# Complex and Hypercomplex-Valued Support Vector Machines: A Survey
## 2 Complex and Hypercomplex Support Vector Machines Extensions
### Division Algebras
#### 2.2.5 Complex Support Vector Machines for Regression and Quaternary Classification  
In [14] the derivation and design of a complex SVM for regression and classification are presented. The authors use two main frameworks to develop their proposal: the widely linear mean square estimation and Wirtinger's calculus.  
In [29] a mean square estimation (MSE) optimization methodology to deal with complex and normal data is presented. The MSE for complex data is not linear as in the real case, but wide sense linear: i.e., the regression of a scalar random variable \(y\) to be estimated is linear, both in terms of a random vector \(x\) that is an observation of \(y\) and regarding the complex conjugate of \(x\), namely \(x^{*}\). This optimization method shows advantages regarding the linear procedure when is used with complex data, and it can yield significant improvements in the estimation performance as it was shown in the examples presented in [29].  
Wirtinger's calculus was introduced in 1927 [30]. The Wirtinger derivatives (or Wirtinger operators) are first-order partial differential operators defined by several complex variables. These derivatives behave analogously to the ordinary derivatives of functions of real variables.  
Thus, in the paper [14], the above frameworks were used in order to work with the SVMs in the complex Reproducing Kernel Hilbert Space (RKHS). Three approaches were presented:  
(1) The _complexification_ of the Hilbert space \(H^{2}=H\times H\): the objective is to enrich \(H^{2}\) with a complex inner product that is equivalent to the definition of the Clifford (or geometric) product of two complex numbers in a Clifford algebra isomorphic to the complex space. Hence, this approach is equivalent to work with this type of hypercomplex number in Clifford or GA algebras of Section 2.1.  
(2) The Dual Real Channel (DRC) approach in which the training data are split into two sets (real and imaginary parts of the input data), and a real regression (or classification) is performed on each data set using a real kernel. It has been proven before that this approach and the complexification are equivalent procedures [31]  
(3) The pure complex SVM, in this paper is derived using [PERSON]'s calculus and widely linear estimation in a very elegant and compact manner. This derivation allows the authors to conclude that the proposed pure complex SVM can be solved by splitting the labels (desired outputs) of data into their real and imaginary parts, and solving two real SVM tasks employing any one of the standard algorithms (they employed SMO in their experiments). Hence, the difference between practical implementations of complexification, DRC and pure complex SVMs is only that the first two mentioned approaches use real kernels meanwhile the pure complex SVM approach uses an induced real kernel \(\kappa_{C}^{r}\).  
One of the main contributions of the paper is the validation of complex kernels to classify the target space into four different categories. This is a very intuitive result, as for real valued kernels the classification features two classes, and the complex kernel features a real and an imaginary part \(K=Kr+jKi\), so there are four possible combinations for the signs of \(Kr\) and \(Ki\), hence yielding four categories naturally.  
The experiments were conducted using pure complex SVR and a complex-induced Gaussian kernel (which is not equal to the real Gaussian RBF) for the function estimation of a sinc function,channel identification, and channel equalization. In addition, a multiclass classification application was demonstrated using the MNIST database of handwritten digits for a 4-classes problem where the problem was solved significantly faster than the one-versus-three and the one-versus-one real SVM strategies: the computational time taken by the pure complex SVM was almost half that of the real SVMs, but the error rate increased.