.
    [160]
    A. Hirose, R. Nakane, and G. Tanaka, “Keynote speech: Information processing hardware, physical reservoir computing and complex-valued neural networks,” in Proc. IEEE Int. Meeting for Future of Electron Devices, Kansai, 2019, pp. 19–24.
    [161]
    A. K. Alexandridis and A. D. Zapranis, “Wavelet neural networks: A practical guide,” Neural Networks, vol. 42, pp. 1–27, 2013. doi: 10.1016/j.neunet.2013.01.008
    [162]
    P. Ö. Bakbak and M. Peker, “Classification of sonar echo signals in their reduced sparse forms using complex-valued wavelet neural network,” Neural Computing and Applications, pp. 1–11, 2018.
    [163]
    L. S. Saoud, F. Rahmoune, V. Tourtchine, and K. Baddari, “Fully complex valued wavelet network for forecasting the global solar irradiation,” Neural Processing Letters, vol. 45, no. 2, pp. 475–505, 2017. doi: 10.1007/s11063-016-9537-7
    [164]
    J. Stankowicz, J. Robinson, J. M. Carmack, and S. Kuzdeba, “Complex neural networks for radio frequency fingerprinting,” in Proc. IEEE Western New York Image and Signal Processing Workshop, 2019, pp. 1–5.
    [165]
    M. Peker, B. Sen, and D. Delen, “A novel method for automated diagnosis of epilepsy using complex-valued classifiers,” IEEE Journal of Biomedical and Health Informatics, vol. 20, no. 1, pp. 108–118, 2016. doi: 10.1109/JBHI.2014.2387795
    [166]
    I. Aizenberg and Z. Khaliq, “Analysis of EEG using multilayer neural network with multi-valued neurons,” in Proc. IEEE 2nd Int. Conf. Data Stream Mining & Processing, 2018, pp. 392–396.
    [167]
    I. Aizenberg, Complex-Valued Neural Networks With Multi-Valued Neurons. Springer, 2011, vol. 353.
    [168]
    S. Hu, S. Nagae, and A. Hirose, “Millimeter-wave adaptive glucose concentration estimation with complex-valued neural networks,” IEEE Trans. Biomedical Engineering, vol. 66, no. 7, pp. 2065–2071, 2018.
    [169]
    D. Hayakawa, T. Masuko, and H. Fujimura, “Applying complex-valued neural networks to acoustic modeling for speech recognition,” in Proc. IEEE Asia-Pacific Signal and Information Processing Association Annu. Summit and Conf., 2018, pp. 1725–1731.
    [170]
    L. Drude, B. Raj, and R. Haeb-Umbach, “On the appropriateness of complex-valued neural networks for speech enhancement,” in Interspeech, 2016, pp. 1745–1749.
    [171]
    S. Gokul, M. Sivachitra, and S. Vijayachitra, “Parkinson’s disease prediction using machine learning approaches,” in Proc. IEEE 5th Int. Conf. Advanced Computing, 2013, pp. 246–252.
    [172]
    H. Gürüler, “A novel diagnosis system for parkinson’s disease using complex-valued artificial neural network with k-means clustering feature weighting method,” Neural Computing and Applications, vol. 28, no. 7, pp. 1657–1666, 2017. doi: 10.1007/s00521-015-2142-2
    [173]
    T. Kitajima and T. Yasuno, “Output prediction of wind power generation system using complex-valued neural network,” in Proc. IEEE SICE Annu. Conf., 2010, pp. 3610–3613.
    [174]
    H. H. Çevik, Y. E. Acar, and M. Çunkaş, “Day ahead wind power forecasting using complex valued neural network,” in Proc. IEEE Int. Conf. Smart Energy Systems and Technologies, 2018, pp. 1–6.
    [175]
    E. Sathish, M. Sivachitra, R. Savitha, and S. Vijayachitra, “Wind profile prediction using a meta-cognitive fully complex-valued neural network,” in Proc. IEEE 4th Int. Conf. Advanced Computing, 2012, pp. 1–6.
