A mathematical motivation for complex-valued convolutional networks
Abstract
A complex-valued convolutional network (convnet) implements the repeated application of the following composition of three operations, recursively applying the composition to an input vector of nonnegative real numbers: (1) convolution with complex-valued vectors, followed by (2) taking the absolute value of every entry of the resulting vectors, followed by (3) local averaging. For processing real-valued random vectors, complex-valued convnets can be viewed as data-driven multiscale windowed power spectra, data-driven multiscale windowed absolute spectra, data-driven multiwavelet absolute values, or (in their most general configuration) data-driven nonlinear multiwavelet packets. Indeed, complex-valued convnets can calculate multiscale windowed spectra when the convnet filters are windowed complex-valued exponentials. Standard real-valued convnets, using rectified linear units (ReLUs), sigmoidal (e.g., logistic or tanh) nonlinearities, or max pooling, for example, do not obviously exhibit the same exact correspondence with data-driven wavelets (whereas for complex-valued convnets, the correspondence is much more than just a vague analogy). Courtesy of the exact correspondence, the remarkably rich and rigorous body of mathematical analysis for wavelets applies directly to (complex-valued) convnets.
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ImageNet classification with deep convolutional neural networks
Krizhevsky, A., Sutskever, I., & Hinton, G. E. (2017) · Communications of the ACM · doi:10.1145/3065386 · Cited by 112,155 · cited 5× · see citing works
“Section 6 briefly compares on a common benchmark the accuracies for the complex-valued convnets of Chintala et al. (2015) to those for the scattering transforms of Mallat (2010) and for the standard real-valued convnets of Krizhevsky et al. (2012).”
BackgroundMethodInfluentialLeCun, Y., Bengio, Y., & Hinton, G. (2015) · Nature · doi:10.1038/nature14539 · Cited by 84,126 · see citing works
Gradient-based learning applied to document recognition
Lecun, Y., Bottou, L., Bengio, Y., & Haffner, P. (1998) · Proceedings of the IEEE · doi:10.1109/5.726791 · Cited by 58,805 · cited 3× · see citing works
“For supervised learning, we optimize according to a specified objective, usually using the multiscale spectra as inputs to a scheme for classification or regression, as detailed by LeCun et al. (1998), for example.”
BackgroundMethodDistinctive Image Features from Scale-Invariant Keypoints
Lowe, D. G. (2004) · International Journal of Computer Vision · doi:10.1023/b:visi.0000029664.99615.94 · Cited by 55,589 · cited 1× · see citing works
“SIFT, HOG, SURF, etc. of Lowe (1999), Lowe (2004), Dalal and Triggs (2005), Bay et al. (2008), and others are more analogous to the multiwavelet architecture of Figure 2 than to the more general wavelet-packet architecture of Figure 3.”
Histograms of Oriented Gradients for Human Detection
Dalal, N., & Triggs, B. (2005) · 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'05) · doi:10.1109/cvpr.2005.177 · Cited by 33,584 · cited 2× · see citing works
“SIFT, HOG, SURF, etc. of Lowe (1999), Lowe (2004), Dalal and Triggs (2005), Bay et al. (2008), and others are more analogous to the multiwavelet architecture of Figure 2 than to the more general wavelet-packet architecture of Figure 3.”
MethodLearning Multiple Layers of Features from Tiny Images
Krizhevsky, A. (2024) · doi:10.57702/zp44cu3g · Cited by 23,890 · see citing works
Object recognition from local scale-invariant features
Lowe, D. G. (1999) · Proceedings of the Seventh IEEE International Conference on Computer Vision · doi:10.1109/iccv.1999.790410 · Cited by 16,472 · cited 1× · see citing works
“SIFT, HOG, SURF, etc. of Lowe (1999), Lowe (2004), Dalal and Triggs (2005), Bay et al. (2008), and others are more analogous to the multiwavelet architecture of Figure 2 than to the more general wavelet-packet architecture of Figure 3.”
Daubechies, I. C., & Heil, C. (1992) · Computers in Physics · doi:10.1063/1.4823127 · Cited by 15,729 · cited 2× · see citing works
“Via the connections discussed below, the rich, rigorous mathematical analysis surveyed by Daubechies (1992), Meyer (1993), Mallat (2008), and others applies directly to complex-valued convnets.”
BackgroundSpeeded-Up Robust Features (SURF)
Bay, H., Ess, A., Tuytelaars, T., & Van Gool, L. (2008) · Computer Vision and Image Understanding · doi:10.1016/j.cviu.2007.09.014 · Cited by 13,557 · cited 1× · see citing works
“SIFT, HOG, SURF, etc. of Lowe (1999), Lowe (2004), Dalal and Triggs (2005), Bay et al. (2008), and others are more analogous to the multiwavelet architecture of Figure 2 than to the more general wavelet-packet architecture of Figure 3.”
