## 1 Introduction  
Machine Learning (ML) has achieved a significant rise in many fields in the past two decades. The rapid growth of data has become a challenging task to find patterns and information from the data. ML has been successfully applied in many disciplines like fraud detection in credit card transaction and object recognition [1, 2, 3], face recognition [4, 5], and decision making systems [6, 7]. ML-based algorithms are developed based on the patterns learned from the data and used these patterns to conclude unseen data samples. SVM is the most prevailing supervised learning model used in many problems related to classification [8], regression [9], computer vision [10], outliers detection [11], to name a few, SVM is one of the most efficient ML algorithms because of its kernel tricks and less computational cost. These kernel tricks transform the data into different feature spaces so that the data are more linearly separable than the original feature space. SVM and its extensions try to set optimal hyperplane with maximum intraclass margin, i.e., the margin between two classes [12]. However, the generalization error of SVM depends on both ratio of the radius and the margin [13]. According to the generalization error function, if feature mapping is given, SVM can minimize the error by maximizing intraclass margin [14]. In feature transformation, radius margin is valuable and cannot be ignored. In the binary classification problem, we have \(\{(x_{i},y_{i}),\ldots(x_{n},y_{n})\}\) training dataset where \(x\in R^{d}\) and \(y\in\{1,-1\}\) Then,  
\[\hat{y}=\beta^{T}x+\beta_{o}, \tag{1}\]  
where \(\beta^{T}\) is weight matrix, \(x\) is a feature matrix and \(\beta_{o}\) is bias, is an equation to separate two classes linearly. SVM tries to find the best hyperplane in a binary classification problem that separates positive samples from negative samples with maximum margin. The shortest distance among positive samples and negative samples, closest to the hyperplane, is called margin [12]. It is more effective to find a hyperplane with the largest margin because it is more restricted to noise than a hyperplane with a smaller margin. The hyperplane with a smaller margin can make mistakes on testing data. SVM without a slack variable \(\xi\) variable is hard margin SVM. In hard margin SVM, all data points satisfy the constraints depicted in (2) and (3) [15]. After passing the linear separability check, SVM can achieve 100% training accuracy but not grantees for testing. \[w.x_{i}+b\geq 1\qquad y_{i}=+1 \tag{2}\]  
\[w.x_{i}+b\leq-1\qquad y_{i}=-1, \tag{3}\]  
where \(w\) is vector, which is normal to the hyperplane. The perpendicular distance from hyperplane to the origin is given by \(\frac{b}{\|w\|}\) where \(\|w\|\) is Euclidean norm of vector \(w\). (2) and (3) can be combined as  
\[y_{i}(w.x_{i}+b)\geq 1\qquad\forall_{i}. \tag{4}\]  
Most of the real-world datasets are not linearly separable; therefore, it is needed to draw a non-linear boundary. In case of non-linear datasets, two types of solutions can be adopted. One is to use soft-margin SVM to accept some errors, and another is to draw a non-linear boundary. Linear separability of the data can be affected by a single point. Even a single sample in the dataset can affect the hyperplane of the SVM. If the data is sensed from sensors, then a single incorrect datum can make the dataset non-linear [16], as shown in Figure 1.  
Figure 1: Non-linearity of the data. Linear separability can also be applied to check the non-linearity of the data before using SVM as a classifier. SVM with the linear kernel can be applied to the data for linear separability check. By getting 1.0 accuracy on training data, the data are linearly separable, and we can go for hard margin SVM. If the accuracy value is less than 1.0 but in an acceptable range, the soft margin SVM can be applied by accepting some errors (5). \[y_{i}(w.x_{i}+b)\geq 1-\xi\qquad\forall_{i} \tag{5}\]  
Here the \(\xi>0\) variable relaxes the constraints given in (4).