    [176]
    D. Wang and J. Huang, “Neural network-based adaptive dynamic surface control for a class of uncertain nonlinear systems in strict-feedback form,” IEEE Trans. Neural Networks, vol. 16, no. 1, pp. 195–202, 2005. doi: 10.1109/TNN.2004.839354
    [177]
    L. Kong, W. He, C. Yang, Z. Li, and C. Sun, “Adaptive fuzzy control for coordinated multiple robots with constraint using impedance learning,” IEEE Trans. Cybernetics, vol. 49, no. 8, pp. 3052–3063, 2019. doi: 10.1109/TCYB.2018.2838573
    [178]
    L. Kong, W. He, Y. Dong, L. Cheng, C. Yang, and Z. Li, “Asymmetric bounded neural control for an uncertain robot by state feedback and output feedback,” IEEE Trans. Systems,Man,and Cybernetics: Systems, vol. 51, no. 3, pp. 1735–1746, 2021.
    [179]
    S. Slama, A. Errachdi, and M. Benrejeb, “Model reference adaptive control for MIMO nonlinear systems using RBF neural networks,” in Proc. IEEE Int. Conf. Advanced Systems and Electric Technologies, 2018, pp. 346–351.
    [180]
    X. Li and J. Cao, “An impulsive delay inequality involving unbounded time-varying delay and applications,” IEEE Trans. Automatic Control, vol. 62, no. 7, pp. 3618–3625, 2017. doi: 10.1109/TAC.2017.2669580
    [181]
    T. Dong, X. Liao, and A. Wang, “Stability and hopf bifurcation of a complex-valued neural network with two time delays,” Nonlinear Dynamics, vol. 82, no. 1, pp. 173–184, 2015.
    [182]
    Y. Kan, J. Lu, J. Qiu, and J. Kurths, “Exponential synchronization of time-varying delayed complex-valued neural networks under hybrid impulsive controllers,” Neural Networks, vol. 114, pp. 157–163, 2019. doi: 10.1016/j.neunet.2019.02.006
    [183]
    H. Zhang, X.-Y. Wang, and X.-H. Lin, “Synchronization of complex-valued neural network with sliding mode control,” Journal of the Franklin Institute, vol. 353, no. 2, pp. 345–358, 2016. doi: 10.1016/j.jfranklin.2015.11.014
    [184]
    X. Wang, Z. Wang, Q. Song, H. Shen, and X. Huang, “A waiting-time-based event-triggered scheme for stabilization of complex-valued neural networks,” Neural Networks, vol. 121, pp. 329–338, 2020. doi: 10.1016/j.neunet.2019.09.032
    [185]
    H. Bao and J. H. Park, “Adaptive synchronization of complex-valued neural networks with time delay,” in Proc. IEEE 8th Int. Conf. Advanced Computational Intelligence, 2016, pp. 283–288.
    [186]
    J. Hu and C. Zeng, “Adaptive exponential synchronization of complex-valued Cohen-Grossberg neural networks with known and unknown parameters,” Neural Networks, vol. 86, pp. 90–101, 2017. doi: 10.1016/j.neunet.2016.11.001
    [187]
    P. Wan, J. Jian, and J. Mei, “Periodically intermittent control strategies for α-exponential stabilization of fractional-order complex-valued delayed neural networks,” Nonlinear Dynamics, vol. 92, no. 2, pp. 247–265, 2018. doi: 10.1007/s11071-018-4053-0
    [188]
    R. Ji, S. Zhang, L. Zheng, Q. Liu, and A. S. Iftikhar, “Prediction of soil moisture with complex-valued neural network,” in Proc. IEEE 29th Chinese Control And Decision Conf., 2017, pp. 1231–1236.
    [189]
    S. Rashid, S. Saraswathi, A. Kloczkowski, S. Sundaram, and A. Kolinski, “Protein secondary structure prediction using a small training set (compact model) combined with a complex-valued neural network approach,” BMC Bioinformatics, vol. 17, no. 1, p. 362, 2016.
    [190]
    B. Yang, W. Zhang, L.-N. Gong, and H.-Z. Ma, “Finance time series prediction using complex-valued flexible neural tree model,” in Proc. IEEE 13th Int. Conf. Natural Computation, Fuzzy Systems and Knowledge Discovery, 2017, pp. 54–58.
    [191]
    A. Y. H. Al-Nuaimi, M. F. Amin, and K. Murase, “Enhancing MP3 encoding by utilizing a predictive complex-valued neural network,” in Proc. IEEE Int. Joint Conf. Neural Networks, 2012, pp. 1–6.