Advances in Neural Information Processing Systems 25
Zhang, Y., Sutton, C. A., Storkey, A., & Zoubin Ghahramani, A. (2012) · Cited by 12,653 · see citing works
A Wavelet Tour of Signal Processing : The Sparse Way
Mallat, S., & Peyré, G. (2008) · Cited by 2,759 · see citing works
Meyer, Y. (1989) · Cambridge University Press eBooks · doi:10.1017/cbo9780511623820 · Cited by 2,056 · see citing works
Translation-Invariant De-Noising
Coifman, R. R., & Donoho, D. L. (1995) · Lecture notes in statistics · doi:10.1007/978-1-4612-2544-7_9 · Cited by 1,943 · cited 2× · see citing works
“Further information is available in the original work of Daubechies (1992), Meyer (1993), Coifman et al. (1994), Coifman and Donoho (1995), Simoncelli and Freeman (1995), Meyer and Coifman (1997), LeCun et al. (1998), Donoho et al. (2003), Srivastava et al. (2003), Rabiner and Schafer (2007), and…”
BackgroundInvariant Scattering Convolution Networks
Bruna, J., & Mallat, S. (2013) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2012.230 · Cited by 1,633 · see citing works
The steerable pyramid: a flexible architecture for multi-scale derivative computation
Simoncelli, E. P., & Freeman, W. T. (2002) · Proceedings - International Conference on Image Processing · doi:10.1109/icip.1995.537667 · Cited by 1,077 · cited 1× · see citing works
“Further information is available in the original work of Daubechies (1992), Meyer (1993), Coifman et al. (1994), Coifman and Donoho (1995), Simoncelli and Freeman (1995), Meyer and Coifman (1997), LeCun et al. (1998), Donoho et al. (2003), Srivastava et al. (2003), Rabiner and Schafer (2007), and…”
A Wavelet Tour of Signal Processing, Third Edition: The Sparse Way
Mallat, S. (2008) · Academic Press eBooks · Cited by 947 · see citing works
On Advances in Statistical Modeling of Natural Images
Srivastava, A., Lee, A. B., Simoncelli, E. P., & Zhu, S. (2003) · Journal of Mathematical Imaging and Vision · doi:10.1023/a:1021889010444 · Cited by 483 · see citing works
Deep roto-translation scattering for object classification
Oyallon, E., & Mallat, S. (2014) · Computer Vision and Pattern Recognition · doi:10.1109/cvpr.2015.7298904 · Cited by 304 · cited 5× · see citing works
“…in its testing set; each image falls into one of ten classes, is full-color, and consists of a 32 × 32 grid of pixels): According to Table 4 of Oyallon and Mallat (2015), the scattering transforms attain an error rate of 18% on the test set, after training their classifiers on the training set.”
BackgroundMethodInfluentialIntroduction to Digital Speech Processing
Rabiner, L. R., & Schafer, R. W. (2007) · now publishers, Inc. eBooks · doi:10.1561/9781601980717 · Cited by 294 · see citing works
Signal processing and compression with wavelet packets
Coifman, R. R., Meyer, Y., Quake, S. R., & Wickerhauser, M. V. (1994) · Springer eBooks · doi:10.1007/978-94-011-1028-0_18 · Cited by 285 · cited 3× · see citing works
“In consonance with the “best-basis” approach of Coifman et al. (1994) and Saito and Coifman (1995), a potentially more efficient possibility is to restrict the convolutional filters in equation 7 to be windowed exponentials that are designed completely a priori, aside from one overall scaling factor…”
MethodLocal discriminant bases and their applications
Saito, N., & Coifman, R. R. (1995) · Journal of Mathematical Imaging and Vision · doi:10.1007/bf01250288 · Cited by 227 · cited 1× · see citing works
“In consonance with the “best-basis” approach of Coifman et al. (1994) and Saito and Coifman (1995), a potentially more efficient possibility is to restrict the convolutional filters in equation 7 to be windowed exponentials that are designed completely a priori, aside from one overall scaling factor…”
MethodWavelets: Calderón-Zygmund and Multilinear Operators
Meyer, Y., & Coifman, R. R. (1997) · CERN Document Server (European Organization for Nuclear Research) · Cited by 200 · see citing works
An exact mapping between the Variational Renormalization Group and Deep LearningPreprint
Mehta, P. D., & Schwab, D. J. (2014) · arXiv (Cornell University) · doi:10.48550/arxiv.1410.3831 · Cited by 142 · cited 1× · see citing works
“A related, more sophisticated connection (to renormalization group theory) is given by Mehta and Schwab (2014).”
BackgroundIntermittent process analysis with scattering moments
Bruna, J., Mallat, S., Bacry, E., & Muzy, J. (2015) · The Annals of Statistics · doi:10.1214/14-aos1276 · Cited by 68 · cited 4× · see citing works
“Further information is available in the original work of Daubechies (1992), Meyer (1993), Coifman, Meyer, Quake, and Wickerhauser (1994), Coifman and Donoho (1995), Simoncelli and Freeman (1995), Meyer and Coifman (1997), LeCun, Bottou, Bengio, and Haffner (1998), Donoho, Mallat, von Sachs, and Samuelides (2003), Sriv…”
BackgroundInfluentialComplex-Valued Convolutional Neural Networks for Object Detection in PolSAR data
Haensch, R., & Hellwich, O. (2010) · Cited by 43 · see citing works
Recursive interferometric representations
Mollai, S. (2010) · Cited by 36 · see citing works
Poggio, T., Mutch, J., Leibo, J. Z., Rosasco, L., & Tacchetti, A. (2012) · DSpace@MIT (Massachusetts Institute of Technology) · Cited by 25 · see citing works
Locally stationary covariance and signal estimation with macrotiles
Donoho, D. L., Mallat, S., von Sachs, R., & Samuelides, Y. (2003) · IEEE Transactions on Signal Processing · doi:10.1109/tsp.2002.808116 · Cited by 15 · cited 2× · see citing works
“…in the original work of Daubechies (1992), Meyer (1993), Coifman et al. (1994), Coifman and Donoho (1995), Simoncelli and Freeman (1995), Meyer and Coifman (1997), LeCun et al. (1998), Donoho et al. (2003), Srivastava et al. (2003), Rabiner and Schafer (2007), and Mallat (2008), for example.”
