# Complex and Hypercomplex-Valued Support Vector Machines: A Survey
## 2 Complex and Hypercomplex Support Vector Machines Extensions
### Division Algebras  
Several efforts using division algebras were made to generalize real SVMs, to deal with complex and hypercomplex numbers. Division algebras are appealing mathematical frameworks to solve the complex, hypercomplex-valued SVM optimization problem due to the fact that all non-zero elements of a vector space have multiplicative inverses. There are four division algebras: real numbers, complex numbers, quaternions, and octonions. .2.1. SVM Multiregression for Nonlinear Channel Estimation in Multiple-Input Multiple-Output Systems  
In 2002, an approach to solve multidimensional function approximation and regression was presented [10], and was fully developed and published in 2004 [11]. The authors addressed the problem of multiple-input, multiple-output (MIMO) system for the frequency nonselective channel estimation. This was an SVM-based approach as it leveraged the multidimensionality of a MIMO channel by using a regression SVM-based algorithm. This study proposed using an iterative reweighted least square (IRWLS) to solve the multivariable SVM problem instead of quadratic programming, and train an SVM multi-regressor to model nonlinearities that affect each transmitter or each receiver module of the transmission-reception chain. These cases were modeled separately using two different channel models of the input information signal represented by the quadrature phase shift keying (QPSK) employed in complex signal processing. On the other hand, the output data are represented as real vectors and the optimization problem defined uses an \(\epsilon\)-insensitive cost function that in turn uses the \(L_{2}\) norm to consider all the multiple dimensions in a unique restriction. When \(\epsilon=0\) the problem is equivalent to solving one independent, regularized real kernel for each dimension, but for \(\epsilon\
eq 0\), the problem becomes an ill-defined one that cannot be solved straightforwardly, and it takes an iterative procedure to obtain the solution which is why they use an IRWLS algorithm. Simulations were conducted to address the problem of nonlinear channel estimation, and their proposal outperforms the applications of one real support vector regression (SVR) and a radial function network (RBFN) for each dimension. Meanwhile, for linear channel models, results equivalent to those obtained using minimum mean square error (MMSE) strategy were achieved. 2.2 Quaternionic and Complex-Valued Support Vector Regression for Equalization and Function Approximation  
In [12], another SVR strategy was designed by [PERSON] et al., this time to deal with quaternionic and complex-valued equalization and function approximation. The outputs of the SVR werebeen interpreted as complex/quaternion-valued vectors to tackle the interconnectedness of outputs, i.e., the fact that these outputs can be coupled and treated independently can lead to a decrease in regressor accuracy. The author proposed the use of a rotationally invariant cost function in order to consider the magnitude, but not the angle of the error in the output. When a real regressor is applied on each dimension of a complex or a quaternionic-valued signal, each one of the regressor estimates one function in its own axis, and when these functions are summed to construct a complex or quaternion output, the overall risk function will not be rotationally symmetric because it will contain the magnitude and the various angles added (even when the angles were computed using different axis for each regressor). Therefore, considering only the magnitude of the error but not the various angles between the axes, will affect the accuracy of the estimated function.