# Complex and Hypercomplex-Valued Support Vector Machines: A Survey
## 2 Complex and Hypercomplex Support Vector Machines Extensions
### Geometric or Clifford Support Vector Machines  
In 2000, the first attempt to develop an extension of SVM to deal with complex and hypercomplex input and output data was published [7, 8]. The approach was designed using the Geometric Algebra (GA) framework [16]. GAs are the Clifford special family of associative algebras built over a field, and a quadratic space of elements called multivectors that is constructed using a special product called the geometric product. In Clifford and GAs we can find embedded concepts of complex, linear, tensor, quaternion, and octonion algebras (for a detailed introduction to Clifford and GAs it is recommended to read [16, 17]. Hence, in [7, 8], a multivector-valued input and output data was used to perform multiclassification and multi-regression, solving one multivector-valued optimization problem. Nevertheless, the approach was not completely developed, because it did not consider the multivector-valued kernel design, or define the Gramm matrix (or kernel matrix) as a multivector matrix. Therefore, the design can be considered to be an application of the idea of [PERSON] presented in 1954 with the proposal of \"Parametron\" [18, 19] wherein the phase of a high-frequency carrier is used to represent binary or multivalued information. Similarly, in [7, 8], the elements of different grades of the multivector are used to represent multivalued information in the input and the output \begin{table}
\begin{tabular}{p{56.9 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline
**Ref./Year** & **Theoretical** & **Advantages of the** **SVM Extension** & **Disadvantages of the** **SVM Extension** \\ \hline
[14]/2015 Section 2.2.5 estimation & Wirtinger’s caculus and widely linear mean square & The Wirtinger calculus includes definitions of complex derivatives and they are used to compute the dual optimization problem of the SVM in an elegant and compact manner. The widely linear mean square estimation is a methodology to deal with complex numbers and when it is used with this kind of data it was proved that it can yield significant improvements in the estimation performance of the proposal was applied to solve problems of function estimation for a sinc function, channel identification and classification. & For classification problems (MNIST classification) the error rate increased with the use of this proposal regarding using one-versus-three and the one-versus-one real SVM strategies. \\ \hline
[15]/2016 Section 2.1 & Quaternion algebra and quadratic programming & It can be performed MIMO classification. & It is an approach restricted to be applied using quaternion-valued input and output data \\ \hline
[15]/2016, & GA and quadratic programming & It can be performed MIMO classification. & It is an exclusive parallelization methodology for CSVM that uses a Gaussian kernel. \\ \hline
[15]/2016 Subsec. 2.1 & GA and quadratic programming & It can be performed MIMO classification. & It is an exclusive parallelization methodology for CSVM that uses a Gaussian kernel. \\ \hline \end{tabular}
\end{table} 
Table 1: Cont. multivector-valued data. It was not until 2010 that the extension presented in [7; 8] was fully developed, with the introduction of Clifford Support Vector Machines (CSVMs) [9]. The author considers this to be the most complete work that generalizes the real-valued SVM into a complex and hypercomplex-valued SVM, as the Clifford Algebra theory includes concepts of the most used and important algebras, such as analytic geometry of [PERSON], the complex algebra of [PERSON] and [PERSON], Hamilton algebra, matrix Cayley algebra, exterior Grassmann algebra, tensor algebra of [PERSON], the algebra of [PERSON] and [PERSON], Lie algebra, etc. Furthermore, all the division algebras such as real, complex, quaternion and octonion can be viewed as isomorphic Clifford algebra.