Affiliation: | (1) Department of Mathematics, Suzhou University, Suzhou, 215006, Peopless Republic of China;(2) School of Science, Jinan University (West), Jinan, 230000, Peopless Republic of China |
Abstract: | In the classical support vector machines, linear polynomials corresponding to the reproducing kernel K(x,y)=xy are used. In many models of learning theory, polynomial kernels K(x,y)=l=0Nal(xy)l generating polynomials of degree N, and dot product kernels K(x,y)=l=0+al(xy)l are involved. For corresponding learning algorithms, properties of these kernels need to be understood. In this paper, we consider their positive definiteness. A necessary and sufficient condition for the dot product kernel K to be positive definite is given. Generally, we present a characterization of a function f:RR such that the matrix [f(xixj)]i,j=1m is positive semi-definite for any x1,x2,...,xmRn, n2.Supported by CERG Grant No. CityU 1144/01P and City University of Hong Kong Grant No. 7001342.AMS subject classification 42A82, 41A05 |