BackgroundMethodScale-invariant learning and convolutional networks
Chintala, S., Ranzato, M., Szlam, A. D., Tian, Y., Tygert, M., & Zaremba, W. (2016) · Applied and Computational Harmonic Analysis · doi:10.1016/j.acha.2016.06.005 · Cited by 10 · see citing works
Scale-invariant learning and convolutional networksPreprint
Tygert, M., Szlam, A. D., Chintala, S., Ranzato, M., Tian, Y., & Zaremba, W. (2015) · arXiv (Cornell University) · Cited by 8 · see citing works
Convolutional networks and learning invariant to homogeneous multiplicative scalingsPreprint
Tygert, M., Szlam, A. D., Chintala, S., Ranzato, M., Tian, Y., & Zaremba, W. (2015) · arXiv (Cornell University) · doi:10.48550/arxiv.1506.08230 · Cited by 4 · cited 11× · see citing works
“Section 6 briefly compares on a common benchmark the accuracies for the complexvalued convnets of Chintala et al. (2015) to those for the scattering transforms of Mallat (2010) and for the standard real-valued convnets of Krizhevsky et al.”
BackgroundMethodInfluential
Flagged first, then by citations
1 of the 32 references couldn’t be matched to a known paper.
Cited by
Sorted by citations
Geometric Deep Learning: Going beyond Euclidean data
Bronstein, M. M., Bruna, J., LeCun, Y., Szlam, A., & Vandergheynst, P. (2017) · IEEE Signal Processing Magazine · doi:10.1109/msp.2017.2693418 · Cited by 3,661 · see citing works
“These statistical properties have been related to physics [16] and formalized in specific classes of convolutional neural networks (CNNs) [17], [18], [19].”
BackgroundSelf-citationDeep learning for computer vision: A brief review
Voulodimos, A., Doulamis, N., Doulamis, A., & Protopapadakis, E. (2018) · Computational Intelligence and Neuroscience · doi:10.1155/2018/7068349 · Cited by 3,403 · see citing works
An overview of deep learning in medical imaging focusing on MRI
Lundervold, A. S., & Lundervold, A. (2019) · Zeitschrift für Medizinische Physik · doi:10.1016/j.zemedi.2018.11.002 · Cited by 2,058 · see citing works
An overview of deep learning in medical imaging focusing on MRI
Lundervold, A. S., & Lundervold, A. (2018) · NORA - Norwegian Open Research Archives · Cited by 1,012 · see citing works
Harmonic Networks: Deep Translation and Rotation Equivariance
Worrall, D. E., Garbin, S. J., Turmukhambetov, D., & Brostow, G. (2017) · IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings - IEEE Computer Society Conference on Computer Vision and Pattern Recognition/Proceedings · doi:10.1109/cvpr.2017.758 · Cited by 585 · see citing works
“, [32]), which use complex filters, but our treatment differs in that the complex phase of the response is explicitly tied to rotation angle.”
BackgroundA Survey on the New Generation of Deep Learning in Image Processing
Jiao, L., & Zhao, J. K. (2019) · IEEE Access · doi:10.1109/access.2019.2956508 · Cited by 307 · see citing works
“To fully explore the rich information embedded in complexvalued (CV) images, CV-CNN has to be developed [49], [50], [82].”
BackgroundTransmission of natural scene images through a multimode fibre
Caramazza, P., Moran, O., Murray-Smith, R., & Faccio, D. (2019) · Nature Communications · doi:10.1038/s41467-019-10057-8 · Cited by 226 · see citing works
Image denoising using complex-valued deep CNN
Quan, Y., Chen, Y., Shao, Y., Teng, H., Xu, Y., & Ji, H. (2020) · Pattern Recognition · doi:10.1016/j.patcog.2020.107639 · Cited by 185 · see citing works
“Then, some mathematical understandings were presented in [42] for a train-able complex-valued CNN.”
BackgroundComprehensive Review of Vision-Based Fall Detection Systems
Gutiérrez, J., Rodríguez, V., & Martín, S. (2021) · Italian National Conference on Sensors · doi:10.3390/s21030947 · Cited by 146 · see citing works
Deep Learning for Iris Recognition: A Survey
Nguyen, K., Proença, H. P. M. C., & Alonso‐Fernandez, F. (2024) · ACM Computing Surveys · doi:10.1145/3651306 · Cited by 144 · see citing works
“Complex-valued backbone networks better retain the phase, are more invariant to multi-scale, multi-resolution, and multi-orientation, have solid correspondence with the classic Gabor wavelets [173], hence are much better suited to iris recognition than their real-valued counterparts.”
