A Geometric Approach to Support Vector Regression

Jinbo Bi and Kristin Bennett

Department of Mathematical Sciences
Rensselaer Polytechnic Institute

We develop an intuitive geometric framework for support vector regression (SVR). By examining when epsilon-tubes exist, we show that SVR can be regarded as a classification problem in the dual space. Hard and soft epsilon-tubes are constructed by separating the convex or reduced convex hulls respectively of the training data with the response variable shifted up and down by epsilon. A novel SVR model is proposed based on choosing the max-margin plane between the two shifted datasets. Maximizing the margin corresponds to shrinking the effective epsilon-tube. In the proposed approach, the effects of the choices of all parameters become clear geometrically. The kernelized model corresponds to separating the convex or reduced convex hulls in feature space. Generalization bounds for classification can be extended to characterize the generalization performance of the proposed approach. We propose a simple iterative nearest-point algorithm that can be directly applied to the reduced convex hull case in order to construct soft epsilon-tubes. Computational comparisons with other SVR formulations are also included.