On the representer theorem and equivalent degrees of freedom of SVR

Support Vector Regression (SVR) for discrete data is considered. An alternative formulation of the representer theorem is derived. This result is based on the newly introduced notion of pseudoresidual and the use of subdifferential calculus. The representer theorem is exploited to analyze the sensitivity properties of epsilon-insensitive SVR and introduce the notion of approximate degrees of freedom. The degrees of freedom are shown to play a key role in the evaluation of the optimism, that is the difference between the expected in-sample error and the expected empirical risk. In this way, it is possible to define a C-p-like statistic that can be used for tuning the parameters of SVR. The proposed tuning procedure is tested on a simulated benchmark problem and on a real world problem (Boston Housing data set).