Convergenceness of a General Formulation for Polynomial Smooth Support Vector Regressions

Lee et al. proposed a smooth ε-support vector regression (ε-SSVR) in 2005, and Xiong et al. proposed a polynomial smooth ε-support vector regression(ε-PSSVR) in 2008, which improved the performance and efficiency of regression. However, problems still exist in looking for a general formulation of the polynomial smooth ε-support vector regressions and proving its convergenceness. Therefore, using a class of polynomial functions as new smoothing functions, the polynomial smooth model ε-PSSVR is extended to a general case; and a dth-order polynomial smooth ε-support vector regression (ε-dPSSVR), which is a general formulation of polynomial smooth ε-support vector regressions, is proposed using the smoothing technique in this paper. The global convergence of ε-dPSSVR is proved by mathematical inductive method. The research concludes that: 1) there are an infinite number of polynomial smooth models for support vector regression, which can be described in a general formulation; 2) the general formulation is global convergent, and the upper bound of the convergence is reduced about half an order of magnitude to ε-SSVR. The convergence problem of general formulation is successfully solved, which supplies a basic theoretical support for researching polynomial smooth ε-support vector regressions.