Maximum margin hyperplane

For any particular set of two-class objects, an SVM finds the unique hyperplane having the maximum margin (denoted with 8 in Figure 1). The hyperplane Hj defines the border with class - -1 objects, whereas the hyperplane H2 defines the border with class —1 objects. Two objects from class -1-1 define the hyperplane Hj, and three objects from class —1 define the hyperplane Fl2. These objects, represented inside circles in Figure 1, are called support vectors. A special characteristic of SVM is that the solution to a classification problem is represented by the support vectors that determine the maximum margin hyperplane. 

Fig. 8. Representation of a support vector machine. There are three different compounds in this simplified SVM representation. The plus (+) symbols represent active, the minus (-) symbols represent nonactive, and the question mark ( ) symbol represents undetermined compounds. The solid line in the hyperplane and the dotted lines represent the maximum margin as defined by the support vectors.

In effect, the maximum margin, i.e. optimal typerplane, is the one that gives the greatest separation between the classes. The data points that are closest to the optimal hyperplane are called support vectors (SV) . In each class, there exists at least one SV very often there are multiple SVs. The optimal hyperplane is uniquely defined by a set of SVs. As a result, all other training data points can be ignored. 

The optimum separation hyperplane (OSFI) is the hyperplane with the maximum margin for a given finite set of learning patterns. The OSH computation with a linear support vector machine is presented in this section. 

The Margin of a training set S is the maximum geometric margin over all hyperplanes. 

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