    [192]
    R. Akhmetsin, V. Giniyatullin, and S. Kirlan, “Identification of structures of organic substances by means of complex-valued perceptron,” Optical Memory and Neural Networks, vol. 21, no. 1, pp. 11–16, 2012. doi: 10.3103/S1060992X1201002X
    [193]
    R. Olanrewaju, O. O. Khalifa, A.-H. Hashim, A. M. Zeki, and A. Aburas, “Forgery detection in medical images using complex valued neural network (CVNN),” Australian Journal of Basic and Applied Sciences, vol. 5, no. 7, pp. 1251–1264, 2011.
    [194]
    L. Xiao, W. Meng, R. Lu, X. Yang, B. Liao, and L. Ding, “A fully complex-valued neural network for rapid solution of complex-valued systems of linear equations,” in Proc. Int. Symp. Neural Networks. Springer, 2015, pp. 444–451.
    [195]
    F. Agostinelli, M. Hoffman, P. Sadowski, and P. Baldi, “Learning activation functions to improve deep neural networks,” in Proc. Int. Conf. Learning Representations, 2015.
    [196]
    J. Inturrisi, S. Y. Khoo, A. Kouzani, and R. Pagliarella, “Piecewise linear units improve deep neural networks,” arXiv preprint arXiv: 2108.00700, 2021.
    [197]
    H. Zhang and D. P. Mandic, “Is a complex-valued stepsize advantageous in complex-valued gradient learning algorithms?” IEEE Trans. Neural Networks and Learning Systems, vol. 27, no. 12, pp. 2730–2735, 2016. doi: 10.1109/TNNLS.2015.2494361
    [198]
    Y. Hua, Q. Liu, K. Hao, and Y. Jin, “A survey of evolutionary algorithms for multi-objective optimization problems with irregular Pareto fronts,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 2, pp. 303–318, 2021. doi: 10.1109/JAS.2021.1003817
    [199]
    P. Wang, Y. Zhou, Q. Luo, C. Han, Y. Niu, and M. Lei, “Complex-valued encoding metaheuristic optimization algorithm: A comprehensive survey,” Neurocomputing, vol. 407, pp. 313–342, 2020. doi: 10.1016/j.neucom.2019.06.112
    [200]
    Y. R. Wang, S. C. Gao, M. C. Zhou, and Y. Yu, “A multi-layered gravitational search algorithm for function optimization and real-world problems,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 1, pp. 94–109, 2021. doi: 10.1109/JAS.2020.1003462
    [201]
    Y. H. Du, L. Wang, L. N. Xing, J. G. Yan, and M. S. Cai, “Data-driven heuristic assisted memetic algorithm for efficient inter-satellite link scheduling in the BeiDou Navigation Satellite System,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 11, pp. 1800–1816, 2021. doi: 10.1109/JAS.2021.1004174
    [202]
    Q. Q. Fan and O. K. Ersoy, “Zoning search with adaptive resource allocating method for balanced and imbalanced multimodal multi-objective optimization,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 6, pp. 1163–1176, 2021. doi: 10.1109/JAS.2021.1004027
    [203]
    Z. Y. Zhao, S. X. Liu, M. C. Zhou, and A. Abusorrah, “Dual-objective mixed integer linear program and memetic algorithm for an industrial group scheduling problem,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 6, pp. 1199–1209, Jun. 2021. doi: 10.1109/JAS.2020.1003539
    [204]
    Y. Wang and X. Q. Zuo, “An effective cloud workflow scheduling approach combining PSO and idle time slot-aware rules,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 5, pp. 1079–1094, May 2021. doi: 10.1109/JAS.2021.1003982
    [205]
    A. H. Khan, X. W. Cao, S. Li, V. N. Katsikis, and L. F. Liao, “BAS-ADAM: An ADAM based approach to improve the performance of beetle antennae search optimizer,” IEEE/CAA J. Autom. Sinica, vol. 7, no. 2, pp. 461–171, Mar. 2020. doi: 10.1109/JAS.2020.1003048
    [206]
    A. Taherkhani, A. Belatreche, Y. Li, G. Cosma, L. P. Maguire, and T. McGinnity, “A review of learning in biologically plausible spiking neural networks,” Neural Networks, vol. 122, pp. 253–272, 2020. doi: 10.1016/j.neunet.2019.09.036
    [207]
    A. Baranski and T. Froese, “Efficient spike timing dependent plasticity rule for complex-valued neurons,” in Proc. Conf. Artificial Life. MIT Press, 2021.