BackgroundDeep Neural Network Approach in EMG-Based Force Estimation for Human–Robot Interaction
Su, H., Qi, W., Li, Z., Chen, Z., Ferrigno, G., & Momi, E. D. (2021) · IEEE Transactions on Artificial Intelligence · doi:10.1109/tai.2021.3066565 · Cited by 105 · see citing works
“It has been well known that the convolutional network is more efficient using the Homogeneous matrix [28].”
Fully Complex-Valued Dendritic Neuron Model
Gao, S., Zhou, M., Wang, Z., Sugiyama, D., Cheng, J., Wang, J., & Todo, Y. (2021) · IEEE Transactions on Neural Networks and Learning Systems · doi:10.1109/tnnls.2021.3105901 · Cited by 95 · see citing works
“…one [46], [49], complex-valued radial basis function one [50], [51], complex-valued Hopfield one [11], complex-valued convolutional networks [8], [52], [53], complex-valued recurrent neural networks [21], complex-valued deep belief networks [54], complex-valued self-organizing map and…”
BackgroundDeep Learning for Computer Vision: A Brief Review
Shetty, K. R., Soorinje, V. S., Dsouza, P., & Swasthik (2022) · International Journal of Advanced Research in Science Communication and Technology · doi:10.48175/ijarsct-2898 · Cited by 91 · see citing works
Oveis, A. H., Giusti, E., Ghio, S., & Martorella, M. (2021) · IEEE Aerospace and Electronic Systems Magazine · doi:10.1109/maes.2021.3117369 · Cited by 77 · see citing works
Transmission of natural scene images through a multimode fibre
Caramazza, P., Moran, O., Murray‐Smith, R., & Faccio, D. (2019) · RePEc: Research Papers in Economics · Cited by 65 · see citing works
Complex-valued unsupervised convolutional neural networks for sleep stage classification
Zhang, J., & Wu, Y. X. (2018) · Computer Methods and Programs in Biomedicine · doi:10.1016/j.cmpb.2018.07.015 · Cited by 61 · see citing works
The universal approximation theorem for complex-valued neural networks
Voigtlaender, F. (2022) · Applied and Computational Harmonic Analysis · doi:10.1016/j.acha.2022.12.002 · Cited by 58 · see citing works
Complex-Valued Full Convolutional Neural Network for SAR Target Classification
Yu, L., Hu, Y., Xie, X., Lin, Y., & Hong, W. Q. (2019) · IEEE Geoscience and Remote Sensing Letters · doi:10.1109/lgrs.2019.2953892 · Cited by 58 · see citing works
“Tygert et al. [12] analyzed the mathematical motivation of CV-CNN.”
BackgroundComplex-valued convolutional neural networks for real-valued image classification
Popa, C.-A. (2017) · International Joint Conference on Neural Networks (IJCNN) · doi:10.1109/ijcnn.2017.7965936 · Cited by 54 · see citing works
“One of the most important papers in this domain is [16], which gives a mathematical motivation for complex-valued convolutional neural networks, showing that they can be seen as nonlinear multiwavelet packets, thus making the mathematical analysis from the signal processing domain available for a…”
BackgroundComplex-Valued Iris Recognition Network
Nguyen, K., Fookes, C., Sridharan, S., & Ross, A. A. (2022) · IEEE Transactions on Pattern Analysis and Machine Intelligence · doi:10.1109/tpami.2022.3152857 · Cited by 47 · see citing works
“• Invariant to multi-scale, multi-resolution and multiorientation for locally stationary stochastic processes: the authors of [43] have provided a solid mathematical framework to prove that complex-valued networks learn representations that are invariant to scale, resolution and orientation variations.”
BackgroundInfluentialA study on generic object detection with emphasis on future research directions
Arulprakash, E., & Aruldoss, M. (2021) · Journal of King Saud University - Computer and Information Sciences · doi:10.1016/j.jksuci.2021.08.001 · Cited by 49 · see citing works
“…utilizing the gradient-based amendment, significant in reaching an excellent outcome in various pattern recognition tasks (LeCun et al., 1989; Tygert et al., 2016) The Convolution Neural Network consists of four primary layers, convolution layers, pooling layers, fully connected layers &…”
An automatic retinal vessel segmentation approach based on Convolutional Neural Networks
Chala, M., Nsiri, B., Alaoui, M. H. E. Y., Soulaymani, A., Mokhtari, A., & Benaji, B. (2021) · Expert Systems with Applications · doi:10.1016/j.eswa.2021.115459 · Cited by 45 · see citing works
Complex input convolutional neural networks for wide angle SAR ATR
Wilmanski, M., Kreucher, C., & Hero, A. O. (2016) · IEEE Global Conference on Signal and Information Processing · doi:10.1109/globalsip.2016.7905999 · Cited by 44 · see citing works
“Our proposed complex-valued CNN is similar to an approach described in [1,2].”
