# Complex and Hypercomplex-Valued Support Vector Machines: A Survey
## 1 Introduction  
The base idea of Support Vector Machines is the soft margin classifier that was first introduced by [PERSON] and [PERSON] in 1995 [PERSON] and [PERSON] in 1995 [PERSON] and [PERSON] (1995), then in the same year, this algorithm was extended for deal with regression problems by [PERSON] (1995). The first formal statistical analysis about boundaries of the generalization of hard margin SVMs were given by [PERSON] (1995) and [PERSON], et al. in 1998 [PERSON] and [PERSON] (2000). In 2000, [PERSON] and [PERSON] continued with the development of the statistical learning theory that supports the SVMs and they present the statistical bounds of the generalization of soft margin algorithms for the regression case [PERSON] et al. (2000). Thus, since its introduction by [PERSON] and his team, SVMs have become an important and very used learning algorithm to solve regression and classification problems. The sheer number of published applications of support vector machines (SVM) for solving problems involving real data is reflected in the high number of citations that the top 100 SVM publications [PERSON] et al. (2000) have recorded; there are more than 46,000 citations, according to Google Scholar.  
SVMs use an intelligent and elegant technique to solve non-linear problems: the kernel trick, which has become a seminal idea to develop several machine learning methods called kernel-based methods. Kernels are symmetric functions that for two vectors of dimension \(n\), return a real number \(\
eq 0\). The use of kernels in SVM allows implicitly mapping the original input data into a higher dimensional Reproducing Kernel Hilbert Space (RKHS) \(H\), in which linear functions that split the mapped data are computed. This is equivalent to solving the non-linear problem in the original input data space.  
Although the RKHS includes complex and hypercomplex spaces in which the original input data can be mapped via kernel functions, the majority of extensions and applications of SVM deal with real data, and therefore use real kernel functions. However, the need for processing complex and hypercomplex data in modern applications has paved the way for some important proposals toprocess complex and hypercomplex-valued input and output data, and kernels. These approaches will be reviewed in more detail in the next section.  
Therefore, since the first proposals of SVM extensions for processing complex and hypercomplex data and kernels, their applications have grown in number, including signal processing, pattern recognition, classification of electromagnetic signals, light, sonic/ultrasonic and quantum waves, chaos in the complex domain, phase and phase-sensitive signal processing and nonlinear filtering, frequency, time-frequency and spatiotemporal domain processing, quantum computation, robotics, control, pattern recognition, time-series prediction, and visual servoing, among others. Due to the variety and importance of the applications of this type of SVM extension, in this survey we present and discuss the importance, recent progress, prospective applications, and future directions of complex, hypercomplex-valued and geometric support vector machines.  
This survey is organized as follows: In Section 2, 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. In Section 3, some of the applications of these extensions of SVMs with the greatest impact are introduced, as well as some of their promising developments with future applicability. The last section is devoted to the conclusions.