Properties of Support Vector Machines for Regression Properties of Support Vector Machines for Regression

In this report we show that the-tube size in Support Vector Machine (SVM) for regression is 2= p 1 + jjwjj 2. By using this result we show that, in the case all the data points are inside the-tube, minimizing jjwjj 2 in SVM for regression is equivalent to maximizing the distance between the approximating hyperplane and the farest points in the training set. Moreover, in the most general setting in which the data points live also outside the-tube, we show that, for a xed value of , minimizing jjwjj 2 is equivalent to maximizing the sparsity of the representation of the optimal approximating hyperplane, that is equivalent to minimizing the number of coeecients diierent from zero in the expression of the optimal w. Then, the solution found by SVM for regression is a tradeoo between sparsity of the representation and closeness to the data. We also include a complete derivation of SVM for regression in the case of linear approximation.