MethodNonblind Image Deblurring via Deep Learning in Complex Field
Quan, Y., Lin, P., Xu, Y., Nan, Y., & Ji, H. (2021) · IEEE Transactions on Neural Networks and Learning Systems · doi:10.1109/tnnls.2021.3070596 · Cited by 38 · see citing works
Surfel convolutional neural network for support detection in additive manufacturing
Huang, J., Kwok, T.-H., Zhou, C., & Xu, W. (2019) · The International Journal of Advanced Manufacturing Technology · doi:10.1007/s00170-019-03792-1 · Cited by 26 · see citing works
“One of the key reasons for the success of the deep neural network is their capability of leaning the statistical properties of the data, and these statistical properties were related to intrinsic physics [23] and formalized with different types of CNNs [5, 27].”
BackgroundDevelopment of a core feature identification application based on the Faster R-CNN algorithm
Jiang, Q., Jia, M., Bi, L., Zhuang, Z., & Gao, K. (2022) · Engineering Applications of Artificial Intelligence · doi:10.1016/j.engappai.2022.105200 · Cited by 25 · see citing works
Zhou, X., Luo, C. L., Ren, P., & Zhang, B. (2023) · IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing · doi:10.1109/jstars.2023.3342986 · Cited by 24 · see citing works
Distributed intrusion detection scheme for next generation networks
Manan, J., Ahmed, A., Ullah, I., Merghem, L., & Gaïti, D. (2019) · Journal of Network and Computer Applications · doi:10.1016/j.jnca.2019.102422 · Cited by 22 · see citing works
A neural network warm-start approach for the inverse acoustic obstacle scattering problem
Zhou, M., Han, J., Rachh, M., & Borges, C. F. (2023) · Journal of Computational Physics · doi:10.1016/j.jcp.2023.112341 · Cited by 21 · see citing works
“Future research considering complex-valued neural networks, such as those discussed in [56, 57], may offer enhanced performance by incorporating the full complex measurement data when available.”
BackgroundImage classification based on quaternion-valued capsule network
Zhou, H., Zhang, C., Zhang, X., & Ma, Q. (2022) · Applied Intelligence · doi:10.1007/s10489-022-03849-x · Cited by 19 · see citing works
“Complex and QV networks have both information-theoretic and biological motivations [38].”
BackgroundFundamentals and recent developments of free-space optical neural networks
McNeil, A. M., Li, Y., Zhang, A., Moebius, M. G., & Liu, Y. (2024) · Journal of Applied Physics · doi:10.1063/5.0215752 · Cited by 20 · see citing works
Physics-Informed Holomorphic Neural Networks (PIHNNs): Solving 2D linear elasticity problems
Calafà, M., Hovad, E., Engsig‐Karup, A. P., & Andriollo, T. (2024) · Computer Methods in Applied Mechanics and Engineering · doi:10.1016/j.cma.2024.117406 · Cited by 18 · see citing works
“The adoption of CVNNs over real-valued ANNs has proven essential in different applications when it is required to treat complex-valued data [22, 23, 25] or when complex techniques can improve some features of ANNs with real-valued data [26, 34].”
BackgroundMethodDing, W., Li, X., Li, G., & Wei, Y. (2020) · Signal Processing Image Communication · doi:10.1016/j.image.2019.115776 · Cited by 18 · see citing works
Enhanced phase recovery in in-line holography with self-supervised complex-valued neural networks
Dou, J., An, Q., Liu, X., Mai, Y., Zhong, L., Di, J., & Qin, Y. (2024) · Optics and Lasers in Engineering · doi:10.1016/j.optlaseng.2024.108685 · Cited by 19 · see citing works
Kumari, V., Kumar, N., Sampath Kumar, K., Kumar, A., Sanagala, S. S., Saxena, S., Khanna, N. N., Laird, J. R., Singh, N., Fouda, M. M., et al. (2023) · Journal of Cardiovascular Development and Disease · doi:10.3390/jcdd10120485 · Cited by 17 · see citing works
Deep Complex-valued Neural Beamformers
Pfeifenberger, L., Zöhrer, M., & Pernkopf, F. (2019) · IEEE International Conference on Acoustics Speech and Signal Processing · doi:10.1109/icassp.2019.8683517 · Cited by 15 · see citing works
“Based on recent contributions on complexvalued neural networks [21, 22, 23], we propose the nonlinear complex-valued activation function g(·) as natural extension of a real-valued tanh unit, i.”
BackgroundWavelet sampling and generalization in neural networks
Zhang, Z., & Kon, M. (2017) · Neurocomputing · doi:10.1016/j.neucom.2017.04.054 · Cited by 14 · see citing works
Hu, W.-S., Li, H.-C., Wang, R., Gao, F., Du, Q., & Plaza, A. (2022) · IEEE Transactions on Geoscience and Remote Sensing · doi:10.1109/tgrs.2022.3188791 · Cited by 14 · see citing works
Mounsey, A., Khan, A., & Sharma, S. K. (2021) · Electronics · doi:10.3390/electronics10243159 · Cited by 14 · see citing works
Yadav, S. K., & Jerripothula, K. R. (2023) · IEEE/CVF International Conference on Computer Vision (ICCV) · doi:10.1109/iccv51070.2023.00981 · Cited by 12 · see citing works
“Complex-valued Learning: In the realm of deep learning, the use of complex numbers as weights presents new opportunities for two-dimensional spectra exploration [14, 38].”
