# Complex and Hypercomplex-Valued Support Vector Machines: A Survey
## 2 Complex and Hypercomplex Support Vector Machines Extensions  
In this section, the most important extensions of SVM are reviewed to deal with complex and hypercomplex-valued input and output data, as well as with kernels that process this type of data. Each approach is studied in the chronological order of publication. In addition, Table 1 is summarizes the information presented in all the subsections of this section, which presents the main features, advantages and disadvantages of each approach.  
\begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline
**Ref./Year** & **Theoretical** & **Advantages of the** & **Disadvantages of the** \\
\begin{tabular}{l} **The GA framework it is not** \\ **popular and it has to be** \\ **taught to propose other GA** \\ **extensions and to design GA** \\ **kernels** \\ \end{tabular} \\ \hline \end{tabular}
\end{table}
Table 1: Part 1-Complex and hypercomplex SVM extensions.  
\begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline
**Ref./Year Section** & **Theoretical Frameworks Used** & **Advantages of the SVM Extension** & **Disadvantages of the SVM Extension** \\ \hline
[10]/2002, [11]/2004, Section 2.2.1 & Division algebras (complex) and IRWLS & MIMO regression can be performed & The output data are represented as real vectors. \\  & & It is applied to frequency & The optimization problem may become an ill-defined one, but it can also be solved using IRWLS \\  & & It deals with complex signals & The optimization problem defined uses an \(\epsilon\)-insensitive cost function that in turn uses the \(L-2\) norm to consider to consider all the multiple dimensions in a unique restriction \\ \hline
[12]/2007, Section 2.2.2 & Division algebras (real,complex and quaternions) and bilevel quadratic optimization & The use of a cost function rotationally invariant was proposed in order to take into account the magnitude and not the angle of the error in the output. & The experimental results are limited to show an equalization of a 4-QAM signal over a complex linear communication channel and complex function approximation \\  & & It presents real, complex and quaternionic-valued derivation of a well-defined dual optimization problem. The outputs of the SVR were interpreted as complex/quaternion-valued vectors to tackle the interconnectedness of outputs. & The derivation of the primal and dual optimization problem must consider the nonassociativity of the octonions and this adds complexity to the derivation of the dual problem, the training machine form and the computation of the kernel matrix \\  & & It deals with octonion-valued regression & octonions and this adds complexity to the derivation of the dual problem, the training machine form and the computation of the kernel matrix \\  & & As the nonassociativity feature adds terms to the loss function, then this increase the dimension of the problem and the curse of dimensionality could lead to computationally intractable problems for real applications \\  & & The output and input data is expressed as pure octonions & dimension of the problem the curse of dimensionality could lead to computationally intractable problems for real applications