Design of semi-tensor product-based kernel function for SVM nonlinear classification

The kernel function method in support vector machine (SVM) is an excellent tool for nonlinear classification. How to design a kernel function is difficult for an SVM nonlinear classification problem, even for the polynomial kernel function. In this paper, we propose a new kind of polynomial kernel functions, called semi-tensor product kernel (STP-kernel), for an SVM nonlinear classification problem by semi-tensor product of matrix (STP) theory. We have shown the existence of the STP-kernel function and verified that it is just a polynomial kernel. In addition, we have shown the existence of the reproducing kernel Hilbert space (RKHS) associated with the STP-kernel function. Compared to the existing methods, it is much easier to construct the nonlinear feature mapping for an SVM nonlinear classification problem via an STP operator.