## 1 Introduction
The objective function becomes  
\[minimize\qquad w^{T}w+C\sum_{i=1}^{m}\xi_{i}^{k}\] \[Subject\ to\qquad y_{i}(w.x_{i}+b)\geq 1-\xi_{i}\qquad and\ \xi_{i}\geq 0 ;\quad(i=1\cdots m) \tag{6}\]  
With the same problem, i.e., non-linear dataset, the classification method can also use kernel tricks to classify the data in kernelized feature space. These kernel tricks transform the data from the original to a kernelized feature space. Among all types of kernels, RBF (7) and polynomial kernels (8) are the most famous [17]. \[K(x_{i},y_{i})=\exp(-\gamma\parallel x_{i}-x_{j}\parallel^{2}) \tag{7}\]  
\[w.x_{i}+b\geq 1\qquad y_{i}=+1 \tag{2}\]0  
In RBF, the parameter \(\gamma\) is used to scale the mapping, and in the polynomial kernel, \(d\) is the degree of the polynomial. Typically, RBF outperforms as compared to polynomial kernel. Feature transformation techniques can also be used in non-linear datasets and SVM to transform data into different feature spaces and try to set a hyperplane [18]. \[w.x_{i}+b\geq 1\qquad y_{i}=+1 \tag{2}\]1  
where \(A\) is a transformation matrix, \(\mu^{T}\) is weights vector, \(x\) is input, and \(b\) is a bias. Along with the feature transformation, another strategy that is used with SVM is metric learning. To learn a linear transformation matrix, which is available more than once, metric learning can be used [19; 20]. The researchers used a straightforward approach to deploy the transformation by combining metric learning and SVM. This approach is unable to produce satisfying results, however. Therefore, to integrate metric learning with SVM, many approaches have been proposed, e.g., Metric Learning with SVM (MSVM) [19] and Support Vector Metric Learning (SVML) [21]. There was a limitation with SVML; it only works with a Gaussian radial basis function kernel (RBF_SVM) and ignores the information of radius, while, on the other hand, MSVM [19] was nonconvex. SVM and its extensions are best to cused lassify the data, but the boundary within a class is not enough to achieve higher prediction accuracy. Each class should be bounded in its limited area. To do so, F-SVM [14] proposed a method to bound the class in its narrow region. Mean strategies were used to calculate the radius of the MEB of the class. However, the mean can shift to a dense area of the class. Real-world datasets are with mixed behavior; some classes may be closed and others or scattered. In this case, the value of radius calculated by the mean will be very high. As a result, the MEB can cross the hyperplane or can overlap samples of other classes. In this paper, a weighted mean approach is proposed to enhance the performance of F-SVM. The F-SVM method computes the MEB of the class by calculating the mean of samples, but the mean can shift to the dense area of the class. If a class contains scattered points along with the dense area, then the radius (maximum distance from mean to samples) will be higher. While the proposed model is based on two steps, the first is the computation of the SVM classifier, and the second is the calculation of the weighted mean. In the first stage, conventional SVM is applied to get the classifier's parameters; after that,the mean of the class is computed. The distance of the sample from the mean is used as the weight of the sample. By assigning the weight to each instance, the samples show different behavior in the weighted feature space. Now the mean is computed in weighed feature space; thus, the mean shifted to the scattered area of the class. The radius calculated by the weighted mean is always less than the radius by mean. Finally, the size of MEB is reduced. The proposed model named WR-SVM is compared with existing margin radius methods, i.e., F-SVM [14]. The proposed model will use the same parameters on which F-SVM was trained. The 10-fold strategy is used to train and test the model. The core contributions of the proposed research study is followed:  
* Calculate a tighter bound of MEB by using weighted mean.