On Margin and Support Vector Separability in Support Vector Machines for Regression on Margin and Support Vector Separability in Support Vector Machines for Regression

In this report we show some simple properties of SVM for regression. In particular we show that for close to zero, minimizing the norm of w is equivalent to maximizing the distance between the optimal approximating hyperplane solution of SVMR and the closest points in the data set. So, in this case, there exists a complete analogy between SVM for regression and classiication, and the-tube plays the same role as the margin between classes. Moreover we show that for every the set of support vectors found by SVMR is linearly separable in the feature space and the optimal approximating hyperplane is a separator for this set. As a consequence, we show that for every regression problem there exists a classiication problem which is linearly separable in the feature space. This is due to the fact that the solution of SVMR separates the set of support vectors in two classes: the support vectors living above and the one living below the optimal approximating hyperplane solution of SVMR. The position of the support vectors with respect to the hyperplane is given by the sign of (i ? i). Finally, we present a simple algorithm for obtaining a sparser representation of the optimal approximating hyperplane by using SVM for classiication.