BackgroundTowards understanding theoretical advantages of complex-reaction networks
Zhang, S.-Q., Gao, W., & Zhou, Z. (2022) · Neural Networks · doi:10.1016/j.neunet.2022.03.024 · Cited by 12 · see citing works
“[40] simulated the complex-valued convolution operations from the perspective of wavelets.”
BackgroundStatistics of Visual Responses to Image Object Stimuli from Primate AIT Neurons to DNN Neurons
Dong, Q., Wang, H., & Hu, Z. (2017) · Neural Computation · doi:10.1162/neco_a_01039 · Cited by 12 · see citing works
Deep Hybrid Real-Complex-Valued Convolutional Neural Networks for Image Classification
Popa, C.-A. (2018) · International Joint Conference on Neural Networks (IJCNN) · doi:10.1109/ijcnn.2018.8489274 · Cited by 12 · see citing works
“One of the most important papers in the domain of complex-valued convolutional neural networks (CVCNNs) is [11], which gives a CVCNNs model that can be seen as nonlinear multiwavelet packets, thus making the mathematical analysis in the signal processing domain available for a rigorous formulation of the properties an…”
BackgroundKachhoria, R., Jaiswal, S., Khairnar, S., Rajeswari, K., Pede, S. V., Kharat, R. S., Galande, S., & Khadse, C. B. (2023) · The International Journal of Advanced Manufacturing Technology · doi:10.1007/s00170-023-10834-2 · Cited by 11 · see citing works
Scale-invariant learning and convolutional networks
Chintala, S., Ranzato, M., Szlam, A. D., Tian, Y., Tygert, M., & Zaremba, W. (2016) · Applied and Computational Harmonic Analysis · doi:10.1016/j.acha.2016.06.005 · Cited by 10 · see citing works
Fourier Transform-Based Image Classification Using Complex-Valued Convolutional Neural Networks
Popa, C.-A., & Cernăzanu-Glăvan, C. (2018) · Lecture notes in computer science · doi:10.1007/978-3-319-92537-0_35 · Cited by 10 · see citing works
Complex-Valued Deep Belief Networks
Popa, C.-A. (2018) · Lecture notes in computer science · doi:10.1007/978-3-319-92537-0_9 · Cited by 10 · see citing works
Sunaga, Y., Natsuaki, R., & Hirose, A. (2020) · International Joint Conference on Neural Networks (IJCNN) · doi:10.1109/ijcnn48605.2020.9207122 · Cited by 8 · see citing works
Artificial and Human Intelligence in Mental Health
Sigman, M., Slezak, D. F., Drucaroff, L. J., Ribeiro, S. T. G., & Carrillo, F. (2021) · AI Magazine · doi:10.1002/j.2371-9621.2021.tb00009.x · Cited by 8 · see citing works
An On-Chip 2-D DFT Accelerator Ultrasonic Wavefront for Convolutional Neural Networks
Teng, K.-H., Raju, S., Zhu, D., Lim, J. L. K., Chen, D. S.-H., Ching, E. W. L., Sharma, J., Lee, J. E.-Y., Ng, E. J., Chuan, K. C. T., et al. (2021) · USNC-URSI Radio Science Meeting · doi:10.23919/usnc-ursi51813.2021.9703556 · Cited by 7 · see citing works
Deep Convolutional Networks for Image Segmentation: Application to Optic Disc detection
Chala, M., Nsiri, B., Soulaymani, A., Mokhtari, A., & Benaji, B. (2020) · International Conference on Electrical and Information Technologies · doi:10.1109/iceit48248.2020.9113204 · Cited by 7 · see citing works
“Later Yann LeCun and his collaborators designed Convolutional Neural Networks employing the error gradient and attaining very good results in a variety of pattern recognition tasks [3, 4].”
BackgroundMethodLiu, L., & Li, Y. (2024) · Remote Sensing · doi:10.3390/rs16061094 · Cited by 7 · see citing works
Li, C., Liu, S., Wang, Z., & Yuan, G. (2023) · Frontiers in Physiology · doi:10.3389/fphys.2022.1085530 · Cited by 6 · see citing works
“Complex-valued CNN (CV-CNN) has been developed and applied to various fi elds (Hirose, 2013; Tygert et al., 2016).”
MethodNeural Networks for Natural Language Processing
Morchid, M. (2019) · Advances in computer and electrical engineering book series · doi:10.4018/978-1-7998-1159-6 · Cited by 6 · see citing works
Aizenberg, I. N., Herman, J. E., & Vasko, A. (2022) · IEEE Annual Ubiquitous Computing, Electronics & Mobile Communication Conference (UEMCON) · doi:10.1109/uemcon54665.2022.9965659 · Cited by 7 · see citing works
“They have been developing for the last 5-6 years since the first CV-CNN was introduced in [8].”
BackgroundConvolutional Neural Network with Multi-Valued Neurons
Aizenberg, I. N., & Vasko, A. (2020) · IEEE International Conference on Data Stream Mining & Processing (DSMP) · doi:10.1109/dsmp47368.2020.9204076 · Cited by 7 · see citing works
“A number of researches pretty recently discussed and introduced complex-valued CNNs based on the complex-valued sigmoidal neurons [11], [12].”
