# Complex and Hypercomplex-Valued Support Vector Machines: A Survey
## 4 Conclusions and Prospective Work  
In this survey, several proposals were reviewed to extend the Support Vector Machine algorithm to deal with complex and hypercomplex-valued inputs, outputs, and kernels. The first attempts to achieve this consisted of splitting the complex and hypercomplex numbers into real and complex (hypercomplex) parts, and then solving the optimization problems independently in the real domain. Then, it was proven that this approach loses the benefits of the complex (hypercomplex) spaces, such as sparsity and geometric and topologic features that the vectors (multivectors) have when are embedded in their spaces. All the proposals that exploit these features by solving the SVM optimization problem in its complex (hypercomplex) space were proven to achieve higher accuracies than those that solve the problem using splitting procedures. The most complete and general extensions of SVMs, that deal with complex/hypercomplex-valued data are those that, besides considering input and output data that is complex/hypercomplex valued, define the optimization problem (primal and dual problems) and the kernel using complex and hypercomplex algebras and calculus frameworks, such as Clifford and GA, widely linear analysis and Wirtinger?s calculus; although it was also proved [14] that the design of a complex/hypercomplex-valued kernel that maps the input data onto a complex/hypercomplex RKHS is the key to obtaining the benefits of solving the problem in these spaces, such as the preservation of the geometric and topological special features of these spaces. Therefore, for dealing with data that is naturally embedded in complex (hypercomplex) spaces the approaches that were presented in the Section 2 of this paper are the best choices to obtain higher accuracies and lower run time in problems of classification and regression. Another important conclusion is about the compatibility of the complex numbers and their frameworks with the wave nature due to the waves have intensity and phase, so two numbers are needed to describe it. In the applications section, it is shown that the extensions of SVM that were designed using complex and hypercomplex mathematical tools can solve efficiently and effectively real-world problems that involve the processing and classification, interpolation, or regression of complex signals, frequency domain data, ECG signals, MRI data, chaotic time series, complex/hypercomplex forecasting, approximation of complex/multiple functions, complex beamforming and DOA, GPR and GPS data, linear and nonlinear complex filtering, pattern recognition, robotics, and computer vision, among others. Regarding the prospective of complex/hypercomplex extensions and applications of SVMs, one promising development is wave informatics because, as it was said before, the natural compatibility between complex numbers and waves. The applications are important and numerous as they can be seen in Section 3. Another area of opportunity for researching is the processing, classification, regression, prediction, identification, and simulation of chaotic systems. All the deterministic aperiodic system that is sensitive to the initial conditions is called a chaotic system. Their study is very important as some chaotic systems examples are the weather, some electrocardiograms, and encephalograms time series, the stock market time series, and fractals, among others. These systems are also naturally modeled using complex-valued functions, and therefore, the extensions of complex-valued SVMs are the best-suited to process them. As was mentioned, perhaps the most general extension of complex/hypercomplex valued SVMs is the one presented in [9]. Here, the use of GA framework allowed the authors to define an SVM problem that processes complex/hypercomplex-valued inputs, outputs, and kernels. Nevertheless, the design of complex/hypercomplex kernels remains a main open issue to explore the extensions of SVM to higher-dimensional hypercomplex spaces (more than quaternions, dual quaternions, or octonions).