    [208]
    K. Zhang, Y. K. Su, X. W. Guo, L. Qi, and Z. B. Zhao, “MU-GAN: Facial attribute editing based on multi-attention mechanism,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 9, pp. 1614–1626, Sept. 2021. doi: 10.1109/JAS.2020.1003390
    [209]
    J. Banzi, I. Bulugu, and Z. F. Ye, “Learning a deep predictive coding network for a semi-supervised 3D-hand pose estimation,” IEEE/CAA J. Autom. Sinica, vol. 7, no. 5, pp. 1371–1379, Sept. 2020.
    [210]
    Z. Zhao, H. Zhou, L. Qi, L. Chang and M. Zhou, “Inductive representation learning via CNN for partially-unseen attributed networks,” IEEE Trans. Network Science and Engineering, vol. 8, no. 1, pp. 695–706, Jan.−Mar. 2021. doi: 10.1109/TNSE.2020.3048902
    [211]
    J. X. Zhang, K. W. Li, and Y. M. Li, “Output-feedback based simplified optimized backstepping control for strict-feedback systems with input and state constraints,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 6, pp. 1119–1132, Jun. 2021. doi: 10.1109/JAS.2021.1004018
    [212]
    E. F. Ohata, G. M. Bezerra, J. V. S. Chagas, A. V. Lira Neto, A. B. Albuquerque, V. H. C. Albuquerque, and P. P. Rebouças Filho, “Automatic detection of COVID-19 infection using chest X-ray images through transfer learning,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 1, pp. 239–248, Jan. 2021. doi: 10.1109/JAS.2020.1003393
    [213]
    T. Wang, X. Xu, F. Shen, and Y. Yang, “A cognitive memory-augmented network for visual anomaly detection,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 7, pp. 1296–1307, Jul. 2021. doi: 10.1109/JAS.2021.1004045
    [214]
    P. Liu, Y. Zhou, D. Peng and D. Wu, “Global-attention-based neural networks for vision language intelligence,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 7, pp. 1243–1252, Jul. 2021.
    [215]
    Z. Huang, X. Xu, H. Zhu and M. Zhou, “An efficient group recommendation model with multiattention-based neural networks,” IEEE Trans. Neural Networks and Learning Systems, vol. 31, no. 11, pp. 4461–4474, Nov. 2020. doi: 10.1109/TNNLS.2019.2955567
    [216]
    W. J. Zhang, J. C. Wang, and F. P. Lan, “Dynamic hand gesture recognition based on short-term sampling neural networks,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 1, pp. 110–120, Jan. 2021. doi: 10.1109/JAS.2020.1003465
    [217]
    S. Liu, Y. Xia, Z. Shi, H. Yu, Z. Li, and J. Lin, “Deep learning in sheet metal bending with a novel theory-guided deep neural network,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 3, pp. 565–581, Mar. 2021. doi: 10.1109/JAS.2021.1003871
    [218]
    M. Ghahramani, Y. Qiao, M. C. Zhou, A. O’Hagan, and J. Sweeney, “AI-based modeling and data-driven evaluation for smart manufacturing processes,” IEEE/CAA J. Autom. Sinica, vol. 7, no. 7, pp. 1026–1037, Jul. 2020.
    [219]
    H. Zhang, L. Jin and C. Ye, “An RGB-D camera based visual positioning system for assistive navigation by a robotic navigation aid,” IEEE/CAA J. Autom. Sinica, vol. 8, no. 8, pp. 1389–1400, Aug. 2021. doi: 10.1109/JAS.2021.1004084
    [220]
    S. Li, M. Zhou and X. Luo, “Modified primal-dual neural networks for motion control of redundant manipulators with dynamic rejection of harmonic noises,” IEEE Trans. Neural Networks and Learning Systems, vol. 29, no. 10, pp. 4791–4801, Oct. 2018. doi: 10.1109/TNNLS.2017.2770172
    [221]
    H. Dong, B. Zou, L. Zhang, and S. Zhang, “Automatic design of CNNs via differentiable neural architecture search for PolSAR image classification,” IEEE Trans. Geoscience and Remote Sensing, vol. 58, no. 9, pp. 6362–6375, 2020. doi: 10.1109/TGRS.2020.2976694

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    • A comprehensive collection of variants of CVNNs are presented to provide their various structures
    • A systematic categorization of the recent applications of CVNNs provides an easy reference
    • Future research prospective on CVNNs are discussed

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