The Differential Geometry Method Of Support Vector Regression Machines

The whole paper includes the four sections. The first chapter contains a short overview over some basic knowledge ,lncluding B-spline function,support vector classification machine ,the regression problem and multicollinearity. Those knowledge are basis for understanding this paper. In chapter 2 ,I will explain the geometry of the support vector machine.I will show systematically how we induce the kernel functions of support vector machine to the Riemannian metric,which makes possible for studying the support vector machine in geometry. Chapter 3 is the keystone of this paper, which narrates the theory of the support vector regression machine. I narrate systematically the problem of SVRM and how to construct the kernel functions. On the basis we propose a method of modifying a kernel function to improve the performance of a support vector regression machine.This is based on the Riemannian geometrical structure induced by the kernel function This idea is to reduce the spatial resolution around the separating boundary surface by a conformal mapping such that the separability between two boundary surface is decreased. In chapter 4,Examples are given specifically for modifying B-spline kernel funtion.Simulation results for both artifical and real data show remarkable improvement of generalization errors ,supporting our idea.