/spl epsi/-SSVR: a smooth support vector machine for /spl epsi/-insensitive regression

Abstract

A new smoothing strategy for solving /spl epsi/-support vector regression (/spl epsi/-SVR), tolerating a small error in fitting a given data set linearly or nonlinearly, is proposed in this paper. Conventionally, /spl epsi/-SVR is formulated as a constrained minimization problem, namely, a convex quadratic programming problem. We apply the smoothing techniques that have been used for solving the support vector machine for classification, to replace the /spl epsi/-insensitive loss function by an accurate smooth approximation. This will allow us to solve /spl epsi/-SVR as an unconstrained minimization problem directly. We term this reformulated problem as /spl epsi/-smooth support vector regression (/spl epsi/-SSVR). We also prescribe a Newton-Armijo algorithm that has been shown to be convergent globally and quadratically to solve our /spl epsi/-SSVR. In order to handle the case of nonlinear regression with a massive data set, we also introduce the reduced kernel technique in this paper to avoid the computational difficulties in dealing with a huge and fully dense kernel matrix. Numerical results and comparisons are given to demonstrate the effectiveness and speed of the algorithm.