BackgroundFrequency-Phase Guided Attention Complex-Valued Network for Ultrasound Image Segmentation
Zhang, W.-B., Zhou, P., Chen, Y., & Zhou, G. (2025) · IEEE Journal of Biomedical and Health Informatics · doi:10.1109/jbhi.2025.3565311 · Cited by 5 · see citing works
“On the other hand, Bruna et al. [29] demonstrated that complex-valued networks can be viewed as data-driven nonlinear multi-wavelet packets.”
An intelligent control approach for defect-free friction stir welding
Cobos, R., Salas, S. D., Angulo, W., & Liao, T. W. (2021) · The International Journal of Advanced Manufacturing Technology · doi:10.1007/s00170-021-07523-3 · Cited by 5 · see citing works
The universal approximation theorem for complex-valued neural networks.Preprint
Voigtländer, F. (2020) · arXiv (Cornell University) · Cited by 5 · see citing works
A deep learning-based neural style transfer optimization approach
Sethi, P., Bhardwaj, R., Sharma, N., Sharma, D. K., & Srivastava, G. (2024) · Intelligent Data Analysis · doi:10.3233/ida-230765 · Cited by 4 · see citing works
“VGG19 is a deeper version of VGG16, with 19 layers instead of 16, and has a higher number of filters in the convolutional layers, which may allow it to capture more complex features [17].”
Deep learning-based MRI reconstruction with Artificial Fourier Transform Network (AFTNet)
Yang, Y., Zhang, Y., Li, Z., Tian, J. S., Dagommer, M., & Guo, J. (2025) · Computers in Biology and Medicine · doi:10.1016/j.compbiomed.2025.110224 · Cited by 4 · see citing works
“Mathematical arguments and implementations have also been discussed [19, 3], enabling the practical application of CVNNs.”
Maxout Networks for Visual Recognition
Castaneda, G., Morris, P., & Khoshgoftaar, T. M. (2019) · International Journal of Multimedia Data Engineering and Management · doi:10.4018/ijmdem.2019100101 · Cited by 4 · see citing works
Kang, M., Yang, Z., Tian, Y., Li, D., Wang, Z., & Chen, L. (2025) · Discover Oncology · doi:10.1007/s12672-025-03970-6 · Cited by 3 · see citing works
CoNO: Complex neural operator for continous dynamical physical systems
Tiwari, K., Krishnan, N. M. A., & Prathosh, A. P. (2025) · APL Machine Learning · doi:10.1063/5.0254013 · Cited by 3 · see citing works
Park, J., & Wojtowytsch, S. (2024) · Applied and numerical harmonic analysis · doi:10.1007/978-3-031-66497-7_3 · Cited by 3 · see citing works
Face Recognition with Inception-Based CNN Models
Patil, L., & Mytri, V. D. (2021) · Algorithms for intelligent systems · doi:10.1007/978-981-15-6707-0_48 · Cited by 3 · see citing works
Complex-Valued Stacked Denoising Autoencoders
Popa, C.-A. (2018) · Lecture notes in computer science · doi:10.1007/978-3-319-92537-0_8 · Cited by 3 · see citing works
Complex-Valued Deep Boltzmann Machines
Popa, C.-A. (2018) · International Joint Conference on Neural Networks (IJCNN) · doi:10.1109/ijcnn.2018.8489359 · Cited by 3 · see citing works
Depth Estimation of Monocular Images using Transfer Learning based Unet Model
Patil, S. S., & Gaikwad, C. J. (2022) · IEEE Bombay Section Signature Conference (IBSSC) · doi:10.1109/ibssc56953.2022.10037503 · Cited by 3 · see citing works
“To avoid utilising only pure linear combinations, the activation function is often a continuously differentiable nonlinear function [13] [14].”
MethodReal to H-Space Autoencoders for Theme Identification in Telephone Conversations
Parcollet, T., Morchid, M., Bost, X., Linarès, G., & Mori, R. D. (2019) · IEEE/ACM Transactions on Audio Speech and Language Processing · doi:10.1109/taslp.2019.2950596 · Cited by 3 · see citing works
$C^{2}N^{2}$: Complex-Valued Contourlet Neural Network
Liu, M., Jiao, L., Liu, X., Li, L., Liu, F., Yang, S., Guo, Y., & Chen, P. (2024) · IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing · doi:10.1109/jstars.2024.3358846 · Cited by 2 · see citing works
Zhang, W., Li, Z., Zhang, X., Wang, F., Wang, S., Lin, N., Li, Y., Wang, J., Yue, J., Dou, C., et al. (2024) · Advanced Intelligent Systems · doi:10.1002/aisy.202300784 · Cited by 2 · see citing works
MLMVN as a Frequency Domain Convolutional Neural Network
Aizenberg, I. N., & Vasko, A. (2023) · International Conference on Computational Science and Computational Intelligence (CSCI) · doi:10.1109/csci62032.2023.00061 · Cited by 2 · see citing works
Gamazo, J., Troncoso, J. M. C., & Rincón, M. (2022) · Lecture notes in computer science · doi:10.1007/978-3-031-06527-9_41 · Cited by 2 · see citing works
Phaseper: A Complex-Valued Transformer for Automatic Speech Recognition
del Rio, D. R., & Jaramillo-Villegas, J. A. (2025) · IEEE Access · doi:10.1109/access.2025.3606844 · Cited by 1 · see citing works
“These frameworks provide a rigorous mathematical foundation for applying complex-valued convolutions and attention mechanisms—essential components of Transformer architectures [28].”
