多项式光滑的支持向量机一般模型研究

A General Formulation of Polynomial Smooth Support Vector Machines

2005年袁玉波等人用一个多项式函数作为光滑函数,提出了一个多项式光滑的支持向量机模型PSSVM(polynomial smooth support vector machine),使分类性能及效率得到了一定提高.2007年熊金志等人用插值函数的方法导出了一个递推公式,得到了一类新的光滑函数,解决了关于是否存在以及如何寻求性能更好的光滑函数的问题.然而,支持向量机是否存在其他多项式光滑模型,以及多项式光滑模型的一般形式是什么等问题依然存在.为此,将一类多项式函数作为新的光滑函数,使用光滑技术,提出了多项式光滑的支持向量机一般模型dPSSVM(dth-order polynomial smooth support vector machine).用数学归纳法证明了该一般模型的全局收敛性,并进行了数值实验.实验结果表明,当光滑阶数等于3时,一般模型的分类性能及效率为最好,并优于PSSVM模型;当光滑阶数大于3后,分类性能基本不变,效率会有所降低.成功解决了多项式光滑的支持向量机的一般形式问题.

Yuan et al. used a polynomial function as smoothing function, and proposed a polynomial smooth support vector machine （PSSVM） in 2005, which improved the performance and efficiency of SVM for classification. Using the technique of interpolation functions, Xiong et al. developed a recursive formula to obtain a new class of smoothing functions, and solved the problems of existence and seeking better smoothing functions in 2007. However, problems still exist in looking for other smooth models and the general formulation of the polynomial smooth support vector machines. A class of polynomial functions is applied as new smoothing functions, and a dth-order polynomial smooth support vector machine （dPSSVM） is proposed using the smoothing technique, which is a general formulation of polynomial smooth support vector machines. The global convergence of dPSSVM is proved by a mathematical inductive method, and experiments are carried out to evaluate dPSSVM. The numerical results show that the performance and efficiency of dPSSVM are best, and better than that of the PSSVM when its smooth order is 3, but after its smooth order is greater than 3, the performance of classification is almost the same while the efficiency becomes worse. The problem of general formulation is successfully solved for polynomial smooth support vector machines.