Complex and Hypercomplex-Valued Support Vector Machines: A Survey

In recent years, the field of complex, hypercomplex-valued and geometric Support Vector Machines (SVM) has undergone immense progress due to the compatibility of complex and hypercomplex number representations with analytic signals, as well as the power of description that geometric entities provide to object descriptors. Thus, several interesting applications can be developed using these types of data and algorithms, such as signal processing, pattern recognition, classification of electromagnetic signals, light, sonic/ultrasonic and quantum waves, chaos in the complex domain, phase and phase-sensitive signal processing and nonlinear filtering, frequency, time-frequency and spatiotemporal domain processing, quantum computation, robotics, control, time series prediction, and visual servoing, among others. This paper presents and discusses the importance, recent progress, prospective applications, and future directions of complex, hypercomplex-valued and geometric Support Vector Machines.

[1]  Michihiko Minoh,et al.  Regression of 3D rigid transformations on real-valued vectors in closed form , 2017, 2017 IEEE International Conference on Robotics and Automation (ICRA).

[2]  Harald Haas,et al.  Harnessing Nonlinearity: Predicting Chaotic Systems and Saving Energy in Wireless Communication , 2004, Science.

[3]  Y. Zhang,et al.  Exploring the complementarity of THz pulse imaging and DCE-MRIs: Toward a unified multi-channel classification and a deep learning framework , 2016, Comput. Methods Programs Biomed..

[4]  Ignacio Santamaria,et al.  Robust array beamforming with sidelobe control using support vector machines , 2004, SPAWC 2004.

[5]  Derek Abbott,et al.  Terahertz Signal Classification Based on Geometric Algebra , 2016, IEEE Transactions on Terahertz Science and Technology.

[6]  F. Petruccione,et al.  An introduction to quantum machine learning , 2014, Contemporary Physics.

[7]  Chee Kheong Siew,et al.  Universal Approximation using Incremental Constructive Feedforward Networks with Random Hidden Nodes , 2006, IEEE Transactions on Neural Networks.

[8]  Sayantan Gupta,et al.  Quantum machine learning-using quantum computation in artificial intelligence and deep neural networks: Quantum computation and machine learning in artificial intelligence , 2017, 2017 8th Annual Industrial Automation and Electromechanical Engineering Conference (IEMECON).

[9]  Simone Scardapane,et al.  Kafnets: kernel-based non-parametric activation functions for neural networks , 2017, Neural Networks.

[10]  Simone Scardapane,et al.  Complex-Valued Neural Networks With Nonparametric Activation Functions , 2018, IEEE Transactions on Emerging Topics in Computational Intelligence.

[11]  Marimuthu Palaniswami,et al.  A Division Algebraic Framework for Multidimensional Support Vector Regression , 2010, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[12]  Pascal Chevalier,et al.  Widely linear estimation with complex data , 1995, IEEE Trans. Signal Process..

[13]  Sergios Theodoridis,et al.  Ieee Transactions on Signal Processing Extension of Wirtinger's Calculus to Reproducing Kernel Hilbert Spaces and the Complex Kernel Lms , 2022 .

[14]  Marimuthu Palaniswami,et al.  A Note on Octonionic Support Vector Regression , 2012, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics).

[15]  R. O. Schmidt,et al.  Multiple emitter location and signal Parameter estimation , 1986 .

[16]  Danilo P. Mandic,et al.  Quaternion Reproducing Kernel Hilbert Spaces: Existence and Uniqueness Conditions , 2014, IEEE Transactions on Information Theory.

[17]  Peter L. Bartlett,et al.  The Sample Complexity of Pattern Classification with Neural Networks: The Size of the Weights is More Important than the Size of the Network , 1998, IEEE Trans. Inf. Theory.

[18]  Eduardo Bayro-Corrochano,et al.  Clifford Support Vector Machines for Classification, Regression, and Recurrence , 2010, IEEE Transactions on Neural Networks.

[19]  Jürgen Schmidhuber,et al.  Long Short-Term Memory , 1997, Neural Computation.

[20]  E. Bayro-Corrochano,et al.  Parallel Clifford Support Vector Machines Using the Gaussian Kernel , 2017 .

[21]  Dong Liang,et al.  Cloud removal of remote sensing image based on multi-output support vector regression , 2014 .

[22]  Fernando Pérez-Cruz,et al.  SVM multiregression for nonlinear channel estimation in multiple-input multiple-output systems , 2004, IEEE Transactions on Signal Processing.

[23]  Juan José Murillo-Fuentes,et al.  Widely Linear Complex-Valued Kernel Methods for Regression , 2016, IEEE Transactions on Signal Processing.

[24]  Jian Li,et al.  On robust Capon beamforming and diagonal loading , 2003, IEEE Trans. Signal Process..

[25]  Eduardo Bayro-Corrochano,et al.  Quaternion support vector classifier , 2016, Intell. Data Anal..

[26]  Yuan-Hwang Chen,et al.  Kalman-based spatial domain forward-backward linear predictor for DOA estimation , 1995 .

[27]  S. Hadjiloucas,et al.  Detection of ECG arrhythmia conditions using CSVM and MSVM classifiers , 2015, 2015 IEEE Signal Processing in Medicine and Biology Symposium (SPMB).

[28]  G. Camps-Valls,et al.  A Support Vector Machine MUSIC Algorithm , 2012, IEEE Transactions on Antennas and Propagation.

[29]  Yide Wang,et al.  Time-Delay Estimation Using Ground-Penetrating Radar With a Support Vector Regression-Based Linear Prediction Method , 2018, IEEE Transactions on Geoscience and Remote Sensing.

[30]  José Luis Rojo-Álvarez,et al.  Support vector machines framework for linear signal processing , 2005, Signal Process..

[31]  John Shawe-Taylor,et al.  Structural Risk Minimization Over Data-Dependent Hierarchies , 1998, IEEE Trans. Inf. Theory.

[32]  G. Bienvenu,et al.  Optimality of high resolution array processing using the eigensystem approach , 1983 .

[33]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[34]  Sergios Theodoridis,et al.  Complex Support Vector Machines for Regression and Quaternary Classification , 2013, IEEE Transactions on Neural Networks and Learning Systems.

[35]  Aleš Procházka,et al.  Cycling Segments Multimodal Analysis and Classification Using Neural Networks , 2017 .

[36]  Jirí Zára,et al.  Geometric skinning with approximate dual quaternion blending , 2008, TOGS.

[37]  Chee Kheong Siew,et al.  Extreme learning machine: Theory and applications , 2006, Neurocomputing.

[38]  E. Bayro-Corrochano,et al.  SVMs using geometric algebra for 3D computer vision , 2001, IJCNN'01. International Joint Conference on Neural Networks. Proceedings (Cat. No.01CH37222).

[39]  Thomas Kailath,et al.  On spatial smoothing for direction-of-arrival estimation of coherent signals , 1985, IEEE Trans. Acoust. Speech Signal Process..