# Complex and Hypercomplex-Valued Support Vector Machines: A Survey
## 3 Applications
### Frequency Domain Processing, Linear and Nonlinear Filtering  
For solving handwritten character recognition using the fast Fourier transform (FFT) coefficients of data from MNIST, fashion MNIST, extended MNIST and Latin OCR benchmark databases a complex-valued neural network with kernel activation functions (Kafs) was presented in [53]. Even though this approach is applied to neural networks instead of SVMs, it is included in this survey to represent one of the proposals to solve one of the main issues of complex or hypercomplex SVMs. It is the design of kernels that process this type of data, and that could be used with all the SVM extensions of the Section 2. Thus, in [53] the ideas of [54; 55; 56] are combined to design a widely linear kernel activation function model (WL-KAF). The WL-KAF allows the NN to learn the shape of a non-parametric kernel activation function defined in the complex domain by modeling each activation function \"with a small number of complex-valued adaptable parameters, representing the linear coefficients in a kernel-based expansion\", and by using widely linear framework; the model was extended to widely linear kernels with more expressiveness than the classic KAF [54]. This is due to the fact that the classic KAF loses two outputs from the complex kernel by forcing the real and imaginary constraints to be equal, and the cross terms are equal with the scale factor of \(-1\). The WL-KAF approach applied for solving the frequency domain representation of the handwritten characters achieved better accuracy than the real-valued NN. Nonlinear channel equalization (NCE) applications using the kernel ridge regression (KRR) with quaternion kernels are shown in [57]. The transmission channel was modeled as a linear filter with a memory-less nonlinearity stage corrupted by noise. Then, a Gaussian quaternion-valued kernel was designed and used in KRR to identify one original message among the noisy measurements. The signals were described as quaternions and, in the comparison analysis performed the quaternion-valued kernel outperformed the real-valued, obtained better accuracy thanks to its ability to capture data geometric and topologic relationships. In [11; 14; 23] the linear and nonlinear channel equalization, channel identification and filtering problems were tackled. See Sections 2.2.1, 2.2.2 and 2.2.5. In [12], the authors consider an application for solving the equalization of a 4-symbol quadrature amplitude modulated (4-QAM) signal over a complex linear communication channel that was added with Gaussian noise, Section 2.2.2. The frequency estimation problem was solved in [58; 59], where a complex-valued support vector autoregressive technique was used. In [59], the autoregressive moving average (ARMA) system \begin{table}
\begin{tabular}{l l l} \hline \hline
**Ref./Year** & **Application** & **SVM Extension Used** \\ \hline \multirow{3}{*}{[23]/2010} & Complex function approximation with & \multirow{3}{*}{\(\epsilon\)-SVR of [23] and complex valued \(\text{SVR of \@@cite[cite]{[\@@bibref{}{SvR}{}{}]}}\)} \\  & Laplacian and Gaussian noises and chaotic & \\  & time-series prediction & \\ \hline \multirow{2}{*}{[9]/2010} & Time-series prediction using Venice Lagoon & Combined recurrent system of \\  & and Mackey-Glass datasets & LSTM-CSVM \\ \hline \multirow{2}{*}{[9]/2010} & Complex function approximation \(sinc()\) & \multirow{2}{*}{CSVM} \\  & and \(osc()\) & \\ \hline \multirow{2}{*}{[14]/2015} & Complex function approximation of \(sinc()\) & Complex-valued SVM [14] \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Applications of complex and hypercomplex SVMs for time-series prediction and function approximation problems. identification and non-parametric spectral analysis were formulated using a framework that allows the authors to identify the Hilbert signal space in the model, define a robust cost function, and minimize a constrained and regularized SVM-like functional with Lagrange multipliers method.