A Succinct Analysis for Deep Learning in Deep Vision and its Applications
Preethi, N., & Singh, W. J. (2021) · International Journal of Current Research and Review · doi:10.31782/ijcrr.2021.13708 · Cited by 1 · see citing works
A Study on Indirect Performance Parameters of Object Detection
Arulprakash, E., Martín, A., & Lakshmi, T. M. (2022) · SN Computer Science · doi:10.1007/s42979-022-01277-9 · Cited by 1 · see citing works
“Features extracted in the initial layers are combined by the deep layers to extract more discriminative features [54, 57].”
BackgroundWhy Convolutional Networks Learn Oriented Bandpass Filters: Theory and Empirical SupportPreprint
Hadji, I., & Wildes, R. P. (2020) · arXiv (Cornell University) · doi:10.48550/arxiv.2011.14665 · Cited by 1 · see citing works
“iances Bruna & Mallat (2013). Other work has noted that purely mathematical considerations show that ConvNets are well suited to realizing filter designs for capturing multiscale, windowed spectra Bruna et al. (2016); however, it did not explicitly established the relationship to eigenfunctions of convolution nor o…”
BackgroundZhang, S., & Xia, J. (2019) · Journal of Physics Conference Series · doi:10.1088/1742-6596/1302/3/032021
Implementation of Optical Deep Neural Networks using the Fabry-Perot InterferometerPreprint
Steel, B. D. (2019) · arXiv (Cornell University) · doi:10.48550/arxiv.1911.10109
“Complex valued ANN have started to gain ground in the field of deep learning [49,50], especially so in CNNs [51, 52].”
BackgroundInfluentialComplex-valued convolutional neural network for terahertz image classification
Hu, N., Qi, F., & Li, Y. (2023) · Advanced topics in optoelectronics, microelectronics, and nanotechnologies. Conference · doi:10.1117/12.2643262
“[6] is one of the most important papers in the field of CVCNN, which gives a CVCNN model that can be seen as nonlinear multiwavelet packets.”
BackgroundCariow, A., & Papliński, J. P. (2023) · Signal Processing Symposium (SPSympo) · doi:10.23919/spsympo57300.2023.10302708
“CV-CNNs are CNNs that process information using complex coefficients and variables [5]–[7].”
BackgroundImproving the Efficiency of Complex Convolution Networks
Sabokpa, M., Mozayani, N., & Garshasbi, M. (2024) · Iranian Conference on Machine Vision and Image Processing · doi:10.1109/mvip62238.2024.10491170
Modelos Deep Learning para Diagnóstico de Covid-19 con Tomografías Computarizadas de Kaggle
Vásquez, R. F. F. (2024) · Ciencia Latina Revista Científica Multidisciplinar · doi:10.37811/cl_rcm.v8i3.11264
Feature Attention Complex-Valued-Based CNN for Ship Target Recognition of SAR Images
Pan, H., Yang, C., Yin, Y., Liang, B., Shi, H., & Wu, G. (2024) · IEEE International Conference on Signal, Information and Data Processing (ICSIDP) · doi:10.1109/icsidp62679.2024.10868562
Lou, J., Yang, J., & Wang, L. (2024) · IEEE International Conference on Software System and Information Processing (ICSSIP) · doi:10.1109/icssip63203.2024.11012541
“SAR images are generally complex valued data that contain both amplitude and phase information, but ordinary CNN are generally real valued networks which can only handle amplitude information and lose phase information [4,5].”
Scaling Continuous Kernels with Sparse Fourier Domain Learning
Harper, C., Wood, L., Gerstoft, P., Thornton, M. A., & Larson, E. C. (2026) · Lecture notes in computer science · doi:10.1007/978-3-032-18474-0_17
- Authors
- Tygert, M., Bruna, J., Chintala, S., LeCun, Y., Piantino, S., & Szlam, A.
- Venue
- Neural Computation
- Publisher
- MIT Press - Journals
- Published
- 2016-05
- Volume
- 28
- Issue
- 5
- Pages
- 815-825
- ISSN
- 0899-7667 (ISSN-L)
- 1530-888X
- OpenAlex
- W2229484904
- Cited by
- 88
- References
- 32
- Citations by year
- 2026: 1 · 2025: 5 · 2024: 13 · 2023: 10 · 2022: 11 · 2021: 13 · 2020: 7 · 2019: 12 · 2018: 9 · 2017: 5 · 2016: 2
- FWCI
- 15.52
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Neural Computation (publishedVersion)
- PubMed (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical Analysis and Transform Methods
- Physical Sciences › Computer Science › Artificial Intelligence › Neural Networks and Applications
- Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Advanced Neural Network Applications
- Keywords
- wavelet theory, local averaging
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