## 2 Related Work
SVM performs well but on a small number of samples. In [35] author proposed a new model for feature extraction and classification using SVM. Different features are extracted, combined, and then the similarity is computed by SVM. In [36] SVM was used for tongue image classification. The main problem of this research was to extract features of the tongue. These features are extracted by the Histograms of Oriented Gradient (HOG) concept [37], and then SVM was applied to classify. Experiment results show good classification and prediction results. Reduced SVM (R-SVM) is proposed by [38] they select features by minimizing \(\frac{R^{2}}{\gamma}\sigma\). They used the SVM radius margin ratio for selecting features. This model was used to optimize the radius margin bound. Moreover, the authors first scaled a feature by multiplying each feature by a vector \(\sqrt{\mu}\), and then SVM is applied to these resulting features. The resulting feature space is not the same as the original feature space. Due to this, in resulting feature space classes are more linearly separable than the actual feature space. The radius of feature space after transformation was denoted by \(R_{\mu}\). In R-SVM, two-step classification was performed; in the first step, SVM is used to select features, and then the radius margin is optimized by using extended feature space. They aimed to achieve a better generalization error. For achieving this aim, they optimize the radius margin, which accomplishes the following optimization problem. \[ \begin{array}{c}min\\ w,b,\mu,R_{\mu}\\ \end{array} =\frac{1}{2}\parallel w\parallel_{2}^{2}R_{2}^{\mu} \tag{11}\]  
where \(R_{\mu}\) is computed by [PERSON] 1998 [39]. The constraints involved to restrict the parameters of (11) are  
\[y_{i}((w,\sqrt{\mu}+b)\geqslant 1,\forall_{i}\]  
where \(y_{i}\) is the label, \(w\) and \(b\) are classification parameters. The value of \(\mu\) is restricted by another constraint which was  
\[\sum_{k=0}^{d}\mu_{k}=1,\mu_{k}\geqslant 0,\forall K\]  
The smooth feature selection was performed through this optimization problem. to transform the features space \(\mathcal{H}\), the radius of MEB according to [41] in that extended feature space is given by  
\[\min_{R,\chi_{0}}R^{2},\quad s.t.\|\Phi(x_{i})-\Phi(x_{0})\|_{2}^{2}\leq R^{2} \quad\forall_{i} \tag{13}\]  
Another solution to minimize the area of MEB is proposed by [42]. The authors proposed two variants of SVM, \(R-SVM_{H}^{+}\), and \(R-SVM_{+}\). \(R-SVM_{+}^{+}\) was a non-convex and difficult to solve because of radius constraints. It was based on R-SVM, in which the ratio of the optimization problem replaces the sum. It first determines the transformation matrix and then projects it into a new feature space. \[\min_{w,\zeta,\mu,b,R_{\mu},x_{0}} \frac{1}{2}\|w\|_{2}^{2}+\lambda R_{\mu}^{2}+C\sum_{i}^{l}\xi_{i} \tag{14}\] \[s.t.\quad\quad y_{i}(\left<w,D_{\sqrt{\mu}X_{i}}\right>+b)\geq 1- \xi_{i},\forall_{i}\] \[\sum_{k=1}^{d}\mu_{k}=1\ \ \mu,\xi\geq 0\] \[\|D_{\sqrt{\mu}X_{i}}-D_{\sqrt{\mu}X_{0}}\|^{2}\leq R_{\mu}^{2}, \forall_{i}\]  
The radius constraints here were non-convex, and the optimization problem was difficult to solve. On the other hand, \(R-SVM_{+}\) was a metric learning-based extension of SVM. In this algorithm, a matrix \(D\) is used to transform features, and SVM is used to classify the data in that new feature space. The extended feature space is also used by [43] in which the author used LogDet regularization with SVM, called LSVM. The methodology was based on the two-step minimization strategy, where the first is to use LogDet to regularize the feature space and then compute the radius. The radius computation method was the same as F-SVM, which was the mean of the class. An extended version of SVM is proposed [14] to reduce the value of the radius of MEB. According to F-SVM, a class area is a maximum distance from the mean to any sample. If we have scattered data, the mean of a class shift to higher values due to which a point is so far from a mean.